business-ma th diannemataba sLa 37677_fm_i-xvi 8/16/07 4:40 PM Page v Dedica tion Just for Ma tthew, Mia, Sa muel, Gracie, Mabel, a nd ...

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Dedication Just for Matthew, Mia, Samuel, Gracie, Mabel, and Maggie Love PaPa Jeff

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Note to Students ROADMAP TO SUCCESS Step 1:

How to use this book and the Total Slater Learning System. Each chapter broken down into Learning Units. You should read one learning unit at a time. How do I know if I understand it? • • •

Step 2:

Try the practice quiz. All the worked out solutions are provided. If you still have questions, watch the author on your DVD (comes with your text) and work each problem out. Need more practice? Try the extra practice quiz provided. Check figures are at the end of the chapter. Your instructor has worked out solutions if needed. Go on to next Learning Unit in chapter.

Review the “Chapter Organizer” at the end of the chapter. How do I know if I understand it? •

Step 3:

Cover over the second or third column and see if you can explain the key points or the examples.

Do assigned problems at the end of the chapter (or Appendix A). These may include discussion questions, drill, word problems, challenge problems, video cases, as well as projects from the Business Math Scrapbook and Kiplinger’s magazine. Can I check my homework? •

Step 4:

Appendix B has check figures for all the odd-numbered problems.

Take the Summary Practice Test. Can I check my progress? •

Appendix B has check figures for all problems.

What do I do if I do not match check figures? •

Review the video tutorial on the student DVD—the author works out each problem.

To aid you in studying the book, I have developed the following color code: Blue: Movement, cancellations, steps to solve, arrows, blueprints Gold: Formulas and steps Green: Tables and forms Red: Key items we are solving for If you have dif ficulty with any text examples, pay special attention to the red and the blue. These will help remind you what you are looking for as well as what the procedures are.

Note to Students

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Note to Students

FEATURES

Features students have told me have helped them the most.

Blueprint Aid Boxes

For the first eight chapters (not in Chapter 4), blueprint aid boxes are available to help you map out a plan to solv e a word problem. I know that often the hardest thing to do in solving word problems is where to start. Use the blueprint as a model to get started.

Business Math Handbook

This reference guide contains all the tables found in the text. It makes homework, exams, etc. easier to deal with than flipping back and forth through the text.

Chapter Organizer

At the end of each chapter is a quick reference guide called the Chapter Organizer and Study Guide. Key points, formulas, and examples are provided. A list of vocabulary terms is also included, as well as Check Figures for Extra Practice Quizzes. All have page references. (A complete glossary is found at the end of the text.) Think of the chapter organizer as your set of notes and use it as a reference when doing homework problems, and to review before exams.

DVD-ROM

The DVD packaged with the te xt includes practice quizzes, links to Web sites listed in the Business Math Internet Resource Guide, the Excel® templates, PowerPoint, videocases, and tutorial videos—which cover all the Learning Unit Practice Quizzes and Summary Practice Tests.

DVD The Business Math Web site

Visit the site at www.mhhe.com/slater9e and find the Internet Resource Guide with hot links, tutorials, practice quizzes, and other study materials useful for the course.

Video Cases

There are seven video cases applying business math concepts to real companies such as Hotel Monaco, Louisville Slugger, American President Lines, Washburn Guitars, Online Banking, Buycostume.com, and Federal Signal Corporation. Video clips are included on the student DVD. Some background case information and assignment problems incorporating information on the companies are included at the end of Chapters 6, 7, 8, 9, 11, 16, and 21.

Compounding/Present Value Overlays

A set of color overlays are inserted in Chapter 13. These color graphics are intended to demonstrate for students the concepts of present value and future value and, even more important, the basic relationship between the two.

Business Math Scrapbook

At the end of each chapter you will f ind clippings from The Wall Street Journal and various other publications. These articles will gi ve you a chance to use the theory pro vided in the chapter to apply to the real world. It allows you to put your math skills to work.

Group activity: Personal Finance, a Kiplinger Approach

In each chapter you can debate a business math issue based on a Kiplinger’s Personal Finance magazine article that is presented. This is great for critical thinking, as well as improving your writing skills.

Spreadsheet Templates

Excel® templates are available for selected end-of-chapter problems. You can run these templates as is or enter your own data. The templates also include an interest table feature that enables you to input an y percentage rate and an y terms. The program will then generate table values for you.

Cumulative Reviews

At the end of Chapters 8 and 13 are w ord problems that test your retention of b usiness math concepts and procedures. Check f igures for all cumulative review problems are in Appendix B.

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Acknowledgments Academic Experts, Contributors Anthony Aiken Justin Barclay Cheryl Bartlett Ben Bean George Bernard Don Boyer Gilbert Cohen Laura Coliton Judy Connell Ronald Cooley Kathleen Crall Patrick Cunningham John Davis Tamra Davis James DeMeuse

Doug Dorsey Acie Earl Rick Elder Marsha Faircloth Tony Franco Bob Grenowski Victor Hall Frank Harber James Hardman Helen Harris Ron Holm William Hubert Christy Isakson Elizabeth Klooster Libby Kurtz

Ken Koerber Jennifer Lopez Bruce MacLean Lynda Mattes Jon Matthews Loretta McAdam Jean McArthur Sharon Meyer Norma Montague Christine Moreno Fran Okoren Roy Peterson Cindy Phipps Anthony Ponder Joseph Reihling

Dana Richardson Denver Riffe David Risch Joel Sacramento Naim Saiti Ellen Sawyer Tim Samolis Marguerite Savage Warren Smock Ray Sparks William Tusang Jennifer Wilbanks Andrea Williams Beryl Wright Denise Wooten

Company/Applications Chapter 1

Chapter 4

Home Depot—Problem solving Girl Scouts—Reading, writing, and rounding numbers McDonald’s—Rounding Tootsie Roll—Rounding all the way Toyota, Honda, Saturn—Rounding Hershey—Subtraction of whole numbers

Bank of America—Personal finance Continental, Amazon—E-checks J.P. Morgan Chase—Online banking eBay—Online banking PayPal—Online banking PNC Financial—Online banking Visa, Mastercard—Electronic bill paying Volkswagon—Banking application

Chapter 2 M&M’s/Mars—Fractions and multiplication Wal-Mart—Type of fractions TiVo—Subraction of fractions M&M’s/Mars—Multiplying and dividing fractions Target, MinuteClinic, RediClinic— Healthcare Exotic Car Share—Fractional ownership

Chapter 3 McDonald’s—Currency application M&M’s/Mars—Fractional decimal conversion Apple—Decimal applications in foreign currency Cingular, T-Mobile—Cost of phone calls Burberry, Tiffany—Currency application

Chapter 5 Calvin Klein, Burberry—Unknown Stanley Consultants—Workforce Snickers—Solving for the Unknown Disney—Solving for the Unknown American Quarter Coach—Personal finance Yacht Smart—Personal finance

Chapter 6 Capital One Financial—Cost of ATMs Ford—Percents Dell, Apple, Gateway—Percents HP, NEC, Sony, IBM—Percents M&M’s/Mars—Percent, percent increase and decrease Kellogg—Converting decimals to percents Wal-Mart—Percent increase, decrease

USA Today, The Wall Street Journal— Portion, base, rate The New York Times, The Washington Post—Portion, base, rate Chicago Tribune, Houston Chronicle— Portion, base, rate UPS—Portion, base, rate

Chapter 7 Google, Overstock, AOL—Online retailers Randall Scott Cycle, Condor Golf— Discounts Lighting Galleries of Sarasota— Discounts DHL, UPS—Freight FedEx—Freight Comcast, AT&T, Time Warner— Personal finance

Chapter 8 Disney, Payless Shoe Source—Licensing Levi-Strauss, Target—Markup H&M, GAP, French Connection, Wal-Mart—Sourcing John Hancock—Long-term care Bennigan’s—Markup

Chapter 9 Delta Airlines—Paycuts Fed Express—Independent contractors

Acknowledgments

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Acknowledgments

Chapter 10

Chapter 15

Chapter 20

Federal Deposit Insurance Company— Liability J.P. Morgan Chase, Citigroup—Late Payment charges Bank of America—Late payment charges Data Trac—Cheaper loans Digital Equipment Corp.—Cheaper loans Pentagon Federal Credit Union— Cheaper loans

Bank for International Settlements— Home price appreciation Credit Suisse First Boston—Monthly payments Lending Tree, Inc.—Cost of refinancing

Mavlife Financial Co.—Long-term care Home Depot, Lowes—Renting a truck AccuQuote.com—Cost of insurance Allstate, Amica Mutual—Cost of insurance Progressive, Youdecide.com—Cost of insurance

Chapter 11 Bank of Internet, Citibank, E-Loan, Prosper.com—Borrowing online Saks Inc.—Notes Small Business Administration—Line of credit U.S. Treasury—Buying treasuries online

Chapter 12 American Express, Bank of America— Saving cash Bankrate.com—Interest rates

Chapter 13 Dunkin’ Donuts—Investing your savings State Lotteries—Annuities D3 Financial Counselors—Roth

Chapter 14 Land Rover—APR Boston Globe—Monthly payments

Chapter 16 Coach, Inc.—Net income Kodak—Accounting errors H. J. Heinz Co.—Profit/Sales L. G. Electronics, Phillips Electronics— Impairment Samsung—Impairment Wal-Mart, Target—Profit margin

Chapter 17 Land Rover—Depreciation Kelley Blue Book—Resale value BMW of North America—Tax breaks

Chapter 18 Wal-Mart—Inventory identification ODW Logistics, Inc.—Outsourcing Ryerson Tull—LIFO, FIFO Global Sources Ltd.—Just-in-time inventory

Chapter 19 Hillerich & Bradsby Co.—Bartering Deloitte & Touche USA—Bartering

Chapter 21 CCH Inc.—Sale of stocks Home Depot—Stock quotations Goodyear—Bonds Putnam Investments—Mutual funds Google—PE ratio Viacom Inc., CBS Corporation— Corporate strategy

Chapter 22 American Institute of Certified Public Accountants—Median Federal Reserve—Retirement Wal-Mart, Sam’s Club—Live graphs, pie charts Apple, Microsoft—Corporate reporting Target, Kmart, Costco—Number reporting Exxon Mobil, General Motors, GE, Ford—Number reporting

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Contents Kiplinger’s Personal Finance Magazine Subscription Form xv

CHAPTER 1

Whole Numbers; How to Dissect and Solve Word Problems 1 LU 1–1 LU 1–2 LU 1–3

CHAPTER 2

Fractions 33 LU 2–1 LU 2–2 LU 2–3

CHAPTER 3

The Checking Account 89 Bank Statement and Reconciliation Process; Trends in Online Banking 93

Solving for the Unknown: A How-to Approach for Solving Equations 113 LU 5–1 LU 5–2

CHAPTER 6

Rounding Decimals; Fraction and Decimal Conversions 65 Adding, Subtracting, Multiplying, and Dividing Decimals 71

Banking 88 LU 4–1 LU 4–2

CHAPTER 5

Types of Fractions and Conversion Procedures 35 Adding and Subtracting Fractions 40 Multiplying and Dividing Fractions 46

Decimals 64 LU 3–1 LU 3–2

CHAPTER 4

Reading, Writing, and Rounding Whole Numbers 2 Adding and Subtracting Whole Numbers 8 Multiplying and Dividing Whole Numbers 12

Solving Equations for the Unknown 114 Solving Word Problems for the Unknown 120

Percents and Their Applications 137 LU 6–1 Conversions 138 LU 6–2 Application of Percents—Portion Formula 144 Video Case: American President Lines 169

CHAPTER 7

Discounts: Trade and Cash 170 LU 7–1 Trade Discounts—Single and Chain (Includes Discussion of Freight) 171 LU 7–2 Cash Discounts, Credit Terms, and Partial Payments 179 Video Case: Hillerich & Bradsby Company “Louisville Slugger” 202

CHAPTER 8

Markups and Markdowns; Perishables and Breakeven Analysis 203 LU 8–1 Markups Based on Cost (100%) 205 LU 8–2 Markups Based on Selling Price (100%) 210 LU 8–3 Markdowns and Perishables 216 LU 8–4 Breakeven Analysis 219 Video Case: Hotel Monaco Chicago 233 Cumulative Review: A Word Problem Approach—Chapters 6, 7, 8

CHAPTER 9

234

Payroll 235 LU 9–1 LU 9–2

Calculating Various Types of Employees’ Gross Pay 236 Computing Payroll Deductions for Employees’ Pay; Employers’ Responsibilities 240 Video Case: Washburn Guitars 257

Contents

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Contents

CHAPTER 10

Simple Interest 258 LU 10–1 Calculation of Simple Interest and Maturity Value 259 LU 10–2 Finding Unknown in Simple Interest Formula 262 LU 10–3 U.S. Rule—Making Partial Note Payments before Due Date 264

CHAPTER 11

Promissory Notes, Simple Discount Notes, and the Discount Process 278 LU 11–1 Structure of Promissory Notes; the Simple Discount Note 279 LU 11–2 Discounting an Interest-Bearing Note before Maturity 282 Video Case: Online Banking 294

CHAPTER 12

Compound Interest and Present Value 295 LU 12–1 Compound Interest (Future Value)—The Big Picture 296 LU 12–2 Present Value—The Big Picture 303

CHAPTER 13

Annuities and Sinking Funds 316 LU 13–1 Annuities: Ordinary Annuity and Annuity Due (Find Future Value) 317 LU 13–2 Present Value of an Ordinary Annuity (Find Present Value) 323 LU 13–3 Sinking Funds (Find Periodic Payments) 326 Cumulative Review: A Word Problem Approach—Chapters 10, 11, 12, 13 339

CHAPTER 14

Installment Buying, Rule of 78, and Revolving Charge Credit Cards 341 LU 14–1 Cost of Installment Buying 342 LU 14–2 Paying Off Installment Loans before Due Date 347 LU 14–3 Revolving Charge Credit Cards 350

CHAPTER 15

The Cost of Home Ownership 365 LU 15–1 Types of Mortgages and the Monthly Mortgage Payment 367 LU 15–2 Amortization Schedule—Breaking Down the Monthly Payment 370

CHAPTER 16

How to Read, Analyze, and Interpret Financial Reports 382 LU 16–1 Balance Sheet—Report As of a Particular Date 383 LU 16–2 Income Statement—Report for a Specific Period of Time 389 LU 16–3 Trend and Ratio Analysis 394 Video Case: Buycostumes.com 410

CHAPTER 17

Depreciation 412 LU 17–1 LU 17–2 LU 17–3 LU 17–4

CHAPTER 18

Concept of Depreciation and the Straight-Line Method 413 Units-of-Production Method 415 Declining-Balance Method 417 Modified Accelerated Cost Recovery System (MACRS) with Introduction to ACRS 418

Inventory and Overhead 429 LU 18–1 Assigning Costs to Ending Inventory—Specific Identification; Weighted Average; FIFO; LIFO 431 LU 18–2 Retail Method; Gross Profit Method; Inventory Turnover; Distribution of Overhead 436

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Contents

CHAPTER 19

Sales, Excise, and Property Taxes 453 LU 19–1 Sales and Excise Taxes 454 LU 19–2 Property Tax 456

CHAPTER 20

Life, Fire, and Auto Insurance 466 LU 20–1 Life Insurance 467 LU 20–2 Fire Insurance 472 LU 20–3 Auto Insurance 475

CHAPTER 21

Stocks, Bonds, and Mutual Funds 490 LU 21–1 Stocks 491 LU 21–2 Bonds 495 LU 21–3 Mutual Funds 497 Video Case: Federal Signal Corporation 509

CHAPTER 22

Business Statistics 510 LU 22–1 Mean, Median, and Mode 511 LU 22–2 Frequency Distributions and Graphs 514 LU 22–3 Measures of Dispersion (Optional) 520

APPENDIX A:

Additional Homework by Learning Unit A

APPENDIX B:

Check Figures B

APPENDIX C:

Glossary C

APPENDIX D:

Metric System D

Index IN

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Because Money Matters... Subscribe to Kiplinger’s at Special Student Rates! Every month, more than three million Americans turn to Kiplinger’s Personal Finance magazine for advice and information about how to manage their money . How to save it. Spend it. Invest it. Protect it. Insure it. And make more of it. If it affects you and your money, then you’ll find it in the pages of Kiplinger’s. From our annual ranking of the nation’s best mutual funds to our yearly rating of new automobiles, we provide you with a different kind of investment publication. We make it easy for you to subscribe with the lowest rates available to students and educators. Just provide your name and address below . Make checks payable to Kiplinger’s Personal Finance. Or, if you prefer we will bill you later.

Student’s Name

Address

City

(

Apt. #

State

Zip

)

Phone

Term: One year for $12.00 After completing the form, please mail it to: Kiplinger’s Personal Finance, P.O. Box 3291, Harlan, Iowa 51593-2471. CODE: J5MCGRAW

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From Jeff’s Desk

One Million Copies Sold...Thank You! DVD with Each Text THE SLATER LEARNING SYSTEM Text

WOW! Thank you for making my book the number one best-

•

Practice Quiz

•

Chapter Organizer

selling text in the country. For this ninth edition, I have spent the

•

Critical Thinking Discussion Questions

•

Drill Problems

•

Word Problems

•

Challenge Problems

•

Summary Practice Tests

is not cosmetic; I have included the latest material from The Wall

•

Personal Finance: A Kiplinger Approach

Street Journal and Kiplinger’s magazine. I continue to write the Test

•

Business Math Scrapbook with Internet Applications

last two years writing new material for the text, doing new videos on the student DVD, as well as updating supplements. This revision

Bank and Instructor’s Manual myself. I am pleased to let you know •

I have created new online tests as well. The following walk-through will show you what I like to call my Total Learning System. But before we do this, I would like to thank you, the customer.

Internet Web site (www.mhhe.com/slater9e)

•

Appendix A Problems by Learning Unit

•

Appendix B Check Figures

•

Video Cases

•

Compounding/Present Value Overlays

My passion: to serve my customers, both instructors and students. I don’t take being number one in sales lightly. You can reach me via e-mail at [email protected] or call me directly on my toll free number: 1-800-484-1341 . . . 8980. I believe you should put your energy into the classroom or your online course. It is my job to provide you with the best and most upto-date text, along with an author-driven supplements package. Now let’s check out this Instructor’s Walk-through.

Best,

Jeffrey Slater

Supplements •

Business Math Handbook

•

Web site and Online Learning Center

•

Business Math Internet Resource Guide

•

DVD-ROM

•

Test Bank

•

Computerized Testing with EZ Test

•

Instructor’s Resource CD-Rom

•

Instructor’s Resource Manual

•

Excel Workbook

•

Financial Calculator Guide

•

Electronic Calculator Guide

•

TI-83/TI-84 Calculator Guide

•

Student Solutions Manual & Study Guide

•

Author support—e-mail

•

Publisher support—sales representatives, e-mail

•

The Wall Street Journal newspaper

l.com

[email protected]

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For All My Loyal Adopters Highlights of Changes for 9E: A Transition Guide For all chapters, I have included new, more recent clips from The Wall Street Journal for the opening, as well as new Kiplinger’s Personal Finance articles and scrapbooks at the end of the chapter. All of these have been chosen to provide valuable and timely real-world topics for you and your students. Chapter 1: Whole Numbers: How to Dissect and Solve Word Problems • Extra practice quiz with check figures added for each learning unit (throughout text) • Deletion of horizontal and vertical addition • New summary practice test, now included on DVD-video tutorial Chapter 2: Fractions • New summary practice test, now included on DVD Chapter 3: Decimals • New foreign currency table updated Chapter 4: Banking • Deletion of credit cards and merchant summary • New clips within chapter and new material on online banking Chapter 5: Solving for the Unknown: A How-to Approach for Solving Equations • New summary practice test, now included on the DVD Chapter 6: Percents and Their Applications • Extra practice quiz with check figures added for each learning unit • New section on calculating percent decrease and increase using Wal-Mart • New summary practice test, now included on the DVD Chapter 7: Discounts: Trade and Cash • Extra practice quiz with check figures added for each learning unit • New clips within chapter Chapter 8: Markups and Markdowns: Perishables and Breakeven Analysis • New chapter opener and clip • New learning unit on breakeven analysis Chapter 9: Payroll • New chapter opener and clip • New payroll tables • FICA now based on 6.2% on $97,500

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Chapter 10: Simple interest • New chapter opener and clip, new clips within chapter • New drill and word problems Chapter 13: Annuities and Sinking Funds • New math formulas added to chapter organizer as alternative to table lookup • New chapter opener and clip, new clips within chapter • Update to plastic overlays, show relationship to tables Chapter 14: Installment Buying, Rule of 78, and Revolving Charge Credit Cards • New chapter opener and clip, new clips within chapter • New drill and word problems Chapter 15: Cost of Home Ownership • New reference to mortgage accelerator loan Chapter 16: How to Read, Analyze, and Interpret Financial Reports • New material on Sarbanes-Oxley • New Video Case Chapter 17: Depreciation • Deleted unit on sum-of-years digits Chapter 19: Sales, Excise, and Property Tax • Updated, new drill and word problems Chapter 20: Life, Fire, and Auto Insurance • Updated with new clips, drill and word problems Chapter 21: Stocks, Bonds, and Mutual Funds • Newspaper insert on How to Read The Wall Street Journal has been updated Chapter 22: Statistics • New chapter opener and clip, new clips within chapter

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Real-World Applications Instructors asked for an even greater emphasis on the applications of business math in the United States and globally. The Ninth Edition includes references to companies such as Google, eBay, TiVo, Wal-Mart, and DHL to illustrate chapter topics. Over 100 actual clippings from The Wall Street Journal and 22 Kiplinger’s Personal Finance magazine articles give students a more complete view of real-world practices from the business press.

Mark Lennihan/AP Wide World

Too often people think their bank is their best friend. You should remember that your bank is a business. The banking industry is very competitive. Note in the Wall Street Journal clipping “Bank of America to Pay $2.5 Billion for China Foothold” how quickly the banking sectors are changing all over the world to be more competitive. An important fixture in today’s banking is the automatic teller machine (ATM). The ability to get instant cash is a convenience many bank customers enjoy. However , more than half of the ATM customers do not like to deposit Wall Street Journal © 2005 checks because they are afraid the checks will not be correctly deposited to their account. Bank of America, Bank One, and Wells Far go are testing new ATMs that accept a check, scan the check, and print a receipt with a photographic image of the check. When these machines are widely available, they will eliminate the fear of depositing checks. The effect of using an ATM card is the same as using a debit card—both transactions

Personal Finance A KIPLINGER APPROACH

Wall Street Journal © 2005

BUSINESS MATH ISSUE Kiplinger’s © 2006

Vegetable oil will not solve our oil problem. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

31

The Wall Street Journal Highlights With over 100 clippings from The Wall Street Journal, students can see the relevance of text topics to the business world. Kiplinger’s Personal Finance Magazine Articles These articles were completely updated this edition and include: 1. Saving the World with French Fries, page 31 2. Checkups on the Run, page 62 3. Call U.S. for Less, page 86 4. Electronic Bill-Paying Snafus, page 111 5. Retire a Millionaire, page 135 6. Live Better and Sell Higher, page 167 7. Save a Bundle on Telecom Services, page 200 8. A Fresh Look at LongTerm Care, page 231 9. Healthy Choices, page 255 10. My Unpaid Debt Still Haunts Me, page 276 11. Last Chance to Lock In, page 292 12. Keep the Change, page 314 13. An Overlooked Way to Shear Your Taxes, page 337 14. Should I Take a 0% Credit Card Offer?, page 363 15. Should You Buy or Sell First?, page 380 16. Pension Protection, page 408 17. Buy a 2006 Car on Sale?, page 427 18. China Go-Between, page 451 19. Cut Your Property Tax, page 464 20. Cut Insurance Costs, page 488 21. What’s Google Worth?, page 507 22. Why Two Tech Titans Will Please Shareholders This Coming Year, page 536

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Favorite Features of the Text You can count on all of the key features developed for this book over the years remaining in the Ninth Edition. I have listened to instructors using the text, as well as my own students, in order to improve the book and make sure it serves you and your students effectively. My goal was to make it as motivating and understandable as possible for both the young, out of high school student and the older, returning student.

Chapter Openers

The chapter openers introduce students to the chapter’s topics, and Learning Objectives for each unit provide them with an overview of the key material that will be covered. Students can see the real-world applications of business math through The Wall Street Journal clips which make the topics relevant to them. (p. 137)

CHAPTER

6

Percents and Their Applications

LEARNING UNIT OBJECTIVES LU 6–1: Conversions • Convert decimals to percents (including rounding percents), percents to decimals, and fractions to percents (pp. 139–141). • Convert percents to fractions (p. 142).

LU 6–2: Application of Percents—Portion Formula • List and define the key elements of the portion formula (pp. 144–145). • Solve for one unknown of the portion formula when the other two key elements are given (pp. 145–148 ). • Calculate the rate of percent decreases and increases (pp. 148–151).

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Clear Explanations

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Explanations are given in a step-by-step format that is easy to follow and remember, followed by understandable examples. (p. 48) Dividing Proper Fractions The division of proper fractions introduces a new term—the reciprocal. To use reciprocals, we must first recognize which fraction in the problem is the divisor—the fraction that we divide by. Let’s assume the problem we are to solve is 18 23. We read this problem as “ 18 divided by 23.” The divisor is the fraction after the division sign (or the second fraction). The steps that follow show how the divisor becomes a reciprocal. DIVIDING PROPER FRACTIONS Step 1.

Invert (turn upside down) the divisor (the second fraction). The inverted number is the reciprocal.

Step 2.

Multiply the fractions.

Step 3.

Reduce the answer to lowest terms or use the cancellation method.

Do you know why the inverted fraction number is a reciprocal? Reciprocals are two num1 2 bers that when multiplied give a product of 1. For example, 2 (which is the same as 1) and 2 are reciprocals because multiplying them gives 1. EXAMPLE

1 2 8 3

1 3 3 8 2 16

Dividing Mixed Numbers Now you are ready to divide mixed numbers by using improper fractions. DIVIDING MIXED NUMBERS

Functional Use of Color

Step 1.

Convert all mixed numbers to improper fractions.

Step 2.

Invert the divisor (take its reciprocal) and multiply. If your final answer is an improper fraction, reduce it to lowest terms. You can do this by finding the greatest common divisor or by using the cancellation technique.

Functional color-coding was first introduced in the Third Edition of the text. While many books use color, I set out from the beginning to use color to teach. I personally color-code each element to enhance the learning process. For example, when a student sees a number in red, they know it is a key item they are solving for. Color Key Blue: Movement, cancellations, steps to solve, arrows, blueprints Gold: Formulas and steps Green: Tables and forms Red: Key items we are solving for Magenta: Worked-out solutions in Teacher’s Edition only

Plastic Overlays

Chapter 13 features plastic overlays that review compounding, present value, ordering annuities, and present value annuities.

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Practice Quizzes follow each Learning Unit in the book. These quizzes provide immediate feedback for students to check their progress and are followed by worked-out solutions. The logo lets students know that videos are available on the student DVD-ROM. In these videos I carefully walk students through the material, reinforcing the content. These are accessible by each Learning Unit so students can go directly to the Practice Quiz they choose without searching cumbersome videotapes. (p. 240) New to this edition are Extra Practice Quizzes which follow the Practice Quizzes. Check figures and page references are included at the bottom of the Chapter Organizer.

Practice Quizzes and New Extra Practice Quizzes

LU 9–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1. 2.

DVD

3.

Jill Foster worked 52 hours in one week for Delta Airlines. Jill earns $10 per hour. What is Jill’s gross pay, assuming overtime is at time-and-a-half? Matt Long had $180,000 in sales for the month. Matt’ s commission rate is 9%, and he had a $3,500 draw. What was Matt’s end-of-month commission? Bob Meyers receives a $1,000 monthly salary . He also receives a variable commission on net sales based on the following schedule (commission doesn’ t begin until Bob earns $8,000 in net sales): $8,000–$12,000 Excess of $12,000 to $20,000

1% 3%

Excess of $20,000 to $40,000 More than $40,000

5% 8%

Assume Bob earns $40,000 net sales for the month. What is his gross pay?

✓ 1.

2.

3.

Blueprint Aid for Dissecting and Solving a Word Problem

Solutions 40 hours $10.00 $400.00 12 hours $15.00 180.00 ($10.00 1.5 $15.00) $580.00 $180,000 .09 $16,200 3,500 $12,700 Gross pay $1,000 ($4,000 .01) ($8,000 .03) ($20,000 .05) $1,000 $40 $240 $1,000 $2,280

Students need help in overcoming their fear of word problems. The first eight chapters (except Chapter 4) provide a “blueprint” format for solving word problems. It shows students how to begin the problem-solving process, gets them actively involved in dissecting the word problem, shows visually what has to be done before calculating, and provides a structure for them to use. (p. 147) The Word Problem Sales of Milk Chocolate M&M’ s® are $320,000. Total sales of Milk Chocolate M&M’ s, Peanut, and other M&M’ s® chocolate candies are $400,000. What percent of Peanut and other M&M’ s® chocolate candies are sold compared to total M&M’s® sales? The facts

Solving for?

Steps to take

Key points

Milk Chocolate M&M’s® sales: $320,000.

Percent of Peanut and other M&M’s® chocolate candies sales compared to total M&M’s® sales.

Identify key elements.

Represents sales of Peanut and other M&M’s® chocolate candies

Total M&M’s® sales: $400,000.

Base: $400,000. Rate: ? Portion: $80,000 ($400,000 $320,000). Portion Rate Base

Portion ($80,000) Base Rate ($400,000) (?) When portion becomes $80,000, the portion and rate now relate to same piece of base.

Steps to solving problem 1. Set up the formula. 2. Calculate rate.

Rate R

Portion Base $80,000 ($400,000 $320,000) $400,000

R 20%

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The Chapter Organizer

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This quick reference guide provides students with a complete set of notes, including color coding consistent with the text. Key points, formulas, examples, vocabulary, and new Check Figures for the Extra Practice Quizzes are included with page references. Widely copied by other textbooks, this tool is useful as a reference for students as well as for reviews before exams. (p. 247)

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

State and federal unemployment, p. 244

Employer pays these taxes. Rates are 6.2% on $7,000 for federal and 5.4% for state on $7,000. 6.2% 5.4% .8% federal rate after credit. If state unemployment rate is higher than 5.4%, no additional credit is taken. If state unemployment rate is less than 5.4%, the full 5.4% credit can be taken for federal unemployment.

Cumulative pay before payroll, $6,400; this week’s pay, $800. What are state and federal unemployment taxes for employer, assuming a 5.2% state unemployment rate?

Biweekly, p. 236 Deductions, p. 237 Differential pay schedule, p. 238 Draw, p. 239 Employee’s Withholding Allowance Certificate (W-4), p. 241 Fair Labor Standards Act, p. 237 Federal income tax withholding (FIT), p. 242

KEY TERMS

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

Critical Thinking Discussion Questions

.052 $600 $31.20

Federal .008 $600 $4.80 ($6,400 $600 $7,000 maximum)

Federal Insurance Contribution Act (FICA), p. 241 Federal Unemployment Tax Act (FUTA), p. 244 Gross pay, p. 237 Medicare, p. 241 Monthly, p. 236 Net pay, p. 242 Overrides, p. 239 Overtime, p. 237 Payroll register, p. 240

LU 9–1a (p. 240) 1. $732 2. $12,800 3. $4,070

State

Percentage method, p. 242 Semimonthly, p. 236 Social Security, p. 241 State income tax (SIT), p. 242 State Unemployment Tax Act (SUTA), p. 244 Straight commission, p. 239 Variable commission scale, p. 239 W-4, p. 241 Weekly, p. 236

LU 9–2a (p. 245) 1. $31; 145; $2,184.43 2. $846.60; $132.80

These thought-provoking questions follow the Chapter Organizer and are designed to get students to think about the larger picture and the “why’s” of business math. They go beyond the typical questions by asking students to explain, define, create, etc. (p. 247)

Critical Thinking Discussion Questions 1. Explain the dif ference between biweekly and semimonthly . Explain what problems may develop if a retail store hires someone on straight commission to sell cosmetics. 2. Explain what each column of a payroll register records (p. 241) and how each number is calculated. Social Security

tax is based on a specific rate and base; Medicare tax is based on a rate but no base. Do you think this is fair to all taxpayers? 3. What taxes are the responsibility of the employer? How can an employer benefit from a merit-rating system for state unemployment?

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Over 50 photos are included to stimulate student interest and help students see business math with imagination and enthusiasm. Whether showing McDonald’s Big Mac prices in various international cities, inventory systems, or online banking and bill paying, business math becomes real to them.

Photos

Freight Terms Do you know how successful the shipping businesses of DHL, UPS, and FedEx are in China? The Wall Street Journal clipping “Faster, Faster . . .” shows that the shipping businesses of these three companies can be quite profitable.

Claro Cortes IV/Reuters/Landov

Wall Street Journal © 2004

End-of-Chapter Problems

At the end of each chapter Drill Problems are followed by Word Problems. I’ve added new problems in each chapter using material from newspapers such as The New York Times and magazines such as BusinessWeek, Consumer Reports, and Smart Money to help students see the relevance of the material. An Excel logo next to a problem indicates an Excel template is available on the DVD-ROM and in the Excel Workbook to help solve that problem. Challenge Problems let your students stretch their understanding and ability to solve more complex problems. I’ve included two per chapter. A Summary Practice Test concludes the problem section and covers all the Learning Objectives in the chapter.

Drill Problems END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Convert the following decimals to percents: 6–1. .74 74%

6–2. .824 82.4%

6–3. .9 90%

6–4. 8.00 800%

6–5. 3.561 356.1%

6–6. 6.006 600.6%

6–8. 14%

3 6–9. 64 % .643 10

Convert the following percents to decimals: 6–7. 8%

.08

6–10. 75.9%

.759

6–11. 119%

.14 1.19

6–12. 89%

Convert the following fractions to percents (round to the nearest tenth percent as needed):

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6–13.

1 .0833 8.3% 12

6–14.

1 .0025 .3% 400

6–15.

7 .875 87.5% 8

6–16.

11 .9166 91.7% 12

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Word Problems WORD PROBLEMS (First of Four Sets) 6–52. At a local Wendy’s, a survey showed that out of 12,000 customers eating lunch, 3,000 ordered Diet Pepsi with their meal. What percent of customers ordered Diet Pepsi? 3,000 Note: Portion and rate must Portion 25% refer to same piece of the base. 12,000 (3,000) Base Rate (12,000) (?)

6–53. What percent of customers in Problem 6–52 did not order Diet Pepsi? Note: Portion and rate must 9,000 75% refer to same piece of the base. 12,000

Portion (9,000) Base Rate (12,000) (?)

(

12,000 –3,000

(

6–54. The Rhinelander Daily News March 4, 2007 issue, ran a story on rising gas prices. Last week, gas was selling for $1.99 a gallon and the world looked rosy. Not so now. The price of a gallon of regular unleaded nosed up to $2.24. What was the

Challenge Problems

CHALLENGE PROBLEMS 6–96. Continental Airlines stock climbed 4% from $18.04. Shares of AMR Corporation, American Airlines’ parent company, closed up 7% at $12.55. AirTran Airways went from $17.27 to $17.96. Round answers to the nearest hundredth. (a) What is the new price of Continental Airlines stock? (b) What had been the price of AMR Corporation stock? (c) What percent did AirTran Airways increase? Round to the nearest percent. a. $18.04 1.04 $18.76 or $18.04 .04 $ .7216 Portion (?) 18.04 $18.7616 $18.76 Base Rate ($18.04) (1.04)

b.

$12.55 $11.728971 $11.73 1.07

Portion ($12.55) Base Rate (?) (1.07)

c.

$.69 3.9953676% 4.00% $17.27

Portion ($.69)

( ( $17.96 – 17.27

Base Rate ($17.27) (?)

Summary Practice Test

The ninth edition DVD now contains video tutorials of all Summary Practice Tests.

15. Target ordered 400 iPods. When Target received the order, 100 iPods were missing. What percent of the order did Target receive? (p. 146) Portion 300 (300) 75% 400 Base Rate (400) (?)

16. Matthew Song, an employee at Putnam Investments, receives an annual salary of $120,000. Today his boss informed him that he would receive a $3,200 raise. What percent of his old salary is the $3,200 raise? Round to the nearest hundredth percent. (p. 146) Portion $3,200 2.67% ($3,200) $120,000 Base Rate ($120,000) (?)

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Personal Finance: A Kiplinger Approach

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A Kiplinger Group Project at the end of each chapter includes an article from Kiplinger’s Personal Finance magazine. Each article presents a business math issue for students to debate and solve. Suggested answers are located in the Instructor’s Resource Manual. This is an excellent tool to develop critical thinking and writing skills. It also provides opportunities for students to become involved in team projects. As stated in the AMATYC standards: “mathematics faculty will foster interactive learning through student writing, reading, speaking, and collaborative activities so that students can learn to work effectively in groups and communicate about mathematics both orally and in writing.” (p. 292)

Personal Finance A KIPLINGER APPROACH

M L H A R R I S /G E T T Y I M A G E S

can apply for a PLUS now and consolidate to lock in this year’s rate, says Mark Brenner, of College Loan Corp. (www.collegeloan.com), which makes such loans. Ask your school’s financialaid office for details.

G Students who take

out Stafford loans after July 1 will pay a fixed interest rate of 6.8%.

CO L L EG E

| To save on student-loan interest rates,

consolidate your debt by July 1. By Jane Bennett Clark

Last chance to LOCK in t se e m s l ik e only yesterday that student-loan rates were sinking faster than a December sun. Alas, the days of magically vanishing—or modestly rising—rates are about to end. Starting July 1, the Deficit Reduction Act of 2005 will set a fixed rate of 6.8% on new Stafford loans, about two percentage points above this past year’s lowest rate. Similarly, PLUS loans for parent borrowers will be fixed at 8.5%, up from the current 6.1%. But the fixed rates won’t apply to outstanding Stafford and PLUS loans. On those loans, rates will continue to change each July 1 based on the 91-day Treasury-bill yield set the last Thursday in May. The T-bill rate is expected to rise, so it pays to consolidate your loans and lock in the lower rate. Things get a little tricky if you con-

I

solidated last spring to take advantage of bottom-cruising rates (as low as 2.87% for Stafford loans and 4.17% for PLUS loans) and have since taken out new loans. You can consolidate the new loans, but you’ll want to keep the two consolidations separate, says Gary Carpenter, executive director of the National Institute of Certified College Planners (www.niccp.com). “If you roll an old consolidation into a new one, you get a blended rate—the lower rate is lost,” says Carpenter. And you may have to shop for a lender; some balk at consolidating loans of less than $7,500. Although financial-aid packages were calculated this spring, next fall’s freshmen will pay the post-July, fixed rate on Staffords; likewise, PLUS loans for parents of incoming freshmen will carry the new fixed rate. However, parents of currently enrolled students

Other options. After July 1, parents choosing between a PLUS loan with an 8.5% fixed rate and a variable-rate home-equity line of credit should take a closer look at the latter, says Carpenter. The average rate for equity lines was recently 7.67%, and interest is deductible. With rates fixed on Stafford loans, private loans, which are issued at variable rates, could someday end up costing less than Staffords. Sallie Mae (www.salliemae.com), the largest of the student-loan companies, offers private loans at the prime rate—lately 7.5%— with no fees for borrowers who have a good credit history. Even if rates head south, borrowers “should exhaust federal loans first,” says Sallie Mae spokeswoman Martha Holler. Unlike private loans, payments on those loans can be extended, deferred or forgiven in certain cases.

As for the other provisions of the Deficit Reduction Act, they represent “a mixed bag” for undergraduates, says Brenner. For Stafford loans, the law boosts the maximum amount you can borrow in each of the first two years of college (the total amount remains the same), phases out origination fees and expands Pell Grants for math and science students. Married couples will no longer be able to consolidate loans taken out separately into a single loan. And, as of July 1, students can no longer consolidate Staffords while they’re still in school. But Brenner says the changes “should in no way discourage American families from applying for the college of their choice.” There’s plenty of money for students who need it, he says, and federally sponsored loans remain “a hell of a deal.”

A mixed bag.

BUSINESS MATH ISSUE Kiplingers © 2006

The Deficit Reduction Act of 2005 is too complicated for students needing loans. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Business Math Scrapbook with Internet Application

The Business Math Scrapbook with Internet Application provides realworld applications at the end of the chapters. They can be assigned at your discretion to give students an opportunity to apply the chapter theory to real life business situations and to see the importance of what they’re learning. (p. 63)

Video Cases on DVD

There are seven video cases applying business math concepts to real companies such as Hotel Monaco, Louisville Slugger, American President Lines, Washburn Guitars, Online Banking, Buycostumes.com, and Federal Signal Corporation. Video clips are included on the student DVD. Some background case information and assignment problems incorporating information on the companies are included at the end of Chapters 6, 7, 8, 9, 11, 14, 16, and 21.

Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A What is the total cost of a Bentley boat? Prove your answer using fractions.

DVD

Video Case BUYCOSTUMES.COM

As a child, Jalem Getz, the CEO of Buycostumes.com put little thought into his Halloween costumes. Now he thinks about costumes all year long. Jalem Getz founded online business Buycostumes in a warehouse in the Milwaukee suburbs in 1999 taking advantage of Wisconsin’s central U.S. location and cheap rent. Getz used to dislike the lack of seasons in his native California. Now, he uses the extreme seasonality of the Halloween business to turn a big profit. The company got its start as brick-and-mortar retail business owned by Getz and partner Jon Majdoch. While still in their early 20s, the two began operating a chain of seasonal Halloween Express franchise stores, and then branched out into a couple of lamp and home accessory shops. In 2001, the company changed its name to Buyseasons, Inc. to reflect its new broader focus. “Rather than just focus on one season, we target consumers in different seasons,” said Jalem Getz. The Buycostumes name is still in broad use. Being an e-tailer means not having to open a retail space for just two months of the year, or stock other items. Money saved on storefronts goes to maintaining a stock of 10,000 Halloween items – 100 times what most retailers carry for the season. “Our selection sets us apart,” Getz said. “A lot of customers are looking for something unique, by having that

large selection we immediately build that additional goodwill.” The key to Buyseasons’ success limiting the choice of merchandise to items that can’t readily be found in neighborhood stores. That means less price competition and higher margins for Byseasons, which has been maintaining a 47.5 percent gross margin rate on the buycostumes.com site. In July 2006, Liberty Media announced plans to acquire Buycostumes.com Inc. 500 company, for an undisclosed sum. Getz will stay on as CEO. Buycostumes.com is the biggest online seller of costumes. It was ranked on October 2005 in Inc. magazine as the 75th –fastest growing U.S. private firm, with revenue of $17.6 million last year and three-year growth of 1,046 percent. Sales this year are expected to hit $25 million to $28 million, according to Getz. Other online Halloween firms also predicted double-digit growth in 2005 – according to a forecast by the National Retail Federation the entire industry would have a 5% gain, leading to a record $3.3 billion in sales for the entire industry. BuySeasons’ sales reached nearly $30 million in 2005, Getz said, up from $17.6 million in 2004. The company’s sales are expected to post 50% annual growth over the next three year. The company bills itself as the world’s largest Internet retailer of Halloween costumes and accessories.

2005 Wall Street Journal ©

site xt Web e : See te Projects later9e) and Th uide. /s Internet eG m rc co ou e. es hh R (www.m Math Internet s Busines

PROBLEM

1

The video stated the shipment of packages will increase from a normal 500 packages per day to as many as 30,000 packages per day. Phone calls would increase from 1,600 per 63 week to 30,000 a week. (a) What is the percent of increase in packages per day? Round to the nearest hundredth percent. (b) What is the percent of increase in phone calls per week? PROBLEM

2

The video advertises the “Mrs. Franklin Adult” costume with a retail value of $149.99 and Buycostumes.com price of $99.99. The “American Revolutionary Adult” costume with a retail value $314.99 and Buycostumes.com price of $239.99. (a) What is the dollar amount of savings achieved by purchasing on-line for each costume? (b) What is the percent savings by purchasing on-line for each costume? Round to the nearest hundredth percent. PROBLEM

3

On March 26, 2006, the Milwaukee Journal Sentinel reported BuySeasons, which now leases 81,000 square feet in a business park, wanted to move to a 200,000-square-foot building. The May 3, 2006 issue reported the Zoning, Neighborhoods & Development Committee voted 3-1 to

recommend the sale of 9.2 acres in Menomonee Valley Industrial Center to house a new headquarters for BuySeasons, operative of Buycostumes.com., at a price of $110,000 per acre. (a) What would be the percent increase in space for BuySeasons? Round to the nearest hundredth percent. (b) What would be the total price for the land? PROBLEM

4

In 2007, the company expects to have just over 600 seasonal employees. The number of seasonal employees is expected to exceed 800 in 2008 and 900 in 2009. In 2007, the company would employ 90 full employees, 126 in 2008, and 161 in 2009. (a) Complete a trend analysis of seasonal employees for the three years. (b) Complete a trend analysis of full time employees for the three years using 2007 as the base year. Round to the nearest whole percent. PROBLEM

5

On October 30, 2006, USA Today reported on the booming number of young adults who treat themselves to Halloween fun. The 18- to 24-year old group is spending an average of $30.38 on costumes this year, a 38% increase over 2005, according to the National Federation and BIGresearch. Those 25 to 34 will spend an average of $31.33 up 17%. National Costumers Association President Debbie Lyn Owens says college students suiting up for parties are fueling much of the

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Supplements Package Business Math Handbook

This reference guide contains all tables found in the text and is included free with the text.

Web site and Online Learning Center

The Business Math Web site at www.mhhe.com/slater9e offers an interactive environment for teachers and students. The instructor section contains text updates, supplement information, and teaching support including the electronic version of the Instructor’s Resource Guide. It includes PageOut—a powerful, easy-to-use tool that allows you to produce professional course Web pages. The Online Learning Center takes the pedagogical features and supplements of the book and places them online. It includes interactive self-grading quizzes, PowerPoint lectures, chapter outlines, teaching tips, and more. Student material includes practice quizzes, glossary, self-paced worksheets, and much more. New to the Ninth Edition—self-grading quizzes, PowerPoint lectures, and chapter summary practice test review videos by the author can be downloaded by students to their iPods. All of this content is also available on cartridges for local use on WebCT or Blackboard.

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Business Math Internet Resource Guide

The Business Math Internet Resource Guide will take students online and show them and you interesting source materials for business math. Following an introduction on how to use the Internet, each chapter of the book has specific sites listed and a description of what students will find there. There are also projects listed for each chapter relating to the Internet. Included on Student DVD-ROM.

DVD-ROM

Students can use this tool on their computers or home DVD player to see and hear how the author solves all the practice quizzes and Summary Practice Tests in the text. Students can also refer to the DVD-ROM for PowerPoint lectures, self-grading practice quizzes, Excel Templates, the Internet Resource Guide, and Web links. Video cases apply business math concepts to real companies such as Hotel Monaco, Louisville Slugger, and others.

Instructor’s Resource CD-ROM

The Instructor’s Resource CD-ROM contains the Test Bank, E-Z Test computerized testing system, PowerPoint Lecture Slides, Instructor’s Resource Guide, and solutions files.

Instructor’s Resource Guide

This resource manual includes: • Syllabus Preparation; Self-Paced Syllabus; Student Progress Chart • Integrating the Electronic Calculator; Suggestions for Using Computers and Videos • Suggestions for Regrouping Chapters • Worked out Solutions for Extra Practice Quizzes • Suggestions on Teaching Using the Business Math Internet Resource Guide • Tips on Teaching Group Activities with Kiplinger’s Personal Finance magazine • Your Course versus Math Anxiety • Sample Civil Service Exam with worked-out solutions • Insight into Proportions supplement • Excel Template Fact Sheet • Check Figures for even-numbered end-of-chapter drill and word problems • Appendix B Solutions (Chapters 13–22) Each chapter includes: • • • • • • •

Teaching Tips from Jeff Slater Lecture Outline The Pocket Calculator Workshop Suggested Solution to Critical Thinking Discussion Questions Teacher’s Guide to Kiplinger Group Activity Additional Word Problems (not in the text) Worked-Out Solutions to Practice Quiz found in the Student Solutions Manual and Study Guide • Vocabulary Crossword Puzzles with solutions

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Excel Workbook

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The Excel Workbook is available as a shrinkwrapped package with the text. This workbook instructs your students in constructing their own spreadsheets. It includes business topics such as inventory, interest, markup, and annuities using problems from the text. The templates are on the student DVD-ROM and are available for selected end-of-chapter problems designated with an Excel logo. Students can run these templates as is or add their own data. The DVD also includes an interest table feature that allows you to input any percentage rate and terms. The program will then generate table values. Included on Student DVD-ROM.

Calculator Guides Financial Calculator Guide

This guide covers using the HP 10BII and TI BAII PLUS financial calculators for Chapters 7, 8, and 10 through 15 in Practical Business Math Procedures. Many of the examples and practical quiz problems are illustrated. Selected end-ofchapter problems are also illustrated. This guide is divided into two sections. One section is devoted to the HP 10BII calculator and the other section covers the TI BAII PLUS calculator, also providing brief introductions to using each model.

Electronic Calculator Guide with Computer Applications

This manual coordinates Practical Business Math Procedures applications with instruction in the 10-key calculator and computer keypad. It also reviews the touch method, includes speed drills, and helps students apply new skills to business math word problems. An introduction to Excel spreadsheets and how to enter data in spreadsheets is included.

TI-83/TI-84 Graphing Calculator Guide

This new updated and enhanced supplement is now found both online and in print, available for packaging with the text. For every chapter covered there are key strokes with notes on how to use the graphing calculator, Practice Sets and Problems, as well as coverage on how to solve the Summary Practice Tests.

Student Solutions Manual and Study Guide

This supplement provides completely worked-out solutions to selected end-ofchapter drill and word problems, plus additional word problems and practice quizzes for student reinforcement. The manual includes the Study Guide which provides self-paced worksheets that review chapter material. The worksheets cover vocabulary, theory and math applications, as well as extra word problem quizzes and a section on how to use the calculator.

ALEKS for Business Math

ALEKS (Assessment and Learning in Knowledge Spaces) is an artificial intelligence based system, which, acting much like a human tutor, can provide individualized assessment, practice, and learning. By assessing your knowledge, ALEKS focuses clearly on what you are ready to learn next and helps you master the course content more quickly and clearly. You can visit ALEKS at www.business.aleks.com.

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McGraw-Hill’s Homework Manager™ and Homework Manager Plus™

Homework Manager is an online homework management system allowing you to assign select end-of-chapter problems and exercises to your students. Homework Manager’s assignments are automatically graded for you and instant feedback is provided directly to your students. Some of Homework Manager’s problems are programmed with algorithms that create new versions of the problem by generating new data for select variables and keeping the structure of the problem intact. In effect, you have an unlimited number of problems. Homework Manager is Web-based, so there is no “setup” as such, no custom software installation is needed, and the application is hosted on our servers, not yours; all you need to do is establish your user account by entering your name and course number. The textbook problems are sorted by chapter, making it easy to browse through and pick the ones you want to assign. The online grade book is also very easy to use.

Comprehensive Testing Package

The Manual of Tests contains four optional, pre-formatted exams per chapter. The computerized testing system featuring E-Z Test Software is networkable for LAN test administration, online, and is included on the Instructor’s CDROM. Tests and Quizzes can also be printed for your standard delivery or posted to a Web site for student access.

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Alternate Choice Practical Business Math Procedures, Brief Ninth Edition

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The Brief Edition of Practical Business Math Procedures is modified, not just shortened. This is the ideal text for a balanced, shorter business math course. The teaching aids have also been revised to ensure your course flows smoothly and all of your teaching objectives are met. The Brief Edition includes Chapters 1–12 from the Ninth Edition, with modifications to Chapter 8. Note: DVD comes with the Brief Edition.

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Whole Numbers; How to Dissect and Solve Word Problems

LEARNING UNIT OBJECTIVES LU 1–1: Reading, Writing, and Rounding Whole Numbers • Use place values to read and write numeric and verbal whole numbers (p. 3). • Round whole numbers to the indicated position (pp. 4–5). • Use blueprint aid for dissecting and solving a word problem (p. 6).

LU 1–2: Adding and Subtracting Whole Numbers • Add whole numbers; check and estimate addition computations (p. 8). • Subtract whole numbers; check and estimate subtraction computations (pp. 9–10).

LU 1–3: Multiplying and Dividing Whole Numbers • Multiply whole numbers; check and estimate multiplication computations (pp. 12–13). • Divide whole numbers; check and estimate division computations (pp. 14–15).

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Page 2

Chapter 1 Whole Numbers; How to Dissect and Solve Word Problems

People of all ages mak e personal b usiness decisions based on the answers to number questions. Numbers also determine most of the b usiness decisions of companies. F or example, click on your computer , go to the website of a compan y such as Home Depot and note the importance of numbers in the compan y’s business decision-making process. The follo wing Wall Street Journal clipping “Home Depot Plans Gas-Mart F ormat in Four-Store Test” announces plans to test con venience stores with gasoline stations located in parking lots of four of its Nashville, Tennessee, stores:

Wall Street Journal © 2005

Companies often follo w a general problem-solving procedure to arri ve at a change in company polic y. Using Home Depot as an e xample, the follo wing steps illustrate this procedure: Step 1. State the problem(s). Step 2. Decide on the best methods

to solve the problem(s). Step 3. Does the solution mak e sense? Step 4. Evaluate the results.

Growth strate gy is to continue dri ve for top-line growth. Add convenience stores to adjacent Home Depot stores (some with car w ashes). Good use of unproducti ve space, and customers can save time shopping. Home Depot will e valuate the four -store test cases.

Your study of numbers be gins with a re view of basic computation skills that focuses on speed and accurac y. You may think, “But I can use my calculator .” Even if your instructor allows you to use a calculator , you still must kno w the basic computation skills. You need these skills to kno w what to calculate, how to interpret your calculations, how to mak e estimates to recognize errors you made in using your calculator , and how to mak e calculations when you do not ha ve a calculator. (The Student Solutions Manual and Study Guide and the text website e xplain how to use calculators.) The United States’ numbering system is the decimal system or base 10 system. Your calculator gives the 10 single-digit numbers of the decimal system—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The center of the decimal system is the decimal point. When you ha ve a number with a decimal point, the numbers to the left of the decimal point are whole numbers and the numbers to the right of the decimal point are decimal numbers (discussed in Chapter 3). When you have a number without a decimal, the number is a whole number and the decimal is assumed to be after the number . This chapter discusses reading, writing, and rounding whole numbers; adding and subtracting whole numbers; and multiplying and di viding whole numbers.

Learning Unit 1–1: Reading, Writing, and Rounding Whole Numbers Girl Scout cookies are bak ed throughout the year . More than 200 million box es of cookies are produced annually. This means that approximately 2 billion, 400 million cookies are produced. Numerically, we can write this as 2,400,000,000. Now let’s begin our study of whole numbers.

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Learning Unit 1–1

Reading and Writing Numeric and Verbal Whole Numbers The decimal system is a place-value system based on the powers o f 10. Any whole number can be written with the 10 digits of the decimal system because the position, or placement, of the digits in a number gives the value of the digits. To determine the value of each digit in a number , we use a place-value chart (Figure 1.1) that divides numbers into named groups of three digits, with each group separated by a comma. To separate a number into groups, you begin with the last digit in the number and insert commas every three digits, moving from right t o left. This divides the number into the named groups (units, thousands, millions, billions, trillions) shown in the place-value chart. Within each group, you have a ones, tens, and hundreds place. Keep in mind that the leftmost group may have fewer than three digits. In Figure 1.1, the numeric number 1,605,743,891,412 illustrates place values. When you study the place-value chart, you can see that the value of each place in the chart is 10 times the value o f the place to the right. We can illustrate this by analyzing the last four digits in the number 1,605,743,891,412 :

Mona Sullivan, Courtesy Girl Scouts USA

1,412 (1 1,000) (4 100) (1 10) (2 1) So we can also say , for example, that in the number 745, the “7” means seven hundred (700); in the number 75, the “7” means 7 tens (70). To read and write a numeric number in verbal form, you begin at the left and read each group of three digits as if it were alone, adding the group name at the end (except the last units group and groups of all zeros). Using the place-value chart in Figure 1.1, the number 1,605,743,891,412 is read as one trillion, six hundred five billion, seven hundred forty-three million, eight hundred ninety-one thousand, four hundred twelve. You do not read zeros. They fill vacant spaces as placeholders so that you can correctly state the number values. Also, the numbers twenty-one to ninety-nine must have a hyphen. And most important, when you read or write whole numbers in verbal form, do not use the word and. In the decimal system, and indicates the decimal, which we discuss in Chapter 3. By reversing this process of changing a numeric number to a verbal number , you can use the place-value chart to change a verbal number to a numeric number . Remember that y ou must keep track of the place value of each digit. The place values of the digits in a number determine its total value. Before we look at how to round whole numbers, we should look at how to convert a n umber indicating parts of a whole number to a whole number . We will use the Girl Scout cooki es as an example.

Whole Number Groups

1.1

Millions

Units

Hundred billions

Ten billions

Billions

Comma

Hundred millions

Ten millions

Millions

Comma

Hundred thousands

Ten thousands

Thousands

Comma

Hundreds

Tens

Ones (units)

Decimal Point

Thousands

Comma

Billions

Trillions

Trillions

Ten trillions

Whole number place-value chart

Hundred trillions

FIGURE

1

,

6

0

5

,

7

4

3

,

8

9

1

,

4

1

2

.

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The 2,400,000,000 Girl Scout cookies could be written as 2.4 billion cookies. This amount is two billion plus four hundred million of an additional billion. The following steps explain how to convert these decimal numbers into a regular whole number: CONVERTING PARTS OF A MILLION, BILLION, TRILLION, ETC., TO A REGULAR WHOLE NUMBER Step 1.

Drop the decimal point and insert a comma.

Step 2.

Add zeros so the leftmost digit ends in the word name of the amount you want to convert. Be sure to add commas as needed.

Convert 2.4 billion to a regular whole number .

EXAMPLE Step 1.

Step 2.

2.4 billion 2,4

Change the decimal point to a comma.

2,400,000,000

Add zeros and commas so the whole number indicates billion.

Rounding Whole Numbers Many of the whole numbers you read and hear are rounded numbers. Government statistics are usually rounded numbers. The financial reports of companies also use rounded numbers. All rounded numbers are approximate numbers. The more rounding you do, the more you approximate the number . Rounded whole numbers are used for many reasons. With rounded whole numbers you can quickly estimate arithmetic results, check actual computations, report numbers that change quickly such as population numbers, and make numbers easier to read and remember . Numbers can be rounded to any identified digit place value, including the first digit of a number (rounding all the way). To round whole numbers, use the following three steps: ROUNDING WHOLE NUMBERS Step 1.

Identify the place value of the digit you want to round.

Step 2.

If the digit to the right of the identified digit in Step 1 is 5 or more, increase the identified digit by 1 (round up). If the digit to the right is less than 5, do not change the identified digit.

Step 3.

Change all digits to the right of the rounded identified digit to zeros.

EXAMPLE 1 Round 9,362 to the nearest hundred. Step 1.

9,362

Step 2.

The digit 3 is in the hundreds place value. The digit to the right of 3 is 5 or more (6). Thus, 3, the identified digit in Step 1, is now rounded to 4. You change the identified digit only if the digit to the right is 5 or more.

9,462 Step 3.

9,400

Change digits 6 and 2 to zeros, since these digits are to the right of 4, the rounded number.

By rounding 9,362 to the nearest hundred, you can see that 9,362 is closer to 9,400 than to 9,300. We can use the following Wall Street Journal clipping “Food for Thought” to illustrate rounding to the nearest hundred. For example, rounded to the nearest hundred, the 560 calories of Big Mac rounds to 600 calories, whereas the 290 calories of McDonald’ s Egg McMuf fin rounds to 300 calories.

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Learning Unit 1–1

The McGraw-Hill Companies, John Flournoy photographer

5

Wall Street Journal © 2005

Next, we show you how to round to the nearest thousand. EXAMPLE 2 Round 67,951 to the nearest thousand. Step 1.

67,951

Step 2.

The digit 7 is in the thousands place value. Digit to the right of 7 is 5 or more (9). Thus, 7, the identified digit in Step 1, is now rounded to 8.

68,951 Step 3.

68,000

Change digits 9, 5, and 1 to zeros, since these digits are to the right of 8, the rounded number.

By rounding 67,951 to the nearest thousand, you can see that 67,951 is closer to 68,000 than to 67,000. Now let’s look at rounding all the way. To round a number all the way , you round to the first digit of the number (the leftmost digit) and have only one nonzero digit remaining in the number. EXAMPLE 3 Round 7,843 all the way. Step 1.

7,843

Step 2.

Identified leftmost digit is 7. Digit to the right of 7 is greater than 5, so 7 becomes 8.

8,843 Step 3.

8,000

Change all other digits to zeros.

Rounding 7,843 all the way gives 8,000. Remember that rounding a digit to a specific place value depends on the degree of accuracy you want in your estimate. For example, 24,800 rounds all the way to 20,000 because the digit to the right of 2 is less than 5. This 20,000 is 4,800 less than the original 24,800. You would be more accurate if you rounded 24,800 to the place value of the identified digit 4, which is 25,000. Before concluding this unit, let’ s look at how to dissect and solve a word problem.

How to Dissect and Solve a Word Problem As a student, your author found solving word problems difficult. Not knowing where to begin after reading the word problem caused the dif ficulty. Today, students still struggle with word problems as they try to decide where to begin. Solving word problems involves organization and persistence. Recall how persistent you were when you learned to ride a two-wheel bike. Do you remember the feeling of suc cess you experienced when you rode the bike without help? Apply this persistence to word problems.

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Chapter 1 Whole Numbers; How to Dissect and Solve Word Problems

Do not be discouraged. Each person learns at a dif ferent speed. Your goal must be to FINISH THE RACE and experience the success of solving word problems with ease. To be organized in solving word problems, you need a plan of action that tells you where to begin—a blueprint aid. Like a builder , you will refer to this blueprint aid constantly until you know the procedure. The blueprint aid for dissecting and solving a word problem fol lows. Note that the blueprint aid serves an important function— it decreases your math anxiety. Blueprint Aid for Dissecting and Solving a Word Problem The facts

Solving for?

Steps to take

Key points

Now let’s study this blueprint aid. The first two columns require that you read the word problem slowly. Think of the third column as the basic information you must know or calc ulate before solving the word problem. Often this column contains formulas that provide the foundation for the step-by-step problem solution. The last column reinforces the key points you should remember . It’s time now to try your skill at using the blueprint aid for dissecting and solving a word problem. The Word Problem On the 100th anniversary of Tootsie Roll Industries, the company reported sharply increased sales and profits. Sales reached one hundred ninety-four million dollars and a record profit of twenty-two million, five hundred fifty-six thousand dollars. The company president requested that you round the sales and profit figures all the way. Study the following blueprint aid and note how we filled in the columns with the information in the word problem. You will find the or ganization of the blueprint aid most helpful. Be persistent! You can dissect and solve word problems! When you are finished with the word problem, make sure the answer seems reasonable.

Teri Stratford

The facts

Solving for?

Steps to take

Key points

Sales: One hundred ninety-four million dollars.

Sales and profit rounded all the way.

Express each verbal form in numeric form. Identify leftmost digit in each number.

Rounding all the way means only the leftmost digit will remain. All other digits become zeros.

Profit: Twenty-two million, five hundred fifty-six thousand dollars. Steps to solving problem

1. Convert verbal to numeric. One hundred ninety-four million dollars Twenty-two million, five hundred fifty-six thousand dollars

$194,000,000 $ 22,556,000

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7

2. Identify leftmost digit of each number. $194,000,000 $22,556,000 3. Round. $200,000,000

$20,000,000

Note that in the final answer , $200,000,000 and $20,000,000 have only one nonzero digit. Remember that you cannot round numbers expressed in verbal form. You must convert these numbers to numeric form. Now you should see the importance of the information in the third column of the blueprint aid. When you complete your blueprint aids for word problems, do not be concerned i f the order of the information in your boxes does not follow the order given in the text boxes. Often you can dissect a word problem in more than one way . Your first Practice Quiz follows. Be sure to study the paragraph that introduces the Practice Quiz.

LU 1–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

At the end of each learning unit, you can check your progress with a Practice Quiz. If you had difficulty understanding the unit, the Practice Quiz will help identify your area of weakness. Work the problems on scrap paper . Check your answers with the worked-out solutions that follow the quiz. Ask your instructor about specific assignments and the videos available on your DVD for each chapter Practice Quiz. Write in verbal form: a. 7,948 b. 48,775 c. 814,410,335,414 Round the following numbers as indicated:

1. 2.

Nearest ten

Nearest hundred

Nearest thousand

Rounded all the way

92 b. 745 c. 8,341 d. 4,752 Kellogg’s reported its sales as five million, one hundred eighty-one thousand dollars.The company earned a profit of five hundred two thousand dollars. What would the sales and profit be if each number were rounded all the way? ( Hint: You might want to draw the blueprint aid since we show it in the solution.) a.

3.

✓ 1.

2. 3.

Solutions a. Seven thousand, nine hundred forty-eight b. Forty-eight thousand, seven hundred seventy-five c. Eight hundred fourteen billion, four hundred ten million, three hundred thirty-five thousand, four hundred fourteen a. 90 b. 700 c. 8,000 d. 5,000 Kellogg’s sales and profit:

The facts

Solving for?

Steps to take

Key points

Sales: Five million, one hundred eightyone thousand dollars.

Sales and profit rounded all the way.

Express each verbal form in numeric form. Identify leftmost digit in each number.

Rounding all the way means only the leftmost digit will remain. All other digits become zeros.

Profit: Five hundred two thousand dollars.

Steps to solving problem 1. Convert verbal to numeric. Five million, one hundred eighty-one thousand Five hundred two thousand 2. Identify leftmost digit of each number. $5,181,000 $502,000 3. Round. $5,000,000

$500,000

$5,181,000 $ 502,000

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Chapter 1 Whole Numbers; How to Dissect and Solve Word Problems

LU 1–1a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 19)

1. 2.

3.

Write in verbal form: a. 8,682 b. 56,295 c. 732,310,444,888 Round the following numbers as indicated: Nearest Nearest Nearest Rounded all ten hundred thousand the way a. 43 b. 654 c. 7,328 d. 5,980 Kellogg’s reported its sales as three million, two hundred ninety-one thousand dollars. The company earned a profit of four hundred five thousand dollars.What would the sales and profit be if each number were rounded all the way?

Learning Unit 1–2: Adding and Subtracting Whole Numbers Did you know that the cost of long-term care in nursing homes varies in different locations? The following Wall Street Journal clipping “Costly Long-Term Care” gives the daily top 1 0 rates and lowest 10 rates of long-care costs reported in various cities. For example, note the difference in daily long-term care costs between Alaska and Shreveport, Louisiana: $561 Alaska: Sherveport: 99 $462

Wall Street Journal © 2006

If you may have long-term nursing care in your future or in the future of someone in your family, be sure to research the cost (and conditions) of long-term care in various locations. This unit teaches you how to manually add and subtract whole numbers. When you least expect it, you will catch yourself automatically using this skill.

Addition of Whole Numbers To add whole numbers, you unite two or more numbers called addends to make one numb er called a sum, total, or amount. The numbers are arranged in a column according to their place values—units above units, tens above tens, and so on. Then, you add the columns of num bers from top to bottom. To check the result, you re-add the columns from bottom to top. This procedure is illustrated in the steps that follow .

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Learning Unit 1–2

9

ADDING WHOLE NUMBERS Step 1.

Align the numbers to be added in columns according to their place values, beginning with the units place at the right and moving to the left (Figure 1.1).

Step 2.

Add the units column. Write the sum below the column. If the sum is more than 9, write the units digit and carry the tens digit.

Step 3.

Moving to the left, repeat Step 2 until all place values are added.

EXAMPLE

2 11

1,362 Adding Checking 5,913 top 8,924 bottom 6,594

bottom to to top

22,793

Alternate check Add each column as a separate total and then combine. The end result is the same. 1,362 5,913 8,924 6,594 13 18 26 20 22,793

How to Quickly Estimate Addition by Rounding All the Way In Learning Unit 1–1, you learned that rounding whole numbers all the way gives quick arithmetic estimates. Using the following Wall Street Journal clipping “Hottest Models,” note how you can round each number all the way and the total will not be rounded all the way . Remember that rounding all the way does not replace actual computations, but it is help ful in making quick commonsense decisions. Hottest Models

Model

Days on Lot

Average Price

5

$25,365

Scion tC

9

$18,278

Scion xB

10

$15,834

BMW 7 Series

13

$80,507

Scion xA

14

$14,532

Lexus RX 400h

14

$50,131

Honda Odyssey

16

$31,001

Toyota Corolla

16

$16,290

Mazda MX-5

17

$25,380

Lexus RX 330

17

$39,467

Saturn VUE

17

$22,553

Toyota Prius

Rounded all the way $ 30,000 Rounding all the way means each number has 20,000 only one nonzero digit. 20,000 80,000 10,000 50,000 30,000 20,000 30,000 40,000 20,000 $350,000

Note: The final answer could have more than one nonzero digit since the total is not rounded all the way.

Subtraction of Whole Numbers Subtraction is the opposite of addition. Addition unites numbers; subtraction takes one number away from another number. In subtraction, the top (lar gest) number is the minuend. The number you subtract from the minuend is the subtrahend, which gives you the difference between the minuend and the subtrahend. The steps for subtracting whole numbers follow .

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SUBTRACTING WHOLE NUMBERS Step 1.

Align the minuend and subtrahend according to their place values.

Step 2.

Begin the subtraction with the units digits. Write the difference below the column. If the units digit in the minuend is smaller than the units digit in the subtrahend, borrow 1 from the tens digit in the minuend. One tens digit is 10 units.

Step 3.

Moving to the left, repeat Step 2 until all place values in the subtrahend are subtracted.

EXAMPLE The following Wall Street Journal clipping “Big Bills Ahead” illustrates the subtraction of whole numbers:

What is the dif ference in cost between the Stamford, Connecticut, and the Miami, Florida, assisted-living facilities? As shown below , you can use subtraction to arrive at the $2,987 difference. 12

3 2 12

$4,3 27 1,340 $2,987 Check

Minuend (larger number) Subtrahend Difference $2,987 1,340 $4,327

Wall Street Journal © 2005

In subtraction, borrowing from the column at the left is often necessary . Remember that 1 ten 10 units, 1 hundred 10 tens, and 1 thousand 10 hundreds. Step 1. In the above example, the 0 in the subtrahend of the rightmost column (ones or units

column) can be subtracted from the 7 in the minuend to give a dif ference of 7. This means we do not have to borrow from the tens column at the left. However , in the tens column, we cannot subtract 4 in the subtrahend from 2 in the minuend, so we move left and borrow 1 from the hundreds column. Since 1 hundred 10 tens, we have 10 2, or 12 tens in the minuend. Now we can subtract 4 tens in the subtrahend from 12 tens in the minuend to give us 8 tens in the dif ference. Step 2. Since we borrowed 1 hundred from our original 3 hundred, we now have 2 hundred in the minuend. The 3 hundred in the subtrahend will not subtract from the 2 hun dred in the minuend, so again we must move left. We take 1 thousand from the 4 thousand in the thousands column. Since 1 thousand is 10 hundreds, we have 10 2, or 12 hundreds in the hundreds column. The 3 hundred in the subtrahend subtracted from the 12 hundred in the minuend gives us 9 hundred in the dif ference. The 1 thousand in the subtrahend subtracted from the 3 thousand in the minuend gives 2 thousand. Our total dif ference between the subtrahend $1,340 and the minuend $4,327 i s $2,987 as proved in the check. Checking subtraction requires adding the difference ($2,987) to the subtrahend ($1,340) to arrive at the minuend ($4,327). The Stamford, Connecticut, assisted-living facility costs $2,987 more than the Miami, Florida, assisted-living facility .

How to Dissect and Solve a Word Problem Accurate subtraction is important in many business operations. In Chapter 4 we discuss the importance of keeping accurate subtraction in your checkbook balance. Now let’ s check your progress by dissecting and solving a word problem.

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11

The Word Problem Hershey’s produced 25 million Kisses in one day . The same day , the

company shipped 4 million to Japan, 3 million to France, and 6 million throughout the United States. At the end of that day , what is the company’ s total inventory of Kisses? What is the inventory balance if you round the number all the way? The facts

Solving for?

Steps to take

Key points

Produced: 25 million.

Total Kisses left in inventory.

Total Kisses produced

Minuend Subtrahend Difference.

Inventory balance rounded all the way.

Total Kisses left in inventory.

Shipped: Japan, 4 million; France, 3 million; United States, 6 million.

Total Kisses shipped

Rounding all the way means rounding to last digit on the left.

Steps to solving problem 1. Calculate the total Kisses shipped.

Teri Stratford

4,000,000 3,000,000 6,000,000 13,000,000

2. Calculate the total Kisses left in inventory.

25,000,000 13,000,000 12,000,000

3. Rounding all the way.

Identified digit is 1. Digit to right of 1 is 2, which is less than 5. Answer: 10,000,000 .

The Practice Quiz that follows will tell you how you are progressing in your study of Chapter 1.

LU 1–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

1.

Add by totaling each separate column: 8,974 6,439 16,941

2.

Estimate by rounding all the way (do not round the total of estimate) and then do the actual computation: 4,241 8,794 3,872

3.

Subtract and check your answer: 9,876 4,967

4.

Jackson Manufacturing Company projected its year 2003 furniture sales at $900,000. During 2003, Jackson earned $510,000 in sales from major clients and $369,100 in sales from the remainder of its clients. What is the amount by which Jackson over - or underestimated its sales? Use the blueprint aid, since the answer will show the completed blueprint aid.

✓ 1.

Solutions 14 14 22 20 22,354

2. Estimate 4,000 9,000 4,000 17,000

Actual 4,241 8,794 3,872 16,907

3.

8 18 6 16

9,876 4,967 4,909

Check 4,909 4,967 9,876

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Chapter 1 Whole Numbers; How to Dissect and Solve Word Problems

Jackson Manufacturing Company over- or underestimated sales:

4.

The facts

Solving for?

Steps to take

Key points

Projected 2003 sales: $900,000.

How much were sales over- or underestimated?

Total projected sales

Projected sales (minuend)

Major clients: $510,000.

Total actual sales Over- or underestimated sales.

Actual sales (subtrahend) Difference.

Other clients: $369,100.

Steps to solving problem 1. Calculate total actual sales.

$510,000 369,100 $879,100

2. Calculate overestimated or underestimated sales.

$900,000 879,100 $ 20,900 (overestimated)

LU 1–2a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 19)

1.

Add by totaling each separate column: 9,853 7,394 8,843

2.

Estimate by rounding all the way (do not round the total of estimate) and then do the actual computation: 3,482 6,981 5,490

3.

Subtract and check your answer: 9,787 5,968

4.

Jackson Manufacturing Company projected its year 2008 furniture sales at $878,000. During 2008, Jackson earned $492,900 in sales from major clients and $342,000 in sales from the remainder of its clients. What is the amount by which Jackson over - or underestimated its sales?

Learning Unit 1–3: Multiplying and Dividing Whole Numbers At the beginning of Learning Unit 1–2, you learned how you would save $462 on the purchase of daily long-term care in a Shreveport, Louisiana, nursing home instead of in an Alaska nursing home. If you stay in the Alaska and Shreveport nursing homes for 5 days, the Alaska nursing home would cost you $2,805, but the Shreveport nursing home would cost you $495, and y ou would save $2,310: Alaska: $561 5 $2,805 Shreveport: 99 5 495 $2,310 If you divide $2,310 by 5, you will get the $462 dif ference in price between Alaska and Shreveport as shown at the beginning of Learning Unit 1–2. This unit will sharpen your skills in two important arithmetic operations—multiplication and division. These two operations frequently result in knowledgeable business decisions.

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Learning Unit 1–3

13

Multiplication of Whole Numbers—Shortcut to Addition From calculating your purchase of 5 days of long-term care in Shreveport, you know that multiplication is a shortcut to addition: $99 5 $495

or

$99 $99 $99 $99 $99 $495

Before learning the steps used to multiply whole numbers with two or more digits, you must learn some multiplication terminology . Note in the following example that the top number (number we want to multiply) is the multiplicand. The bottom number (number doing the multiplying) is the multiplier. The final number (answer) is the product. The numbers between the multiplier and the product are partial products. Also note how we positioned the partial product 2090. This number is the result of multiplying 418 by 50 (the 5 is in the tens position). On each line in the partial products, we placed the first digit directly below the digit we used in the multiplication process. EXAMPLE

Partial products

Top number (multiplicand) Bottom number (multiplier) 2 418 836 50 418 20,900 21,736 Product answer

418 52 836 20 90 21,736

We can now give the following steps for multiplying whole numbers with two or more digits: MULTIPLYING WHOLE NUMBERS WITH TWO OR MORE DIGITS Step 1.

Align the multiplicand (top number) and multiplier (bottom number) at the right. Usually, you should make the smaller number the multiplier.

Step 2.

Begin by multiplying the right digit of the multiplier with the right digit of the multiplicand. Keep multiplying as you move left through the multiplicand. Your first partial product aligns at the right with the multiplicand and multiplier.

Step 3.

Move left through the multiplier and continue multiplying the multiplicand. Your partial product right digit or first digit is placed directly below the digit in the multiplier that you used to multiply.

Step 4.

Continue Steps 2 and 3 until you have completed your multiplication process. Then add the partial products to get the final product.

Checking and Estimating Multiplication We can check the multiplication process by reversing the multiplicand and multiplier and then multiplying. Let’s first estimate 52 418 by rounding all the way . EXAMPLE

50 400 20,000

52 418 416 52 20 8 21,736

By estimating before actually working the problem, we know our answer should be about 20,000. When we multiply 52 by 418, we get the same answer as when we multiply 418 52—and the answer is about 20,000. Remember , if we had not rounded all the way , our estimate would have been closer . If we had used a calculator , the rounded estimate would have helped us check the calculator ’s answer . Our commonsense estimate tells us our answer is near 20,000—not 200,000. Before you study the division of whole numbers, you should know (1) the multiplication shortcut with numbers ending in zeros and (2) how to multiply a whole number by a power of 10.

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MULTIPLICATION SHORTCUT WITH NUMBERS ENDING IN ZEROS Step 1.

When zeros are at the end of the multiplicand or the multiplier, or both, disregard the zeros and multiply.

Step 2.

Count the number of zeros in the multiplicand and multiplier.

Step 3.

Attach the number of zeros counted in Step 2 to your answer.

EXAMPLE

65,000 420

65 42 1 30 26 0 27,300,000

3 zeros 1 zero 4 zeros

No need to multiply rows of zeros 65,000 420 00 000 1 300 00 26 000 0 27,300,000

MULTIPLYING A WHOLE NUMBER BY A POWER OF 10 Step 1.

Count the number of zeros in the power of 10 (a whole number that begins with 1 and ends in one or more zeros such as 10, 100, 1,000, and so on).

Step 2.

Attach that number of zeros to the right side of the other whole number to obtain the answer. Insert comma(s) as needed every three digits, moving from right to left.

EXAMPLE 99 10

99 100

99 0 990 9,9 00 9,900

99 1,000 99, 000 99,000

Add 1 zero Add 2 zeros Add 3 zeros

When a zero is in the center of the multiplier , you can do the following: EXAMPLE

658 403 1 974 263 2䊐 265,174

3 658 1,974 400 658 263,200 265,174

Division of Whole Numbers Division is the reverse of multiplication and a time-saving shortcut related to subtraction. F or example, in the introduction to this learning unit, you determined that you would save $2,310 by staying for 5 days in a nursing home in Shreveport, Louisiana, versus Alaska. If you su btract $462—the difference between the cost of Alaska and Shreveport—5 times from the dif ference of $2,310, you would get to zero. You can also multiply $462 times 5 to get $2,310. Since division is the reverse of multiplication, you can say that $2,310 5 = $462. Division can be indicated by the common symbols and 冄 , or by the bar — in a fraction and the forward slant / between two numbers, which means the first number is divided b y the second number . Division asks how many times one number (divisor) is contained in another number (dividend). The answer, or result, is the quotient. When the divisor (numb er used to divide) doesn’ t divide evenly into the dividend (number we are dividing), the result i s a partial quotient, with the leftover amount the remainder (expressed as fractions in later chapters). The following example illustrates even division (this is also an example of long division because the divisor has more than one digit).

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Learning Unit 1–3

Quotient Dividend

18 15冄270 15 120 120

EXAMPLE

Divisor

15

This example divides 15 into 27 once with 12 rem aining. The 0 in the dividend is brought down to 12. Dividing 120 by 15 equals 8 with no remainder; that is, even division. The following example illustrates uneven division with a remainder (this is also an example of short division because the divisor has only one digit). 24 R1 7冄 169 14 29 28 1

EXAMPLE

Remainder

Check (7 24) 1 169 Divisor Quotient Remainder Dividend

Note how doing the check gives you assurance that your calculation is correct. When the divisor has one digit (short division) as in this example, you can often calculate the division mentally as illustrated in the following examples: EXAMPLES

108 8冄864

16 R6 7冄118

Next, let’s look at the value of estimating division. Estimating Division Before actually working a division problem, estimate the quotient by rounding. This estimate helps check the answer . The example that follows is rounded all the way . After you make an estimate, work the problem and check your answer by multiplication. EXAMPLE

36 R111 138冄 5,079 4 14 939 828 111

Estimate 50 100冄5,000

Check 138 36 828 4 14 4,968 111 5,079

Add remainder

Now let’s turn our attention to division shortcuts with zeros. Division Shortcuts with Zeros The steps that follow show a shortcut that you can use when you divide numbers with zeros. DIVISION SHORTCUT WITH NUMBERS ENDING IN ZEROS Step 1.

When the dividend and divisor have ending zeros, count the number of ending zeros in the divisor.

Step 2.

Drop the same number of zeros in the dividend as in the divisor, counting from right to left.

Note the following examples of division shortcut with numbers ending in zeros. Since two of the symbols used for division are and 冄 , our first examples show the zero shortc ut method with the symbol.

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Chapter 1 Whole Numbers; How to Dissect and Solve Word Problems

One ending zero

EXAMPLES

Dividend

Divisor

Drop 1 zero in dividend

95,000 10

95,000 9,500

95,000 100 95,000 1,000

95,000 95,000

950 Drop 2 zeros 95 Drop 3 zeros

In a long division problem with the 冄 symbol, you again count the number of ending zeros in the divisor. Then drop the same number of ending zeros in the dividend and divide as usual. EXAMPLE

6,500冄 88,000

Drop 2 zeros

65冄 880

13 R35 65冄 880 65 230 195 35

You are now ready to practice what you learned by dissecting and solving a word problem.

How to Dissect and Solve a Word Problem The blueprint aid that follows will be your guide to dissecting and solving the following word problem. The Word Problem Dunkin’ Donuts sells to four dif ferent companies a total of $3,500

worth of doughnuts per week. What is the total annual sales to these companies? What is the yearly sales per company? (Assume each company buys the same amount.) Check your answer to show how multiplication and division are related. The facts

Solving for?

Steps to take

Key points

Sales per week: $3,500.

Total annual sales to all four companies. Yearly sales per company.

Sales per week Weeks in year (52) Total annual sales.

Division is the reverse of multiplication.

Companies: 4.

Total annual sales Total companies Yearly sales per company.

Steps to solving problem 1. Calculate total annual sales.

$3,500 52 weeks $182,000

2. Calculate yearly sales per company,

$182,000 4 $45,500 Check $45,500 4 $182,000

It’s time again to check your progress with a Practice Quiz.

LU 1–3

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

1.

Estimate the actual problem by rounding all the way check: Actual Estimate Check 3,894 18

, work the actual problem, and

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Learning Unit 1–3

17

2.

Multiply by shortcut method: 77,000 1,800

4.

Divide by rounding all the way , complete the actual calculation, and check, showing remainder as a whole number. 26冄5,325 Divide by shortcut method: 4,000冄 96,000 Assume General Motors produces 960 Chevrolets each workday (Monday through Friday). If the cost to produce each car is $6,500, what is General Motors’ total cost for the year? Check your answer.

5. 6.

✓

3.

Multiply by shortcut method: 95 10,000

Solutions

1.

Estimate 4,000 20 80,000

2. 4.

77 18 1,386 5 zeros 138,600,000 Rounding Actual 204 R21 166 R20 26冄5,325 30冄 5,000 30 52 2 00 125 1 80 104 200 21 180 20

5. 6.

Actual 3,894 18 31 152 38 94 70,092

Check 8 3,894 31,152 10 3,894 38,940 70,092

3. 95 4 zeros 950,000 Check 26 204 5,304 21 5,325

24 Drop 3 zeros 4冄 96 General Motors’ total cost per year: The facts

Solving for?

Steps to take

Key points

Cars produced each workday: 960.

Total cost per year.

Cars produced per week 52 Total cars produced per year.

Whenever possible, use multiplication and division shortcuts with zeros. Multiplication can be checked by division.

Workweek: 5 days. Cost per car: $6,500.

Total cars produced per year Total cost per car Total cost per year.

Steps to solving problem 1. Calculate total cars produced per week.

5 960 4,800 cars produced per week

2. Calculate total cars produced per year.

4,800 cars 52 weeks 249,600 total cars produced per year

3. Calculate total cost per year.

249,600 cars $6,500 $1,622,400,000 (multiply 2,496 65 and add zeros) Check $1,622,400,000 249,600 $6,500 (drop 2 zeros before dividing)

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Chapter 1 Whole Numbers; How to Dissect and Solve Word Problems

LU 1–3a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 19)

1.

Estimate the actual problem by rounding all the way check: Actual Estimate Check 4,938 19

2.

Multiply by shortcut method: 86,000 1,900

4.

Divide by rounding all the way , complete the actual calculation, and check, showing remainder as a whole number. 26冄6,394 Divide by the shortcut method: 3,000冄99,000 Assume General Motors produces 850 Chevrolets each workday (Monday through Friday). If the cost to produce each car is $7,000, what is General Motors’s total cost for the year? Check your answer.

5. 6.

3.

, work the actual problem, and

Multiply by shortcut method: 86 10,000

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Reading and writing numeric and verbal whole numbers, p. 3

Placement of digits in a number gives the value of the digits (Figure 1.1). Commas separate every three digits, moving from right to left. Begin at left to read and write number in verbal form. Do not read zeros or use and. Hyphenate numbers twenty-one to ninety-nine. Reverse procedure to change verbal number to numeric.

462 6,741

Rounding whole numbers, p. 4

1. Identify place value of the digit to be rounded. 2. If digit to the right is 5 or more, round up; if less than 5, do not change. 3. Change all digits to the right of rounded identified digit to zeros.

643 to nearest ten

Round to first digit of number. One nonzero digit remains. In estimating, you round each number of the problem to one nonzero digit. The final answer is not rounded.

468,451

Rounding all the way, p. 5

Adding whole numbers, p. 8

1. Align numbers at the right. 2. Add units column. If sum more than 9, carry tens digit. 3. Moving left, repeat Step 2 until all place values are added. Add from top to bottom. Check by adding bottom to top or adding each column separately and combining.

Four hundred sixty-two Six thousand, seven hundred forty-one

4 in tens place value.

3 is not 5 or more

Thus, 643 rounds to 640 . 500,000

The 5 is the only nonzero digit remaining. 1

65 47 112

12 10 112

Checking sum of each digit

(continues)

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19

Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Subtracting whole numbers, p. 9

1. Align minuend and subtrahend at the right. 2. Subtract units digits. If necessary, borrow 1 from tens digit in minuend. 3. Moving left, repeat Step 2 until all place values are subtracted. Minuend less subtrahend equals difference.

Check

1. Align multiplicand and multiplier at the right. 2. Begin at the right and keep multiplying as you move to the left. First partial product aligns at the right with multiplicand and multiplier. 3. Move left through multiplier and continue multiplying multiplicand. Partial product right digit or first digit is placed directly below digit in multiplier. 4. Continue Steps 2 and 3 until multiplication is complete. Add partial products to get final product. Shortcuts: (a) When multiplicand or multiplier, or both, end in zeros, disregard zeros and multiply; attach same number of zeros to answer. If zero in center of multiplier, no need to show row of zeros. (b) If multiplying by power of 10, attach same number of zeros to whole number multiplied.

223 32 446 6 69

Multiplying whole numbers, p. 12

Dividing whole numbers, p. 14

1. When divisor is divided into the dividend, the remainder is less than divisor. 2. Drop zeros from dividend right to left by number of zeros found in the divisor. Even division has no remainder; uneven division has a remainder; divisor with one digit is short division; and divisor with more than one digit is long division.

KEY TERMS

addends, p. 8 decimal point, p. 2 decimal system, p. 2 difference, p. 9 dividend, p. 14 divisor, p. 14

CHECK FIGURE FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 1–1a (p. 8) 1. A. Eight thousand, six hundred eighty-two; B. Fifty-six thousand, two hundred ninety-five; C. Seven hundred thirty two billion, three hundred ten million, four hundred forty-four thousand, eight hundred eighty-eight 2. A. 40; B. 700; C. 7,000; D. 6,000 3. 3,000,000; 400,000

5 18

685 492 193

7,136 a. 48,000 40

3 zeros 1 zero 3 1,920,000 4 zeros 104

b. 14

10 140 (attach 1 zero)

48 4

524 206 144 8

107,944 14 1,000 14,000 (attach 3 zeros)

1.

5 R6 14冄76 70 6

2. 5,000 100

50 1 50

5,000 1,000 5 1 5

minuend, p. 9 multiplicand, p. 13 multiplier, p. 13 partial products, p. 13 partial quotient, p. 14 product, p. 13

1. 2. 3. 4.

193 492 685

LU 1–2a (p. 12) 26,090 15,000; 15,953 3,819 43,100 (over)

quotient, p. 14 remainder, p. 14 rounding all the way, p. 5 subtrahend, p. 9 sum, p. 8 whole number, p. 2

1. 2. 3. 4. 5. 6.

LU 1–3a (p. 18) 100,000; 93,822 163,400,000 860,000 255 R19 33 $1,547,000,000

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Chapter 1 Whole Numbers; How to Dissect and Solve Word Problems

Critical Thinking Discussion Questions 1. List the four steps of the decision-making process. Do you 4. Explain how you can check multiplication. If you visit a local think all companies should be required to follow these steps? supermarket, how could you show multiplication as a shortGive an example. cut to addition? 2. Explain the three steps used to round whole numbers. Pick a 5. Explain how division is the reverse of multiplication. Using whole number and explain why it should not be rounded. the supermarket example, explain how division is a timesaving shortcut related to subtraction. 3. How do you check subtraction? If you were to attend a movie, explain how you might use the subtraction check method.

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Add the following: 1–1.

88 16

1–5.

6,251 7,329

1–2.

855 699

1–3.

79 79

1–4.

1–6.

59,481 51,411 70,821

1–9.

80 42

1–10.

287 199

1–12.

9,800 8,900

1–13.

1,622 548

66 9

1–15.

510 61

1–16.

677 503

1–18.

309 850

1–19.

1–7.

66 92

78,159 15,850 19,681

Subtract the following: 1–8.

1–11.

68 19 9,000 5,400

Multiply the following: 1–14.

1–17.

900 300

450 280

Divide the following by short division: 1–20. 6冄 1,200

1–21.

9冄810

1–22.

4冄164

Divide the following by long division. Show work and remainder. 1–23. 6冄 520

1–24. 62冄8,915

Add the following without rearranging: 1–25. 99 210

1–26. 1,055 88

1–27. 666 950

1–28. 1,011 17

21

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1–29. Add the following and check by totaling each column individually without carrying numbers: Check

8,539 6,842 9,495

Estimate the following by rounding all the way and then do actual addition: 1–30.

Actual 7,700 9,286 3,900

Estimate 1–31.

Actual 6,980 3,190 7,819

Estimate

Subtract the following without rearranging: 1–32. 190 66

1–33. 950 870

1–34. Subtract the following and check answer: 591,001 375,956 Multiply the following horizontally: 1–35. 16 9

1–36. 84 8

1–37.

27 8

1–40.

46冄1,950

1–38.

Divide the following and check by multiplication: 1–39. 45冄876

Check

Check

Complete the following: 1–41.

9,200 1,510

1–42.

700

3,000,000 769,459

68,541

1–43. Estimate the following problem by rounding all the way and then do the actual multiplication: Actual 870 81

Estimate

Divide the following by the shortcut method: 1–44. 1,000冄850,000

22

1–45. 100冄 70,000

17 6

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1–46. Estimate actual problem by rounding all the way and do actual division: Actual

Estimate

695冄 8,950

WORD PROBLEMS 1–47. The January 8, 2007 issue of Retailing Today, reported on price cuts implemented by Wal-Mart ahead of the holi day season. A Panasonic 42-inch HD plasma TV was reduced to one-thousand, two hundred ninety-four dollars from onethousand, seven hundred ninety-four dollars. A Polaroid 37-inch LDC HDTV priced at nine hundred ninety-seven d ollars was reduced from one-thousand, two hundred ninety-seven dollars. (a) In numerical form, how much was saved by purchasing a HD plasma TV? (b) How much was saved by purchasing the LCD HDTV? (c) Prior to price reduction, how much more was the HDTV compared to the LCD HDTV?

1–48. On February 3, 2007, The Boston Globe reported on ticket reseller Admit One Ticket Agency’s ticket price for games the Red Sox played against the New York Yankees and Baltimore Orioles. Admit One paid $135,550, in 2005, for 14 loge box seat season tickets with a face value of $90,720.The average price per ticket was $120. The transaction netted the company 1,134 tickets to 81 games. Admit One Ticket resold 1,084 of the tickets for a total of $231,976. (a) How much did Admit One pay over the face value for the tickets? (b) What was the average price for the resold tickets? (c) What was the difference between the average price paid and average reselling price?

1–49. The Buffalo News on January 25, 2007, reported season-ticket prices for the Buf falo Bills 15-yard line on the lower bowl. Tickets will rise from $480 for a 10-game package to $600. Fans sitting in the best seats in the upper deck will pay an increase from $440 to $540. Don Manning plans to purchase 2 season tickets for either lower bowl or upper deck.(a) How much more will 2 tickets cost for lower bowl? (b) How much more will 2 tickets cost for upper deck? (c) What will be his total cost for a 10-game package for lower bowl? (d) What will be his total cost for a 10-game package for upper deck?

1–50. The Billboard reported that ticket prices for the Old Friends concert tour of Paul Simon and Art Garfunkel are $251 (VIP), $126, $86, and $51. For a family of four, estimate the cost of the $86 tickets by rounding all the way and then do the actual multiplication:

23

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1–51. USA Today reports that Walt Disney World Resort and United Vacations got together to create a special deal. The airinclusive package features accommodations for three nights at Disney’s All-Star Resort, hotel taxes, and a four-day unlimited Magic Pass. Prices are $609 per person traveling from Washington, DC, and $764 per person traveling from Los Angeles. (a) What would be the cost for a family of four leaving from Washington, DC? (b) What would be the cost for a family of four leaving from Los Angeles? (c) How much more will it cost the family from Los Angeles?

1–52. NTB Tires bought 910 tires from its manufacturer for $36 per tire. What is the total cost of NTB’s purchase? If the store can sell all the tires at $65 each, what will be the store’ s gross profit, or the dif ference between its sales and costs (Sales Costs Gross profit)?

1–53. What was the total average number of visits for these Internet Web sites? Web site

Average daily unique visitor

1. Orbitz.com

1,527,000

2. Mypoints.com

1,356,000

3. Americangreetings.com

745,000

4. Bizrate.com

503,000

5. Half.com

397,000

1–54. Lee Wong bought 5,000 shares of GE stock. She held the stock for 6 months. Then Lee sold 190 shares on Monday, 450 shares on Tuesday and again on Thursday, and 900 shares on Friday. How many shares does Lee still own? The average share of the stock Lee owns is worth $48 per share. What is the total value of Lee’s stock?

1–55. USA Today reported that the Center for Science in the Public Interest—a consumer group based in Washington, DC— released a study listing calories of various ice cream treats sold by six of the lar gest ice cream companies. The worst treat tested by the group was 1,270 total calories. People need roughly 2,200 to 2,500 calories per day . Using a daily average, how many additional calories should a person consume after eating the ice cream?

24

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1–56. At Rose State College, Alison Wells received the following grades in her online accounting class: 90, 65, 85, 80, 75, and 90. Alison’s instructor, Professor Clark, said he would drop the lowest grade. What is Alison’s average? 1–57. Lee Wills, professor of business, has 18 students in Accounting I, 26 in Accounting II, 22 in Introduction to Computers, 23 in Business Law, and 29 in Introduction to Business. What is the total number of students in Professor Wills’s classes? If 12 students withdraw, how many total students will Professor Wills have?

1–58. Ron Alf, owner of Alf’s Moving Company, bought a new truck. On Ron’s first trip, he drove 1,200 miles and used 80 gallons of gas. How many miles per gallon did Ron get from his new truck? On Ron’ s second trip, he drove 840 miles and used 60 gallons. What is the difference in miles per gallon between Ron’s first trip and his second trip?

1–59. Office Depot reduced its $450 Kodak digital camera by $59. What is the new selling price of the digital camera? If Office Depot sold 1,400 cameras at the new price, what were the store’s digital camera dollar sales?

1–60. Barnes and Noble.com has 289 business math texts in inventory. During one month, the online bookstore ordered and received 1,855 texts; it also sold 1,222 on the Web. What is the bookstore’s inventory at the end of the month? If each text costs $59, what is the end-of-month inventory cost?

1–61. Cabot Company produced 2,115,000 cans of paint in August. Cabot sold 2,011,000 of these cans. If each can cost $18, what were Cabot’s ending inventory of paint cans and its total ending inventory cost?

1–62. Long College has 30 faculty members in the business department, 22 in psychology, 14 in English, and 169 in all other departments. What is the total number of faculty at Long College? If each faculty member advises 30 students, how many students attend Long College?

1–63. Hometown Buffet had 90 customers on Sunday, 70 on Monday, 65 on Tuesday, and a total of 310 on Wednesday to Saturday. How many customers did Hometown Buf fet serve during the week? If each customer spends $9, what were the total sales for the week?

If Hometown Buffet had the same sales each week, what were the sales for the year?

1–64. Longview Agency projected its year 2006 sales at $995,000. During 2006, the agency earned $525,960 sales from its major clients and $286,950 sales from the remainder of its clients. How much did the agency overestimate its sales?

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1–65. Jim Floyd works at US Airways and earned $61,000 last year before tax deductions. From Jim’s total earnings, his company subtracted $1,462 for federal income taxes, $3,782 for Social Security, and $884 for Medicare taxes. What was Jim’s actual, or net, pay for the year?

1–66. Macy’s received the following invoice amounts from suppliers. How much does the company owe? Per item 22 paintings

$210

39 rockers

75

40 desk lamps 120 coffee tables

65 155

1–67. Roger Company produces beach balls and operates three shifts. Roger produces 5,000 balls per shift on shifts 1 and 2. On shift 3, the company can produce 6 times as many balls as on shift 1. Assume a 5-day workweek. How many beach balls does Roger produce per week and per year?

1–68. The New York Times reported on the changes in the prices of Disneyland tickets. Disneyland lowered the age limit for adult tickets from 12 years old to 10 years old. This raised the cost of admission from $31 to $41. If 125 children attending the park each day are in this age bracket, how much additional revenue will Disneyland receive each day?

1–69. Moe Brink has a $900 balance in his checkbook. During the week, Moe wrote the following checks: rent, $350; telephone, $44; food, $160; and entertaining, $60. Moe also made a $1,200 deposit. What is Moe’s new checkbook balance?

1–70. Sports Authority, an athletic sports shop, bought and sold the following merchandise:

Tennis rackets Tennis balls Bowling balls Sneakers

Cost

Selling price

$ 2,900

$ 3,999

70

210

1,050

2,950

8,105

14,888

What was the total cost of the merchandise bought by Sports Authority? If the shop sold all its merchandise, what were the sales and the resulting gross profit (Sales Costs Gross profit)?

26

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1–71. Matty Kaminsky, the bookkeeper for Maggie’s Real Estate, and his manager are concerned about the company’s telephone bills. Last year the company’s average monthly phone bill was $34. Matty’s manager asked him for an average of this year’s phone bills. Matty’s records show the following:

January

$ 34

July

$ 28

February

60

August

23

March

20

September

29

April

25

October

25

May

30

November

22

June

59

December

41

What is the average of this year’s phone bills? Did Matty and his manager have a justifiable concern?

1–72. The Associated Press reported that bankruptcy filings were up for the first three months of the year . Filings reached 366,841 in the January–March period, the highest ever for a first quarter , up from 312,335 a year earlier. How much was the increase in quarterly filings?

1–73. On Monday, True Value Hardware sold 15 paint brushes at $3 each, 6 wrenches at $5 each, 7 bags of grass seed at $3 each, 4 lawn mowers at $119 each, and 28 cans of paint at $8 each. What were True Value’s total dollar sales on Monday?

1–74. While redecorating, Pete Allen went to Sears and bought 125 square yards of commercial carpet. The total cost of the carpet was $3,000. How much did Pete pay per square yard? 1–75. Washington Construction built 12 ranch houses for $1 15,000 each. From the sale of these houses, Washington received $1,980,000. How much gross profit (Sales Costs Gross profit) did Washington make on the houses?

The four partners of Washington Construction split all profits equally. How much will each partner receive?

CHALLENGE PROBLEMS 1–76. The St. Paul Pioneer Press reported that after implementing a new parking service called e-Park, the Minneapolis–St. Paul International Airport reduced the number of its parking garage cashiers. E-Park is expected to allow the airport to cut 35 parking cashiers from its force of 130. Cashiers make about $11 an hour plus benefits. For a 40-hour week, (a) what has been the yearly cost of salaries? and (b) what will be the savings in labor costs for a year?

27

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1–77. Paula Sanchez is trying to determine her 2009 finances. Paula’s actual 2008 finances were as follows:

Income: Gross income

Assets: $69,000

Interest income Total

450 $69,450

$24,500

Insurance premium Taxes

350 14,800

Medical

585

Investment Total

$ 1,950

Savings account

8,950

Automobile

1,800

Personal property

Expenses: Living

Checking account

Total

14,000 $26,700

Liabilities: Note to bank Net worth

4,500 $22,200

($26,700 $4,500)

4,000 $44,235

Net worth Assets Liabilities (own) (owe) Paula believes her gross income will double in 2009 but her interest income will decrease $150. She plans to reduce her 2009 living expenses by one-half. Paula’s insurance company wrote a letter announcing that her insurance premiums would triple in 2009. Her accountant estimates her taxes will decrease $250 and her medical costs will increase $410. Paula also hopes to cut her investments expenses by one-fourth. Paula’ s accountant projects that her savings and checking accounts will each double in value. On January 2, 2009, Paula sold her automobile and began to use public transportation. Paula forecasts that her personal property will decrease by one-seventh. She has sent her bank a $375 check to reduce her bank note. Could you give Paula an updated list of her 2009 finances? If you round all the way each 2008 and 2009 asset and liability, what will be the difference in Paula’s net worth?

28

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SUMMARY PRACTICE TEST

DVD SUMMARY PRACTICE TEST 1. Translate the following verbal forms to numbers and add. (p. 3) a.

Four thousand, eight hundred thirty-nine

b.

Seven million, twelve

c.

Twelve thousand, three hundred ninety-two

2. Express the following number in verbal form. (p. 3) 9,622,364 3. Round the following numbers. (p. 4) Nearest ten a. 68

Nearest hundred b. 888

Nearest thousand c. 8,325

Round all the way d. 14,821

4. Estimate the following actual problem by rounding all the way, work the actual problem, and check by adding each column of digits separately. (pp. 5, 8) Actual 1,886 9,411 6,395

Estimate

Check

5. Estimate the following actual problem by rounding all the way and then do the actual multiplication. (pp. 5, 12) Actual 8,843 906

Estimate

6. Multiply the following by the shortcut method. (p. 14) 829,412 1,000 7. Divide the following and check the answer by multiplication. (p. 15) Check 39冄14,800

8. Divide the following by the shortcut method. (p. 15) 6,000 60 9. Ling Wong bought a $299 ipod that was reduced to $205. Ling gave the clerk 3 $100 bills. What change will Ling receive? (p. 9) 10. Sam Song plans to buy a $16,000 Ford Saturn with an interest charge of $4,000. Sam figures he can afford a monthly payment of $400. If Sam must pay 40 equal monthly payments, can he afford the Ford Saturn? (p. 14)

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11. Lester Hal has the oil tank at his business filled 20 times per year. The tank has a capacity of 200 gallons. Assume (a) the price of oil fuel is $3 per gallon and (b) the tank is completely empty each time Lester has it filled.What is Lester’s average monthly oil bill? Complete the following blueprint aid for dissecting and solving the word problem. (pp. 6, 12, 15) The facts

Steps to solving problem

30

Solving for?

Steps to take

Key points

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Personal Finance A KIPLINGER APPROACH

BUSINESS MATH ISSUE Kiplinger’s © 2006

Vegetable oil will not solve our oil problem. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A Show the math using the shortcut method of multiplication to prove the million dollars.

2005 Wall Street Journal ©

32

b site text We he e e S : s t T t Projec /slater9e) and e. Interne m ce Guid r o u .c o e s h e h R t .m e (www Intern ss Math Busine

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CHAPTER

2

Fractions

LEARNING UNIT OBJECTIVES LU 2–1: Types of Fractions and Conversion Procedures • Recognize the three types of fractions (pp. 35–36). • Convert improper fractions to whole or mixed numbers and mixed numbers to improper fractions (p. 36). • Convert fractions to lowest and highest terms (pp. 36–38).

LU 2–2: Adding and Subtracting Fractions • Add like and unlike fractions (pp. 40–41). urnal © 2006 Wall Street Jo

• Find the least common denominator (LCD) by inspection and prime numbers (pp. 41–42). • Subtract like and unlike fractions (p. 43). • Add and subtract mixed numbers with the same or different denominators (pp. 42–45).

LU 2–3: Multiplying and Dividing Fractions • Multiply and divide proper fractions and mixed numbers (pp. 46–48). • Use the cancellation method in the multiplication and division of fractions (pp. 47–48).

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Chapter 2 Fractions

The following two Wall Street Journal clippings “Product Piracy Rises in China, U.S. Says” and “Fruitcake Makers See a Way to Boost Sales: Slice the Serving Size” illustrate the use of 2 fractions. For example, from the first clipping you learn that almost two-thirds ( 3 ) of all seizures of fake products come from China.

Wall Street Journal © 2005

Associated Press © 2005

Now let’ s look at Milk Chocolate M&M’ s® candies as another example of using fractions. As you know, M&M’s® candies come in different colors. Do you know how many of each color are in a bag of M&M’ s®? If you go to the M&M’ s website, you learn that a typical bag of M&M’s® contains approximately 17 brown, 11 yellow, 11 red, and 5 each of orange, blue, and green M&M’ s®.1 The 1.69-ounce bag of M&M’ s® shown here contains 55 M&M’ s®. In this bag, you will find the following colors: 18 yellow 10 red

9 blue 7 orange

6 brown 5 green

55 pieces in the bag

The number of yellow candies in a bag might suggest that yellow is the favorite color of many people. Since this is a business math text, however , let’s look at the 55 M&M’ s® in terms of fractional arithmetic. Of the 55 M&M’ s® in the 1.69-ounce bag, 5 of these M&M’ s® are green, so we can say that 5 parts of 55 represent green candies. We could also say that 1 out of 1 1 M&M’s® is green. Are you confused? For many people, fractions are dif ficult. If you are one of these people, this chapter is for you. First you will review the types of fractions and the fraction conversion procedures. Then you will gain a clear understanding of the addition, subtraction, multiplication, and division of fractions.

Off 1 due to rounding.

1

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Learning Unit 2–1

35

Learning Unit 2–1: Types of Fractions and Conversion Procedures This chapter explains the parts of whole numbers called fractions. With fractions you can divide any object or unit—a whole—into a definite number of equal parts. For example, the bag of 55 M&M’ s® shown at the beginning of this chapter contains 6 brown candies. If you eat only the brown M&M’ s®, you have eaten 6 parts of 55, or 6 parts of the whole bag of M&M’s®. We can express this in the following fraction: 6 is the numerator, or top of the fraction. The numerator describes the number of equal parts of the whole bag that you ate. 6 55 55 is the denominator, or bottom of the fraction. The denominator gives the total number of equal parts in the bag of M&M’ s®. Before reviewing the arithmetic operations of fractions, you must recognize the three types of fractions described in this unit. You must also know how to convert fractions to a workable form.

Types of Fractions

Wall Street Journal © 2005

When you read the Wall Street Journal clipping “Wal-Mart Buys Stake in Retailer in Latin America,” you see that Wal-Mart is buying a one-third (13) stake in Central America’s largest retailer. The fraction 13 is a proper fraction. PROPER FRACTIONS A proper fraction has a value less than 1; its numerator is smaller than its denominator.

EXAMPLES

1 , 1 , 1 , 1 , 4 , 9 , 12 , 18 2 10 12 3 7 10 13 55 IMPROPER FRACTIONS

An improper fraction has a value equal to or greater than 1; its numerator is equal to or greater than its denominator.

EXAMPLES

14 , 7 , 15 , 22 14 6 14 19

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Chapter 2 Fractions

MIXED NUMBERS A mixed number is the sum of a whole number greater than zero and a proper fraction.

EXAMPLES

1 9 7 5 9 5 , 5 , 8 , 33 , 139 6 10 8 6 11

Conversion Procedures In Chapter 1 we worked with two of the division symbols ( and 冄 ). The horizontal line (or the diagonal) that separates the numerator and the denominator of a fraction also indicates division. The numerator, like the dividend, is the number we are dividing into. The denominator, like the divisor, is the number we use to divide. Then, referring to the 6 brown M&M’s® in the bag of 55 M&M’ s® (556 ) shown at the beginning of this unit, we can say that we are dividing 55 into 6, or 6 is divided by 55. Also, in the fraction 43, we can say that we are dividing 4 into 3, or 3 is divided by 4. Working with the smaller numbers of simple fractions such as 34 is easier, so we often convert fractions to their simplest terms. In this unit we show how to convert improper fractions to whole or mixed numbers, mixed numbers to improper fractions, and fractions to lowest and highest terms. Converting Improper Fractions to Whole or Mixed Numbers Business situations often make it necessary to change an improper fraction to a whole number or mixed number . You can use the following steps to make this conversion: CONVERTING IMPROPER FRACTIONS TO WHOLE OR MIXED NUMBERS Step 1. Divide the numerator of the improper fraction by the denominator. Step 2. a. If you have no remainder, the quotient is a whole number. b. If you have a remainder, the whole number part of the mixed number is the quotient. The remainder is placed over the old denominator as the proper fraction of the mixed number.

EXAMPLES

15 1 15

16 1 3 5 5

3 R1 5冄 16 15 1

Converting Mixed Numbers to Improper Fractions By reversing the procedure of converting improper fractions to mixed numbers, we can change mixed numbers to improper fractions. CONVERTING MIXED NUMBERS TO IMPROPER FRACTIONS Step 1. Multiply the denominator of the fraction by the whole number. Step 2. Add the product from Step 1 to the numerator of the old fraction. Step 3. Place the total from Step 2 over the denominator of the old fraction to get the improper fraction.

EXAMPLE

1 (8 6) 1 49 6 8 8 8

Note that the denominator stays the same.

Converting (Reducing) Fractions to Lowest Terms When solving fraction problems, you always reduce the fractions to their lowest terms. This reduction does not change the value of the fraction. For example, in the bag of M&M’ s®, 5 5 out of 55 were green. The fraction for this is 55. If you divide the top and bottom of the

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Learning Unit 2–1

37

fraction by 5, you have reduced the fraction to 111 without changing its value. Remember , we said in the chapter introduction that 1 out of 1 1 M&M’s® in the bag of 55 M&M’ s® represents green candies. Now you know why this is true. To reduce a fraction to its lowest terms, begin by inspecting the fraction, looking for the largest whole number that will divide into both the numerator and the denominator without leaving a remainder . This whole number is the greatest common divisor, which cannot be zero. When you find this lar gest whole number , you have reached the point where the fraction is reduced to its lowest terms. At this point, no number (except 1) can divide evenly into both parts of the fraction. REDUCING FRACTIONS TO LOWEST TERMS BY INSPECTION Step 1. By inspection, find the largest whole number (greatest common divisor) that will divide evenly into the numerator and denominator (does not change the fraction value). Step 2. Now you have reduced the fraction to its lowest terms, since no number (except 1) can divide evenly into the numerator and denominator.

EXAMPLE

24 24 6 4 30 30 6 5

Using inspection, you can see that the number 6 in the above example is the greatest common divisor. When you have lar ge numbers, the greatest common divisor is not so obvious. For large numbers, you can use the following step approach to find the greatest common divisor: STEP APPROACH FOR FINDING GREATEST COMMON DIVISOR Step 1. Divide the smaller number (numerator) of the fraction into the larger number (denominator). Step 2. Divide the remainder of Step 1 into the divisor of Step 1. Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue this division process until the remainder is a 0, which means the last divisor is the greatest common divisor.

EXAMPLE

Step 1 1 24冄 30 24 6

24 30

Step 2 4 6 冄 24 24 0

24 6 4 30 6 5

Reducing a fraction by inspection is to some extent a trial-and-error method. Sometimes you are not sure what number you should divide into the top (numerator) and bottom (denominator) of the fraction. The following reference table on divisibility tests will be helpful. Note that to reduce a fraction to lowest terms might result in more than one division. Will divide evenly into number if

Examples

2

3

4

5

6

10

Last digit is 0, 2, 4, 6, 8.

Sum of the digits is divisible by 3.

Last two digits can be divided by 4.

Last digit is 0 or 5.

The number is even and 3 will divide into the sum of the digits.

The last digit is 0.

12 6 14 7

36 12 69 23

140 1(40) 160 1(60)

15 3 20 4

12 2 18 3

90 9 100 10

36933 6 9 15 3 5

35 7 40 8

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Chapter 2 Fractions

Converting (Raising) Fractions to Higher Terms Later, when you add and subtract fractions, you will see that sometimes fractions must be raised to higher terms. Recall that when you reduced fractions to their lowest terms, you looked for the largest whole number (greatest common divisor) that would divide evenly into both the numerator and the denominator . When you raise fractions to higher terms, you do the opposite and multiply the numerator and the denominator by the same whole number . For example, if you want to raise the fraction 14, you can multiply the numerator and denominator by 2. EXAMPLE

1 2 2 4 2 8

1 The fractions and are28 equivalent in value. By converting 41 to 28, you only divided it into 4 more parts. Let’s suppose that you have eaten 47 of a pizza. You decide that instead of expressing the amount you have eaten in 7ths, you want to express it in 28ths. How would you do this? To find the new numerator when you know the new denominator (28), use the steps that follow.

RAISING FRACTIONS TO HIGHER TERMS WHEN DENOMINATOR IS KNOWN Step 1. Divide the new denominator by the old denominator to get the common number that raises the fraction to higher terms. Step 2. Multiply the common number from Step 1 by the old numerator and place it as the new numerator over the new denominator.

EXAMPLE 4

7

? 28

Step 1. Divide 28 by 7 = 4. Step 2. Multiply 4 by the numerator 4 = 16. Result: 4 16 4 4 aNote: This is the same as multiplying .b 7 28 7 4 Note that the 47 and 16 28 are equivalent in value, yet they are dif ferent fractions. Now try the following Practice Quiz to check your understanding of this unit.

LU 2–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1.

2.

DVD 3.

4.

5.

Identify the type of fraction—proper, improper, or mixed: 4 6 1 20 a. b. c. 19 d. 5 5 5 20 Convert to a mixed number: 160 9 Convert the mixed number to an improper fraction: 5 9 8 Find the greatest common divisor by the step approach and reduce to lowest terms: 24 91 a. b. 40 156 Convert to higher terms: 14 8 a. b. 20 200 10 60

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Learning Unit 2–1

✓

Solutions

1.

a. Proper b. Improper c. Mixed d. Improper

4.

1 24冄 40 24 16 3 24 8 40 8 5 a.

1 16冄 24 16 8

1 1 91冄156 65冄 91 91 65 65 26 91 13 7 156 13 12 b.

5.

LU 2–1a

3. (9 8) 5 77 1779 8 8 9冄160 9 70 63 7 2 8 is greatest 8 冄 16 common divisor. 16 0 2.

2 26冄 65 52 13

2 13 冄 26 26 0

a.

10 20冄 200

10 14 140

14 140 20 200

b.

6 10冄60

6 8 48

48 8 10 60

13 is greatest common divisor.

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 52)

1.

2.

3.

4.

5.

Identify the type of fraction—proper, improper, or mixed: 2 7 1 40 a. b. c. 18 d. 5 6 3 40 Convert to a mixed number (do not reduce): 155 7 Convert the mixed number to an improper fraction: 7 8 9 Find the greatest common divisor by the step approach and reduce to lowest terms: 42 96 a. b. 70 182 Convert to higher terms: 16 9 a. b. 30 300 20 60

39

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Chapter 2 Fractions

Learning Unit 2–2: Adding and Subtracting Fractions The Wall Street Journal clipping “TiVo Slashes Recorder Price in Half, to $50” states that TiVo cut the price of its 1 recorder in half ( 2 ). Since a whole is 2 2 2 (2 1), you can determine the new selling price of the recorder by subtracting the numerator of the fraction 21 from the numerator of the fraction 22. You can make this subtraction because you are working with like fractions—fractions with the same denominators. Then you can prove that you are correct by adding the numerators of the fractions 12 and 12. In this unit you learn how to add and subtract fractions with the same denominators (like fractions) and fractions with different denominators (unlike fractions). We have also included how to add and subtract mixed numbers.

Wall Street Journal © 2005

Addition of Fractions When you add two or more quantities, they must have the same name or be of the same denomination. You cannot add 6 quarts and 3 pints unless you change the denomination of one or both quantities. You must either make the quarts into pints or the pints into quarts. The same principle also applies to fractions. That is, to add two or more fractions, they must have a common denominator. Adding Like Fractions In our TiVo clipping at the beginning of this unit we stated that because the fractions had the same denominator , or a common denominator , they were like fractions. Adding like fractions is similar to adding whole numbers. ADDING LIKE FRACTIONS Step 1. Add the numerators and place the total over the original denominator. Step 2. If the total of your numerators is the same as your original denominator, convert your answer to a whole number; if the total is larger than your original denominator, convert your answer to a mixed number.

EXAMPLE

1 4 5 7 7 7

The denominator, 7, shows the number of pieces into which some whole was divided. The two numerators, 1 and 4, tell how many of the pieces you have. So if you add 1 and 4, you get 5, or 57. Adding Unlike Fractions Since you cannot add unlike fractions because their denominators are not the same, you must change the unlike fractions to like fractions—fractions with the same denominators. To do this, find a denominator that is common to all the fractions you want to add. Then look for the least common denominator (LCD).2 The LCD is the smallest nonzero whole number into which all denominators will divide evenly . You can find the LCD by inspection or with prime numbers. Often referred to as the lowest common denominator.

2

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Learning Unit 2–2

41

Finding the Least Common Denominator (LCD) by Inspection The example that

follows shows you how to use inspection to find an LCD (this will make all the denominators the same).

EXAMPLE

3 5 7 21

Inspection of these two fractions shows that the smallest number into which denominators 7 and 21 divide evenly is 21. Thus, 21 is the LCD. You may know that 21 is the LCD of 73 215 , but you cannot add these two fractions until you change the denominator of 73 to 21. You do this by building (raising) the equivalent of 37, as explained in Learning Unit 2–1. You can use the following steps to find the LCD by inspection: Divide the new denominator (21) by the old denominator (7): 21 7 = 3. Step 2. Multiply the 3 in Step 1 by the old numerator (3): 3 3 = 9. The new numerator is 9. Step 1.

Result: 3 9 7 21 Now that the denominators are the same, you add the numerators. 5 14 2 9 21 3 21 21 2 Note that 14 21 is reduced to its lowest terms 3 . Always reduce your answer to its lowest terms. You are now ready for the following general steps for adding proper fractions with dif ferent denominators. These steps also apply to the following discussion on finding LCD by prime numbers.

ADDING UNLIKE FRACTIONS Step 1. Find the LCD. Step 2. Change each fraction to a like fraction with the LCD. Step 3. Add the numerators and place the total over the LCD. Step 4. If necessary, reduce the answer to lowest terms.

Finding the Least Common Denominator (LCD) by Prime Numbers When you cannot determine the LCD by inspection, you can use the prime number method. First you must understand prime numbers. PRIME NUMBERS A prime number is a whole number greater than 1 that is only divisible by itself and 1. The number 1 is not a prime number.

EXAMPLES

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43

Note that the number 4 is not a prime number . Not only can you divide 4 by 1 and by 4, but you can also divide 4 by 2. A whole number that is greater than 1 and is only divisible by itself and 1 has become a source of interest to some people. These people are curious as to what is the lar gest known prime number. The accompanying newspaper clipping answers this question. This number, of course, is the known number at the time of the writing of this clipping. Probably by the time you become impressed with this lar ge prime number , someone will have discovered a lar ger prime number. Time 2003

EXAMPLE

1 1 1 1 3 8 9 12

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Chapter 2 Fractions

Step 1.

Copy the denominators and arrange them in a separate row .

Step 2.

Divide the denominators in Step 1 by prime numbers. Start with the smallest number that will divide into at least two of the denominators. Bring down any number that is not divisible. Keep in mind that the lowest prime number is 2. 2 l 3 8 9 12 3 4 9 6

3

8

9

12

Note: The 3 and 9 were brought down, since they were not divisible by 2. Continue Step 2 until no prime number will divide evenly into at least two numbers. Note: The 3 is used, 2 l 3 8 9 12 since 2 can no longer 2l3 4 9 6 divide evenly into at 3l3 2 9 3 least two numbers. 1 2 3 1 Step 4. To find the LCD, multiply all the numbers in the divisors (2, 2, 3) and in the last row (1, 2, 3, 1). Step 3.

2 2 3 1 2 3 1 72 (LCD) Divisors Step 5.

Last row

Raise each fraction so that each denominator will be 72 and then add fractions. 1 ? 3 72

72 3 24 24 1 24

1 ? 8 72

72 8 9 919

24 9 8 6 47 72 72 72 72 72

The above five steps used for finding LCD with prime numbers are summarized as follows: FINDING LCD FOR TWO OR MORE FRACTIONS Step 1. Copy the denominators and arrange them in a separate row. Step 2. Divide the denominators by the smallest prime number that will divide evenly into at least two numbers. Step 3. Continue until no prime number divides evenly into at least two numbers. Step 4. Multiply all the numbers in divisors and last row to find the LCD. Step 5. Raise all fractions so each has a common denominator and then complete the computation.

Adding Mixed Numbers The following steps will show you how to add mixed numbers: ADDING MIXED NUMBERS Step 1. Add the fractions (remember that fractions need common denominators, as in the previous section). Step 2. Add the whole numbers. Step 3. Combine the totals of Steps 1 and 2. Be sure you do not have an improper fraction in your final answer. Convert the improper fraction to a whole or mixed number. Add the whole numbers resulting from the improper fraction conversion to the total whole numbers of Step 2. If necessary, reduce the answer to lowest terms.

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Learning Unit 2–2

Using prime numbers to find LCD of example 2 l 20 5 4 2 l 10 5 2 5 l 5 5 1 1 1 1 2 2 5 20 LCD

EXAMPLE

7 20 3 6 5 1 7 4

7 20 12 6 20 5 7 20

4

43

3 ? 5 20 20 5 4 3 12

4

24 4 1 20 20 17 4 1 18 18 20 5

Step 1 Step 2 Step 3

Subtraction of Fractions The subtraction of fractions is similar to the addition of fractions. This section explains how to subtract like and unlike fractions and how to subtract mixed numbers. Subtracting Like Fractions To subtract like fractions, use the steps that follow . SUBTRACTING LIKE FRACTIONS Step 1. Subtract the numerators and place the answer over the common denominator. Step 2. If necessary, reduce the answer to lowest terms.

EXAMPLE

1 82 4 9 10 10 10 2 5 Step 1

Step 2

Subtracting Unlike Fractions Now let’s learn the steps for subtracting unlike fractions. SUBTRACTING UNLIKE FRACTIONS Step 1. Find the LCD. Step 2. Raise the fraction to its equivalent value. Step 3. Subtract the numerators and place the answer over the LCD. Step 4. If necessary, reduce the answer to lowest terms.

EXAMPLE

5 8 2 64

40 64 2 64 38 19 64 32

By inspection, we see that LCD is 64. Thus 64 8 8 5 40.

Subtracting Mixed Numbers When you subtract whole numbers, sometimes borrowing is not necessary . At other times, you must borrow . The same is true of subtracting mixed numbers.

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Chapter 2 Fractions

SUBTRACTING MIXED NUMBERS

When Borrowing Is Not Necessary

When Borrowing Is Necessary

Step 1. Subtract fractions, making sure to find the LCD.

Step 1. Make sure the fractions have the LCD.

Step 2. Subtract whole numbers.

Step 2. Borrow from the whole number of the minuend (top number).

Step 3. Reduce the fraction(s) to lowest terms.

Step 3. Subtract the whole numbers and fractions. Step 4. Reduce the fraction(s) to lowest terms.

EXAMPLE Where borrowing is not necessary:

Find LCD of 2 and 8. LCD is 8.

1 2 3 8

4 6 8 3 8 1 6 8

6

EXAMPLE

Where borrowing is necessary: 1 3 2 3 1 4 LCD is 4.

2 3 4 3 1 4

6 4 3 1 4 2

4 2 a b 4 4

3 4

1

Since 43 is larger than 24, we must borrow 1 from the 3. This is the same as borrowing 44. Afraction with the same numerator and denominator represents a whole. When we add 44 24, we get 64. Note how we subtracted the whole number and fractions, being sure to reduce the final answer if necessary .

How to Dissect and Solve a Word Problem Let’s now look at how to dissect and solve a word problem involving fractions. 1

The Word Problem The Albertsons grocery store has 550 4; total square feet of floor space.

11512

Albertsons’ meat department occupies square feet, and its deli department occupies 145 square feet. If the remainder of the floor space is for groceries, what square footage remains for groceries? The facts

Solving for?

Steps to take

Key points

Total square footage: 55014 sq. ft.

Total square footage for groceries.

Total floor space Total meat and deli floor space Total grocery floor space.

Denominators must be the same before adding or subtracting fractions.

Meat department: 11512 sq. ft. Deli department: 14578 sq. ft.

8 1 8 Never leave improper fraction as final answer.

Steps to solving problem 1. Calculate total square footage of the meat and deli departments. 4 1 Meat: 115 115 2 8 7 7 Deli: 145 145 8 8 11 3 260 261 sq. ft. 8 8

7 8

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45

2. Calculate total grocery square footage. 1 2 10 550 550 549 4 8 8 3 3 3 261 261 261 8 8 8

2 8 a b 8 8

288 7 sq. ft. 8

Check 3 261 8 7 288 8 10 2 1 549 550 550 sq. ft. 8 8 4

Note how the above blueprint aid helped to gather the facts and identify what we were looking for. To find the total square footage for groceries, we first had to sum the areas for meat and deli. Then we could subtract these areas from the total square footage. Also note that in Step 1 above, we didn’ t leave the answer as an improper fraction. In Step 2, we borrowed from the 550 so that we could complete the subtraction. It’s your turn to check your progress with a Practice Quiz.

LU 2–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1. 2.

DVD 3.

4.

Find LCD by the division of prime numbers: 12, 9, 6, 4 Add and reduce to lowest terms if needed: 3 2 3 1 a. b. 2 6 40 5 4 20 Subtract and reduce to lowest terms if needed: 6 1 1 9 3 a. b. 8 3 c. 4 1 7 4 4 28 4 Computerland has 660 41 total square feet of floor space. Three departments occupy this floor space: hardware, 201 18 square feet; software, 242 14 square feet; and customer ser vice, __________ square feet. What is the total square footage of the customer service area? You might want to try a blueprint aid, since the solution will show a completed blueprint aid.

✓ Solutions 1.

2 l 12 9 6 4 2 l 6 9 3 2 3 l 3 9 3 1 1 3 1 1

2.

a.

b.

3.

LCD 2 2 3 1 3 1 1 36

2 3 16 19 3 40 5 40 40 40 3 4 1 6 20

15 2 20 1 6 20 16 4 8 8 20 5

2

a.

6 24 7 28 1 7 4 28 17 28

c.

4 3 4 3 1 4 1 2 4

b.

? 2 ° 5 40 ¢ 40 5 8 2 16

3 ? 4 20 20 4 5 3 15 1 7 35 8 8 7 4 28 28 9 9 9 3 3 3 28 28 28 26 13 4 4 28 14

4 Note how we showed the 4 as 3 . 4

a

28 7 b 28 28

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Chapter 2 Fractions

Computerland’s total square footage for customer service:

4.

The facts

Solving for?

Steps to take

Key points

Total square footage: 66014 sq. ft.

Total square footage for customer service.

Total floor space ⫺ Total hardware and software floor space ⫽ Total customer service floor space.

Denominators must be the same before adding or subtracting fractions.

Hardware: 20118 sq. ft. Software: 24214 sq. ft.

Steps to solving problem 1. Calculate the total square footage of hardware and software.

1 1 201 ⫽ 201 (hardware) 8 8 1 2 ⫹ 242 ⫽ ⫹ 242 (software) 4 8 3 443 8

2. Calculate the total square footage for customer service. 1 2 10 660 ⫽ 660 ⫽ 659 (total square footage) 4 8 8 3 3 3 ⫺443 ⫽ ⫺ 443 ⫽ ⫺ 443 (hardware plus software) 8 8 8 7 216 sq. ft. (customer service) 8

LU 2–2a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 52)

1. 2.

3.

4.

Find the LCD by the division of prime numbers: 10, 15, 9, 4 Add and reduce to lowest terms if needed: 2 3 3 1 ⫹ a. b. 3 ⫹ 6 25 5 8 32 Subtract and reduce to lowest terms if needed: 5 1 1 7 2 ⫺ a. b. 9 ⫺ 3 c. 6 ⫺ 1 6 3 8 32 5 Computerland has 98514 total square feet of floor space. Three departments occupy this floor space: hardware, 209 81 square feet; software, 38214 square feet; and customer service, ____________ square feet. What is the total square footage of the customer service area?

Learning Unit 2–3: Multiplying and Dividing Fractions The following recipe for Coconutty “M&M’ s”® Brownies makes 16 brownies. What would you need if you wanted to triple the recipe and make 48 brownies?

© 2000 Mars, Incorporated

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Learning Unit 2–3

47

Preheat oven to 350 F. Grease 8 8 2-inch pan; set aside. In small saucepan combine chocolate, butter, and sugar over low heat; stir constantly until smooth. Remove from heat; let cool. In bowl beat eggs, oil, and vanilla; stir in chocolate mixture until blended. Stir in flour , cocoa powder, baking powder, and salt. Stir in 1 cup “M&M’ s”® Chocolate Mini Baking Bits. Spread batter in prepared pan. Bake 35 to 40 minutes or until toothpick inserted in center 1 comes out clean. Cool. Prepare a coconut topping. Spread over brownies; sprinkle with 2 cup ® “M&M’s” Chocolate Mini Baking Bits. In this unit you learn how to multiply and divide fractions.

Multiplication of Fractions Multiplying fractions is easier than adding and subtracting fractions because you do not have to find a common denominator . This section explains the multiplication of proper fractions and the multiplication of mixed numbers. MULTIPLYING PROPER FRACTIONS3 Step 1. Multiply the numerators and the denominators. Step 2. Reduce the answer to lowest terms or use the cancellation method.

First let’s look at an example that results in an answer that we do not have to reduce. EXAMPLE

5 1 5 7 8 56

In the next example, note how we reduce the answer to lowest terms. EXAMPLE

5 1 4 20 10 1 6 7 42 21

We can reduce

20 42

Keep in mind

5 is equal to 5. 1

by the step approach as follows: 10 2冄20 20 0

2 20冄42 40 2

We could also have found the greatest common divisor by inspection.

10 20 2 42 2 21 As an alternative to reducing fractions to lowest terms, we can use the cancellation technique. Let’s work the previous example using this technique. 2

EXAMPLE

5 1 4 10 1 6 7 21 3

2 divides evenly into 4 twice and into 6 three times.

Note that when we cancel numbers, we are reducing the answer before multiplying. We know that multiplying or dividing both numerator and denominator by the same number gives an equivalent fraction. So we can divide both numerator and denominator by any number that divides them both evenly . It doesn’ t matter which we divide first. Note that this division reduces 10 21 to its lowest terms. Multiplying Mixed Numbers The following steps explain how to multiply mixed numbers: MULTIPLYING MIXED NUMBERS Step 1. Convert the mixed numbers to improper fractions. Step 2. Multiply the numerators and denominators. Step 3. Reduce the answer to lowest terms or use the cancellation method.

You would follow the same procedure to multiply improper fractions.

3

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Chapter 2 Fractions

1

EXAMPLE

1 1 7 3 7 1 2 1 3 3 2 3 2 2 2 1

Step 1

Step 2

Step 3

Division of Fractions When you studied whole numbers in Chapter 1, you saw how multiplication can be checked by division. The multiplication of fractions can also be checked by division, as you will see in this section on dividing proper fractions and mixed numbers. Dividing Proper Fractions The division of proper fractions introduces a new term—the reciprocal. To use reciprocals, we must first recognize which fraction in the problem is the divisor—the fraction that we divide by. Let’s assume the problem we are to solve is 18 23. We read this problem as “ 18 divided by 23.” The divisor is the fraction after the division sign (or the second fraction). The steps that follow show how the divisor becomes a reciprocal. DIVIDING PROPER FRACTIONS Step 1.

Invert (turn upside down) the divisor (the second fraction). The inverted number is the reciprocal.

Step 2.

Multiply the fractions.

Step 3.

Reduce the answer to lowest terms or use the cancellation method.

Do you know why the inverted fraction number is a reciprocal? Reciprocals are two num1 2 bers that when multiplied give a product of 1. For example, 2 (which is the same as 1) and 2 are reciprocals because multiplying them gives 1. EXAMPLE

1 2 8 3

1 3 3 8 2 16

Dividing Mixed Numbers Now you are ready to divide mixed numbers by using improper fractions. DIVIDING MIXED NUMBERS Step 1.

Convert all mixed numbers to improper fractions.

Step 2.

Invert the divisor (take its reciprocal) and multiply. If your final answer is an improper fraction, reduce it to lowest terms. You can do this by finding the greatest common divisor or by using the cancellation technique.

EXAMPLE

3 5 8 2 4 6

Step 1. 35 17 4 6 3 Step 2. 35 6 105 3 3 4 17 34 34

Here we used the cancellation technique.

2

How to Dissect and Solve a Word Problem Jamie Slater ordered 5 21 cords of oak. The cost of each cord is $150. He also ordered cords of maple at $120 per cord. Jamie’s neighbor, Al, said that he would share the wood and pay him 51 of the total cost. How much did Jamie receive from Al? Note how we filled in the blueprint aid columns. We first had to find the total cost of all the wood before we could find Al’s share— 15 of the total cost. The Word Problem

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49

Learning Unit 2–3

The facts

Solving for?

Cords ordered: 512 at $150 per cord; 241 at $120 per cord.

What will Al pay Jamie?

Steps to take

Key points

Total cost of wood Al’s cost.

Convert mixed numbers to improper fractions when multiplying.

1 5

Al’s cost share: the total cost.

1 5

Cancellation is an alternative to reducing fractions.

Steps to solving problem 1. Calculate the cost of oak.

$75 1 11 5 $150 $150 $825 2 2

2. Calculate the cost of maple.

$30 1 9 2 $120 $120 270 4 4

1

1

$1,095 (total cost of wood) $219 1 $1,095 $219 5

3. What Al pays.

1

You should now be ready to test your knowledge of the final unit in the chapter

LU 2–3

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

.

1.

2.

3.

4.

Multiply (use cancellation technique): 4 4 4 a. b. 35 8 6 7 Multiply (do not use canceling; reduce by finding the greatest common divisor): 14 7 15 10 Complete the following. Reduce to lowest terms as needed. 1 5 51 5 a. b. 9 6 5 9 Jill Estes bought a mobile home that was 8 18 times as expensive as the home her brother bought. Jill’s brother paid $16,000 for his mobile home. What is the cost of Jill’ s new home?

✓ Solutions 1 2

1.

a.

2 1

2.

3. 4.

1

4 4 1 8 6 3

5

b.

3

35

4 20 7 1

14 7 98 2 49 15 10 150 2 75 1 1 1 98冄 150 52冄 98 46冄52 98 52 46 52 46 6 1 6 63 2 a. 9 5 45 3 15

7 6冄 46 42 4 b.

1 2 4冄6 2冄 4 4 4 2 0 51 9 459 9 18 5 5 25 25

Total cost of Jill’s new home: The facts

Solving for?

Steps to take

Key points

Jill’s mobile home: 818 as expensive as her brother’s.

Total cost of Jill’s new home.

818 Total cost of Jill’s brother’s mobile home Total cost of Jill’s new home.

Canceling is an alternative to reducing.

Brother paid: $16,000.

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Chapter 2 Fractions

Steps to solving problem 1. Convert 818 to a mixed number.

65 8

2. Calculate the total cost of Jill’s home.

$2,000 65 $16,000 $130,000 8 1

LU 2–3a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 52)

1.

2.

3.

4.

Multiply (use cancellation technique): 6 3 1 a. b. 42 8 6 7 Multiply (do not use canceling; reduce by finding the greatest common divisor): 13 9 117 5 Complete the following. Reduce to lowest terms as needed. 1 4 61 6 a. b. 8 5 6 7 Jill Estes bought a mobile home that was 10 18 times as expensive as the home her brother brought. Jill’s brother paid $10,000 for his mobile home. What is the cost of Jill’ s new home?

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Types of fractions, p. 35

Proper: Value less than 1; numerator smaller than denominator. Improper: Value equal to or greater than 1; numerator equal to or greater than denominator. Mixed: Sum of whole number greater than zero and a proper fraction.

3 7 8 , , 5 9 15 14 19 , 14 18

Improper to whole or mixed: Divide numerator by denominator; place remainder over old denominator. Mixed to improper: Whole number Denominator Numerator Old denominator

1 17 4 4 4

Reducing fractions to lowest terms, p. 37

1. Divide numerator and denominator by largest possible divisor (does not change fraction value). 2. When reduced to lowest terms, no number (except 1) will divide evenly into both numerator and denominator.

18 2 9 46 2 23

Step approach for finding greatest common denominator, p. 37

1. Divide smaller number of fraction into larger number. 2. Divide remainder into divisor of Step 1. Continue this process until no remainder results. 3. The last divisor used is the greatest common divisor.

Fraction conversions, p. 36

Raising fractions to higher terms, p. 38

Multiply numerator and denominator by same number. Does not change fraction value.

3 8 6 ,9 8 9

1 32 1 33 4 8 8 8

15 65

4 15冄65 60 5

3 5冄 15 15 0 5 is greatest common divisor.

15 ? 41 410 410 41 10 15 150

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

51

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (Continued) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Adding and subtracting like and unlike fractions, p. 40

When denominators are the same (like fractions), add (or subtract) numerators, place total over original denominator, and reduce to lowest terms. When denominators are different (unlike fractions), change them to like fractions by finding LCD using inspection or prime numbers. Then add (or subtract) the numerators, place total over LCD, and reduce to lowest terms.

4 1 5 9 9 9 4 1 3 1 9 9 9 3 4 2 28 10 38 3 1 5 7 35 35 35 35

Prime numbers, p. 41

Whole numbers larger than 1 that are only divisible by itself and 1.

2, 3, 5, 7, 11

LCD by prime numbers, p. 42

1. Copy denominators and arrange them in a separate row. 2. Divide denominators by smallest prime number that will divide evenly into at least two numbers. 3. Continue until no prime number divides evenly into at least two numbers. 4. Multiply all the numbers in the divisors and last row to find LCD. 5. Raise fractions so each has a common denominator and complete computation.

1 1 1 1 1 3 6 8 12 9 2 l 3 6 8 12 9 2 l3 3 4 6 9 3 l3 3 2 3 9 1 1 2 1 3 2 2 3 1 1 2 1 3 72

Adding mixed numbers, p. 42

1. Add fractions. 2. Add whole numbers. 3. Combine totals of Steps 1 and 2. If denominators are different, a common denominator must be found. Answer cannot be left as improper fraction.

3 4 1 1 7 7 4 3 7 Step 1: 7 7 7 Step 2: 1 1 2 7 Step 3: 2 3 7

Subtracting mixed numbers, p. 44

1. Subtract fractions. 2. If necessary, borrow from whole numbers. 3. Subtract whole numbers and fractions if borrowing was necessary. 4. Reduce fractions to lowest terms.

3 2 12 7 5 5 7 3 11 7 5 5

If denominators are different, a common denominator must be found. Multiplying proper fractions, p. 47

Multiplying mixed numbers, p. 47

Dividing proper fractions, p. 48

1. Multiply numerators and denominators. 2. Reduce answer to lowest terms or use cancellation method. 1. Convert mixed numbers to improper fractions. 2. Multiply numerators and denominators. 3. Reduce answer to lowest terms or use cancellation method. 1. Invert divisor. 2. Multiply. 3. Reduce answer to lowest terms or use cancellation method.

Due to borrowing 5 from number 12 5 5 2 7 5 5 5

4 4 5

The whole number is now 11. 1

4 7 4 7 9 9 1

5 1 1 2 8 8 9 21 189 61 2 8 8 64 64 2

1 1 1 8 2 4 8 4 1 1

(continues)

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Chapter 2 Fractions

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (Concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Dividing mixed numbers, p. 48

1. Convert mixed numbers to improper fractions. 2. Invert divisor and multiply. If final answer is an improper fraction, reduce to lowest terms by finding greatest common divisor or using the cancellation method.

5 3 13 1 1 1 2 8 2 8

KEY TERMS

Cancellation, p. 47 Common denominator, p. 40 Denominator, p. 40 Equivalent, p. 38 Fraction, p. 35 Greatest common divisor, p. 37

CHECK FIGURE FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 2–1a (p. 39) 1. a. P b. I c. M d. I 1 2. 22 7 79 3. 9 3 48 4. a. 14; b.2; 5 91 5. a. 160; b. 27

Higher terms, p. 38 Improper fraction, p. 35 Least common denominator (LCD), p. 40 Like fractions, p. 40 Lowest terms, p. 37 Mixed numbers, p. 36 LU 2–2a (p. 46) 1. 180 17 13 2. a. b. 9 25 32 1 29 3 3. a. b. 5 c. 4 2 32 5 7 4. 393 ft. 8

4

3 8 2 13 1

12 13 Numerator, p. 35 Prime numbers, p. 41 Proper fractions, p. 35 Reciprocal, p. 48 Unlike fractions, p. 40

LU 2–3a (p. 50) 3 b. 6 8 1 2. 117; 5 31 5 3. a. b. 11 32 36 4. $101,250 1. a.

Note: For how to dissect and solve a word problem, see page 44.

Critical Thinking Discussion Questions 1. What are the steps to convert improper fractions to whole or 5. Explain the steps of adding or subtracting unlike fractions. mixed numbers? Give an example of how you could use this Using a ruler , measure the heights of two dif ferent-size conversion procedure when you eat at Pizza Hut. cans of food and show how to calculate the dif ference in height. 2. What are the steps to convert mixed numbers to improper fractions? Show how you could use this conversion proce6. What is a prime number? Using the two cans in question dure when you order doughnuts at Dunkin’ Donuts. 5, show how you could use prime numbers to calculate the LCD. 3. What is the greatest common divisor? How could you use the greatest common divisor to write an advertisement showing 7. Explain the steps for multiplying proper fractions and mixed that 35 out of 60 people prefer MCI to AT&T? numbers. Assume you went to Staples (a stationery superstore). Give an example showing the multiplying of proper 4. Explain the step approach for finding the greatest common fractions and mixed numbers. divisor. How could you use the MCI–AT&T example in question 3 to illustrate the step approach?

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Identify the following types of fractions: 2–1.

11 10

1 8

2–2. 12

2–3.

2 9

Convert the following to mixed numbers: 2–4.

79 8

2–5.

921 15

Convert the following to improper fractions: 7 8

2–6. 8

2 3

2–7. 19

Reduce the following to the lowest terms. Show how to calculate the greatest common divisor by the step approach. 2–8.

16 38

2–9.

44 52

Convert the following to higher terms: 2–10.

9 10 70

Determine the LCD of the following (a) by inspection and (b) by division of prime numbers: 2–11.

3 7 5 1 , , , 4 12 6 5

Check

Inspection

2–12.

5 7 5 2 , , , 6 18 9 72

Check

Inspection

2–13.

1 3 5 1 , , , 4 32 48 8

Check

Inspection

Add the following and reduce to lowest terms: 2–14.

3 3 9 9

1 3 2–16. 6 4 8 8

2–15.

3 4 7 21

3 1 2–17. 6 9 8 24

7 9 2–18. 9 6 10 10

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Subtract the following and reduce to lowest terms: 2–19.

11 1 12 12

3 5 2–20. 14 10 8 8

1 2 2–21. 12 4 9 3

Multiply the following and reduce to lowest terms. Do not use the cancellation technique for these problems. 2–22. 17

4 2

2–23.

5 3 6 8

7 2–24. 8 64 8 Multiply the following. Use the cancellation technique. 2–25.

4 30 6 10 60 10

3 8 9 2–26. 3 4 4 9 12

Divide the following and reduce to lowest terms. Use the cancellation technique as needed. 2–27.

12 4 9

2 2–29. 4 12 3

2–28. 18

1 5

5 1 2–30. 3 3 6 2

WORD PROBLEMS 2–31. The Baltimore Sun on January 10, 2007, ran a story about Cal Ripken being inducted in the Baseball Hall of Fame with 9812 percent of the votes cast. Ripken was named on 537 of the 545 ballots submitted by the Baseball Writers’ Association of America, the largest number of votes ever received. In order to be named to the Hall, a former player must receive at least 34 of the votes cast. (a) What are the minimum votes needed to be inducted? (b) How many votes did Ripken receive over the total needed?

2–32. The February 2007 issue of Taunton’s Fine Woodworking has measurements for constructing a country hutch. The measurements for the upper portion, in inches, were: 934, 12169 , 1038, and 16167 . The total height of the hutch is 8238 inches. (a) What is the height of the upper portion? (b) What is the height of the lower portion?

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2–33. Jet Blue pays Paul Lose $140 per day to work in the maintenance department at the airport. Paul became ill on Monday and went home after 41 of a day. What did he earn on Monday? Assume no work, no pay.

2–34. Britney Summers visited Curves and lost 214 pounds in week 1, 134 pounds in week 2, and 85 pound in week 3. What is the total weight loss for Britney?

2–35. Joy Wigens, who works at Putnam Investments, received a check for $1,600. She deposited account. How much money does Joy have left after the deposit?

1 4

of the check in her Citibank

2–36. Pete Hall worked the following hours as a manager for Ne ws.com: 1241, 541, 812, and 714. How many total hours did Pete work?

2–37. Woodsmith magazine tells ho w to b uild a country w all shelf. The two side panels are 34 ⫻ 712 ⫻ 3158 inches long. (a) What is the total length of board you will need? (b) If you ha ve a board 74 31 inches long, how much of the board will remain after cutting?

2–38. Lester bought a piece of property in Vail, Colorado. The sides of the land measure 115 21 feet, 6614 feet, 10681 feet, and 11014 feet. Lester wants to know the perimeter (sum of all sides) of his property. Can you calculate the perimeter for Lester?

2–39. The February 2007 issue of Woodsmith provided measurements to construct a storage center . The measurements were, in inches, 3112, 1112, 512, 434, 3138, and 3412. The Home Depot has boards in 7 foot lengths. (a) What is the total length needed? (b) After cutting, how much of the board will be left o ver?

2–40. From Lowes, Pete Wong ordered 76 of a ton of crushed rock to mak e a patio. If Pete used only crushed rock remains unused?

3 4

of the rock, how much

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2–41. At a Wal-Mart store, a Coke dispenser held 19 14 gallons of soda. During working hours, 1234 gallons were dispensed. How many gallons of Coke remain?

2–42. Matt Kaminsky bought a home from Century 21 in San Antonio, Texas, that is 8 12 times as expensive as the home his parents bought. Matt’s parents paid $20,000 for their home. What is the cost of Matt’s new home?

2–43. Ajax Company charges $150 per cord of w ood. If Bill Ryan orders 3 12 cords, what will his total cost be?

2–44. Learning.com bought 90 pizzas at Pizza Hut for their holiday party . Each guest ate 61 of a pizza and there w as no pizza left over. How many guests did Learning.com have for the party?

2–45. Marc, Steven, and Daniel entered into a Subw ay sandwich shop partnership. Marc owns 19 of the shop and Steven owns 14. What part does Daniel own?

2–46. Lionel Sullivan works for Burger King. He is paid time and one-half for Sundays. If Lionel w orks on Sunday for 6 hours at a regular pay of $8 per hour, what does he earn on Sunday?

2–47. Hertz pays Al Davis, an employee, $125 per day. Al decides to donate 51 of a day’s pay to his church. How much will Al donate?

2–48. A trip to the White Mountains of New Hampshire from Boston will tak e you 2 34 hours. Assume you have traveled 111 of the way. How much longer will the trip tak e?

2–49. Andy, who loves to cook, makes apple cobbler for his f amily. The recipe (serves 6) calls for 1 12 pounds of apples, 341 cups of flour, 14 cup of margarine, 238 cups of sugar, and 2 teaspoons of cinnamon. Since guests are coming, Andy wants to make a cobbler that will serv e 15 (or increase the recipe 2 12 times). How much of each ingredient should Andy use?

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2–50. Mobil allocates 1,692 43 gallons of gas per month to Jerry’ s Service Station. The first week, Jerry sold 275 21 gallons; second week, 28041 gallons; and third week, 189 81 gallons. If Jerry sells 582 21 gallons in the fourth week, how close is Jerry to selling his allocation?

2–51. A marketing class at North Shore Community College conducted a viewer preference survey . The survey showed that 65 of the people surveyed preferred DVDs to videotapes. Assume 2,400 responded to the survey. How many favored using traditional tapes?

2–52. The price of a new Ford Explorer has increased to 1 14 times its earlier price. If the original price of the Ford Explorer was $28,000, what is the new price?

2–53. Chris Rong felled a tree that was 299 feet long. Chris decided to cut the tree into pieces 3 41 feet long. How many pieces can Chris cut from this tree?

2–54. Tempco Corporation has a machine that produces 1212 baseball gloves each hour. In the last 2 days, the machine has run for a total of 22 hours. How many baseball gloves has Tempco produced?

2–55. McGraw-Hill/Irwin publishers stores some of its inventory in a warehouse that has 14,500 square feet of space. Each book requires 221 square feet of space. How many books can McGraw-Hill/Irwin keep in this warehouse?

2–56. Alicia, an employee of Dunkin’ Donuts, receives 23 41 days per year of vacation time. So far this year she has taken 3 18 days in January, 5 12 days in May, 6 41 days in July, and 4 14 days in September. How many more days of vacation does Alicia have left?

2–57. Amazon.com offered a new portable color TV for $250 with a rebate of 15 off the regular price. What is the final cost of the TV after the rebate?

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2–58. Shelly Van Doren hired a contractor to refinish her kitchen. The contractor said the job would take 49 12 hours. To date, the contractor has worked the following hours: 1 Monday 4 4 1 Tuesday 9 8 1 Wednesday 4 4 1 Thursday 3 2 5 Friday 10 8

How much longer should the job take to be completed?

ADDITIONAL SET OF WORD PROBLEMS 2–59. An issue of Taunton’s Fine Woodworking included plans for a hall stand. The total height of the stand is 81 12 inches. If the base is 36 165 inches, how tall is the upper portion of the stand?

2–60. Albertsons grocery planned a big sale on apples and received 750 crates from the wholesale market. Albertsons will bag these apples in plastic. Each plastic bag holds 19 of a crate. If Albertsons has no loss to perishables, how many bags of apples can be prepared?

2–61. Frank Puleo bought 6,625 acres of land in ski country . He plans to subdivide the land into parcels of 13 14 acres each. Each parcel will sell for $125,000. How many parcels of land will Frank develop? If Frank sells all the parcels, what will be his total sales?

If Frank sells 53 of the parcels in the first year , what will be his total sales for the year?

2–62. A local Papa Gino’s conducted a food survey. The survey showed that 91 of the people surveyed preferred eating pasta to hamburger. If 5,400 responded to the survey, how many actually favored hambur ger?

2–63. Tamara, Jose, and Milton entered into a partnership that sells men’ s clothing on the Web. Tamara owns 83 of the company, and Jose owns 41. What part does Milton own?

2–64. Quilters Newsletter Magazine gave instructions on making a quilt. The quilt required 4 12 yards of white-on-white print, 2 yards blue check, 12 yard blue-and-white stripe, 243 yards blue scraps, 43 yard yellow scraps, and 478 yards lining. How many total yards are needed?

58

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2–65. A trailer carrying supplies for a Krispy Kreme from Virginia to New York will take 3 41 hours. If the truck traveled 15 of the

way, how much longer will the trip take?

2–66. Land Rover has increased the price of a FreeLander by 15 from the original price. The original price of the FreeLander was $30,000. What is the new price?

2–67. Norman Moen, an employee at Subway, prepared a 90-foot submarine sandwich for a party. Norman decided to cut the submarine into sandwiches of 1 12 feet. How many sandwiches can Norman cut from this submarine?

CHALLENGE PROBLEMS 2–68. Woodsmith magazine gave instructions on how to build a pine cupboard. Lumber will be needed for 2 shelves 10 14 inches long, 2 base sides 1221 inches long, and 2 door stiles 2918 inches long. Your lumber comes in six-foot lengths. (a) How many feet of lumber will you need? (b) If you want 21 a board left over, is this possible with two boards?

2–69. Jack MacLean has entered into a real estate development partnership with Bill L yons and June Reese. Bill owns 41 of the partnership, while June has a 15 interest. The partners will divide all profits on the basis of their fractional ownership. The partnership bought 900 acres of land and plans to subdivide each lot into 214 acres. Homes in the area have been selling for $240,000. By time of completion, Jack estimates the price of each home will increase by 13 of the current value. The partners sent a survey to 12,000 potential customers to see whether they should heat the homes with oil or gas. One-fourth of the customers responded by indicating a 5-to-1 preference for oil. From the results of the survey , Jack now plans to install a 270-gallon oil tank at each home. He estimates that each home will need 5 fills per year . Current price of home heating fuel is $1 per gallon. The partnership estimates its profit per home will be 18 the selling price of each home. From the above, please calculate the following: a. Number of homes to be built.

b.

Selling price of each home.

c. Number of people responding to survey.

d.

Number of people desiring oil.

59

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d. Average monthly cost to run oil heat per house.

e. Amount of profit Jack will receive from the sale of homes.

DVD SUMMARY PRACTICE TEST Identify the following types of fractions. (p. 35) 1 8

1. 5

2.

2 7

4. Convert the following to a mixed number. (p. 36) 163 9

3.

20 19

5. Convert the following to an improper fraction. (p. 36) 1 8

8

6. Calculate the greatest common divisor of the following by the step approach and reduce to lowest terms. (p. 37) 63 90 7. Convert the following to higher terms. (p. 38) ? 16 94 376 8. Find the LCD of the following by using prime numbers. Show your work. (p. 41) 1 1 1 1 8 3 2 12

9. Subtract the following. (p. 43) 4 5 19 8 20 15

Complete the following using the cancellation technique. (p. 47) 10.

3 2 6 4 4 9

1 6 11. 7 9 7

12.

3 6 7

13. A trip to Washington from Boston will take you 5 43 hours. If you have traveled 13 of the way, how much longer will the trip take? (p. 49)

14. Quizno produces 640 rolls per hour. If the oven runs 12 14 hours, how many rolls will the machine produce? (p. 49)

60

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15. A taste-testing survey of Zing Farms showed that 32 of the people surveyed preferred the taste of veggie bur gers to regular burgers. If 90,000 people were in the survey , how many favored veggie bur gers? How many chose regular bur gers? (p. 48)

16. Jim Janes, an employee of Enterprise Co., worked 9 14 hours on Monday, 421 hours on Tuesday, 914 hours on Wednesday, 7 21 hours on Thursday, and 9 hours on Friday. How many total hours did Jim work during the week? (p. 41)

17. JCPenney offered a 13 rebate on its $39 hair dryer. Joan bought a J.C. Penney hair dryer . What did Joan pay after the rebate? (p. 48)

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Personal Finance A KIPLINGER APPROACH

Is financial

independence

a reachable dream?

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t ` `ÃVÛiÀ Ü ÞÕ V> Ài>V ÞÕÀ `Ài>Ãt

£nääÓÎ{Î{{x

| Retail kiosks offer routine services at a

fraction of physician prices. By Thomas M. Anderson

CHECKUPS on the run

Y

o u r child wakes up tor’s office, $125 at an urgent-care cenwith an earache—and ter and $406 at an emergency room, you take him to Target. according to the Minnesota Council of You suspect your nagHealth Plans. ging cough may signal Clinics generally accept cash and bronchitis, so you have it checked—at major credit cards. You can be reimWal-Mart. You don’t need an appointbursed with money you’ve contributed ment for either visit, the to an employercost is a fraction of what sponsored flexiyou would have paid if ble-spending you had cooled your heels account or to a in your doctor’s office all health savings morning, and your insuraccount, and ance might even cover it. now some Walk-in clinics are health insurers coming soon to a retailer, are picking up pharmacy or grocery store the tab. near you. MinneapolisMinutebased MinuteClinic, the Clinic, for country’s largest chain of example, has retail clinics, expects to signed agreehave 250 facilities in 20 ments with states by year-end. Aetna, Cigna Customers appreciate and Unitedthe convenience of oneHealthcare. If stop shopping. Stores get G MinuteClinic posts prices for services, you’re covered which are often covered by insurance. a boost in sales of drugs by one of those and other health-related insurers, you’ll products. And patients and insurers pay your plan’s co-payment rather than save money as routine care, which acthe full cost of the clinic visit. Some counts for one-fourth of U.S. health employers, including Best Buy, Black spending, moves out of doctor’s offices & Decker and Carlson Cos., offer lower and into settings with lower overhead. co-payments to encourage employees to At express medical clinics, one nurse use MinuteClinics. practitioner usually runs the whole opRetail clinics generally won’t treat eration, from reception to diagnosis to children younger than 18 months old. prescription. A visit takes about 15 If you’re on multiple medications or are minutes per patient. older than 65, it’s better to visit an urRetail clinics put a clear price tag gent-care center or your doctor’s office. on your health care. RediClinic, with If you have chest pain, head straight for kiosks in three Wal-Marts, charges a the emergency room. flat fee of $45 for all its basic services. Have questions about using a clinic? A sore-throat checkup with a strep test “Call your family doctor,” advises Larry costs $62 at a Minneapolis MinuteFields, president of the American AcadClinic, compared with $109 at a docemy of Family Physicians. “It’s free.” C O U R T E SY M I N U T E C L I N I C

ÃÌÜiÀÃÛ>ÌÃ°VÉÓ«äÈ°>Ã«

H E A LT H

Reduce Wait for refund e-file or tell your tax preparer to do it for you Refunds in half the time 60 million people are already e-filing*

aboutefile.com

*IRS IMF Database (2003).

BUSINESS MATH ISSUE Kiplinger’s © 2006

Retail clinics are a fad and will not solve the health care problem. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

62

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A What is the total cost of a Bentley boat? Prove your answer using fractions.

2005 Wall Street Journal ©

site ext Web e t e e S : ts Th t Projec /slater9e) and e. Interne ce Guid m r o u .c o e s h e h R t .m (www Interne ss Math Busine

63

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CHAPTER

3

Decimals

LEARNING UNIT OBJECTIVES LU 3–1: Rounding Decimals; Fraction and Decimal Conversions • Explain the place values of whole numbers and decimals; round decimals (pp. 65–67). • Convert decimal fractions to decimals, proper fractions to decimals, mixed numbers to decimals, and pure and mixed decimals to decimal fractions (pp. 67–70).

LU 3–2: Adding, Subtracting, Multiplying, and Dividing Decimals • Add, subtract, multiply, and divide decimals (pp. 71–73). • Complete decimal applications in foreign currency (p. 73). • Multiply and divide decimals by shortcut methods (p. 74).

WHAT THING S COST McDonald’s B ig Mac New York Cit y: $3.29 Buenos Aires: $1.58 (4.80 Argentine peso Johannesburg s) : $2.30 (13.9 5 rand) London: $3.6 1 (1.94 poun ds) Moscow: $1.7 8 (48 rubles) Paris: $4.09 (3.20 euros) Shanghai: $1 .31 (10.50 re nminbi) Tel Aviv: $3.5 0 (15.50 shek els) Tokyo: $2.26 (250 yen) BusinessWeek Online © 2006

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Learning Unit 3–1

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Wall Street Journal © 2005

Are you looking to buy health insurance? As you can see from the Wall Street Journal clipping “What It Costs to Buy Health Insurance,” health insurance is $280.09 cheaper in Long Beach, California, than in New York: New York: $334.09 Long Beach: 54.00 $280.09 If you plan to move to another city , you might consider the cost of health insurance in that city. Also, remember that health insurance is a cost that continues to increase. Chapter 2 introduced the 1.69-ounce bag of M&M’ s® shown in Table 3.1. In Table 3.1 (p. 66), the six colors in the 1.69-ounce bag of M&M’ s® are given in fractions and their values expressed in decimal equivalents that are rounded to the nearest hundredths. This chapter is divided into two learning units. The first unit discusses rounding decimals, converting fractions to decimals, and converting decimals to fractions. The second unit shows you how to add, subtract, multiply , and divide decimals, along with some shortcuts for multiplying and dividing decimals. Added to this unit is a global application of decimals dealing with foreign exchange rates. The increase in the United States of the cost of a stamp from $.39 to $.41 is indicated by decimals. If you think $.41 is high, compare this with Norway ($.87), Italy ($.73), Japan ($.57), and the United Kingdom ($.53). One of the most common uses of decimals occurs when we spend dollars and cents, which is a decimal number. A decimal is a decimal number with digits to the right of a decimal point, indicating that decimals, like fractions, are parts of a whole that are less than one. Thus, we can interchange the terms decimals and decimal numbers. Remembering this will avoid confusion between the terms decimal, decimal number, and decimal point.

Learning Unit 3–1: Rounding Decimals; Fraction and Decimal Conversions Remember to read the decimal point as and.

In Chapter 1 we stated that the decimal point is the center of the decimal numbering system. So far we have studied the whole numbers to the left of the decimal point and the parts of whole numbers called fractions. We also learned that the position of the digits in a whole number gives the place values of the digits (Figure 1.1, p. 3). Now we will study the position (place values) of the digits to the right of the decimal point (Figure 3.1, p. 66). Note that the words to the right of the decimal point end in ths. You should understand why the decimal point is the center of the decimal system. If you move a digit to the left of the decimal point by place (ones, tens, and so on), you increase its value 10 times for each place (power of 10). If you move a digit to the right of the decimal point by place (tenths, hundredths, and so on), you decrease its value 10 times for each place.

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TABLE

3.1

Color*

Analyzing a bag of M&M’s®

Fraction 18 55 10 55 9 55 7 55 6 55 5 55

Yellow Red Blue Orange Brown Green

.33 .18 .16 .13 .11 .09

55 ⫽1 55

Total

1.00

*The color ratios currently given are a sample used for educational purposes. They do not represent the manufacturer’s color ratios.

Sharon Hoogstraten

EXAMPLES

Decimal

$.06

The 6 is in the hundred ths place value.

1.527

The 5 is in the ten ths place value.

2.8394

The 4 is in the ten thousand ths place value.

.33 1.69 oz.

The thirty-three hundred ths represents the yellow M&M’s® in our M&M’ s® bag of 55 M&M’ s®. The one ounce and sixty-nine hundred ths of another ounce is the weight of our bag of M&M’ s®.

Do you recall from Chapter 1 how you used a place-value chart to read or write whole numbers in verbal form? To read or write decimal numbers, you read or write the decimal number as if it were a whole number . Then you use the name of the decimal place of the last digit as given in Figure 3.1. For example, you would read or write the decimal .0796 as seven hundred ninety-six ten thousandths (the last digit, 6, is in the ten thousandths place). To read a decimal with four or fewer whole numbers, you can also refer to Figure 3.1. For larger whole numbers, refer to the whole-number place-value chart in Chapter 1 (Figure 1.1, p. 3). For example, from Figure 3.1 you would read the number 126.2864 as one hundred twenty-six and two thousand eight hundred sixty-four ten thousandths. Remember that the and is the decimal point. Now let’s round decimals. Rounding decimals is similar to the rounding of whole numbers that you learned in Chapter 1. FIGURE

Whole Number Groups

3.1

Decimal Place Values

Thousands

Hundreds

Tens

Ones (units)

Decimal point (and)

Tenths

Hundredths

Thousandths

Ten thousandths

Hundred thousandths

Decimal place-value chart

1,000

100

10

1

and .

1 10

1 100

1 1,000

1 10,000

1 100,000

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Rounding Decimals From Table 3.1, you know that the 1.69-ounce bag of M&M’ s® introduced in Chapter 2 ® contained 18 55 , or .33, yellow M&M’ s . The .33 was rounded to the nearest hundredth. Rounding decimals involves the following steps: ROUNDING DECIMALS TO A SPECIFIED PLACE VALUE Step 1.

Identify the place value of the digit you want to round.

Step 2.

If the digit to the right of the identified digit in Step 1 is 5 or more, increase the identified digit by 1. If the digit to the right is less than 5, do not change the identified digit.

Step 3.

Drop all digits to the right of the identified digit.

® Let’s practice rounding by using the 18 55 yellow M&M’s that we rounded to .33 in Table 18 3.1. Before we rounded 55 to .33, the number we rounded was .32727. This is an example of a repeating decimal since the 27 repeats itself.

EXAMPLE Step 1.

Round .3272727 to nearest hundredth.

.3272727

The identified digit is 2, which is in the hundredths place (two places to the right of the decimal point). The digit to the right of 2 is more than 5 (7). Thus, 2, the identified digit in Step 1, is changed to 3.

Step 2.

Step 3.

.3372727 .33

Drop all other digits to right of the identified digit 3.

We could also round the .3272727 M&M’ follows: Tenth .3272727

.3

s® to the nearest tenth or thousandth as

or .3272727

Thousandth

.327

OTHER EXAMPLES

Round to nearest dollar: Round to nearest cent: Round to nearest hundredth: Round to nearest thousandth:

$166.39 $1,196.885 $38.563 $1,432.9981

$166 $1,196.89 $38.56 $1,432.998

The rules for rounding can dif fer with the situation in which rounding is used. For example, have you ever bought one item from a supermarket produce department that was marked “3 for $1” and noticed what the cashier char ged you? One item marked “3 for $1” would not cost you 3313 cents rounded to 33 cents. You will pay 34 cents. Many retail stores 1 round to the next cent even if the digit following the identified digit is less than 2 of a penny. In this text we round on the concept of 5 or more.

Fraction and Decimal Conversions In business operations we must frequently convert fractions to decimal numbers and decimal numbers to fractions. This section begins by discussing three types of fraction-todecimal conversions. Then we discuss converting pure and mixed decimals to decimal fractions. Converting Decimal Fractions to Decimals From Figure 3.1 you can see that a decimal fraction (expressed in the digits to the right 1 of the decimal point) is a fraction with a denominator that has a power of 10, such as 10 , 23 17 100 , and 1,000 . To convert a decimal fraction to a decimal, follow these steps:

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CONVERTING DECIMAL FRACTIONS TO DECIMALS Step 1.

Count the number of zeros in the denominator.

Step 2.

Place the numerator of the decimal fraction to the right of the decimal point the same number of places as you have zeros in the denominator. (The number of zeros in the denominator gives the number of digits your decimal has to the right of the decimal point.) Do not go over the total number of denominator zeros.

Now let’s change

3 10

and its higher multiples of 10 to decimals.

EXAMPLES

Decimal fraction 3 10

Verbal form a. Three tenths

3 100 3 1,000

b. Three hundredths c. Three thousandths d. Three ten thousandths

3 10,000

Decimal1

Number of decimal places to right of decimal point

.3

1

.03

2

.003

3

.0003

4

Note how we show the dif ferent values of the decimal fractions above in decimals. The zeros after the decimal point and before the number 3 indicate these values. If you add zeros after the number 3, you do not change the value. Thus, the numbers .3 , .30 , and .300 have the same value. So 3 tenths of a pizza, 30 hundredths of a pizza, and 300 thousandths of a pizza are the same total amount of pizza. The first pizza is sliced into 10 pieces. The second pizza is sliced into 100 pieces. The third pizza is sliced into 1,000 pieces. Also, we don’t need to place a zero to the left of the decimal point. Converting Proper Fractions to Decimals Recall from Chapter 2 that proper fractions are fractions with a value less than 1. That is, the numerator of the fraction is smaller than its denominator . How can we convert these proper fractions to decimals? Since proper fractions are a form of division, it is possible to convert proper fractions to decimals by carrying out the division. CONVERTING PROPER FRACTIONS TO DECIMALS Step 1.

Divide the numerator of the fraction by its denominator. (If necessary, add a decimal point and zeros to the number in the numerator.)

Step 2.

Round as necessary.

EXAMPLES

.75 3 4冄 3.00 4 28 20 20

.375 3 8冄 3.000 8 24 60 56 40 40

.333 1 3冄 1.000 3 9 10 9 10 9 1

From .3 to .0003, the values get smaller and smaller , but if you go from .3 to .3000, the values remain the same.

1

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69

Note that in the last example 13 , the 3 in the quotient keeps repeating itself (never ends). The short bar over the last 3 means that the number endlessly repeats. Converting Mixed Numbers to Decimals A mixed number , you will recall from Chapter 2, is the sum of a whole number greater than zero and a proper fraction. To convert mixed numbers to decimals, use the following steps: CONVERTING MIXED NUMBERS TO DECIMALS Step 1.

Convert the fractional part of the mixed number to a decimal (as illustrated in the previous section).

Step 2.

Add the converted fractional part to the whole number.

EXAMPLE

8.00

.4 5冄 2.0 20

2 8 (Step 1) 5

(Step 2) .40 8.40

Now that we have converted fractions to decimals, let’ s convert decimals to fractions. Converting Pure and Mixed Decimals to Decimal Fractions A pure decimal has no whole number(s) to the left of the decimal point (.43, .458, and so on). A mixed decimal is a combination of a whole number and a decimal. An example of a mixed decimal follows. 737.592 Seven hundred thirty-seven and five hundred ninety-two thousandths

EXAMPLE

Note the following conversion steps for converting pure and mixed decimals to decimal fractions: CONVERTING PURE AND MIXED DECIMALS TO DECIMAL FRACTIONS Step 1.

Place the digits to the right of the decimal point in the numerator of the fraction. Omit the decimal point. (For a decimal fraction with a fractional part, see examples c and d below.)

Step 2.

Put a 1 in the denominator of the fraction.

Step 3.

Count the number of digits to the right of the decimal point. Add the same number of zeros to the denominator of the fraction. For mixed decimals, add the fraction to the whole number.

Step 1

EXAMPLES If desired, you can reduce the fractions in Step 3.

a.

.3

b.

.24

c.

.24

1 2

Step 2

Places

Step 3

3

3 1

1

3 10

24

24 1

2

24 100

245

245 1

3

245 1,000

Before completing Step 1 in example c, we must remove the fractional part, convert it to a decimal ( 12 .5), and multiply it by .01 (.5 .01 .005). We use .01 because the 4 of .24 is in the hundredths place. Then we add .005 .24 .245 (three places to right of the decimal) and complete Steps 1, 2, and 3. 1 4

d. .07

725

725 1

4

725 10,000

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In example d, be sure to convert 41 to .25 and multiply by .01. This gives .0025. Then add .0025 to .07, which is .0725 (four places), and complete Steps 1, 2, and 3. e.

45

17.45

45 1

45 45 17 100 100

2

Example e is a mixed decimal. Since we substitute and for the decimal point, we read this mixed decimal as seventeen and forty-five hundredths. Note that after we converted the .45 of the mixed decimals to a fraction, we added it to the whole number 17. The Practice Quiz that follows will help you check your understanding of this unit.

LU 3–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Write the following as a decimal number . 1. Four hundred eight thousandths Name the place position of the identified digit: 2. 6.8241 3. 9.3942 Round each decimal to place indicated: Tenth Thousandth 4. .62768 a. b. 5. .68341 a. b. Convert the following to decimals: 9 14 6. 7. 10,000 100,000 Convert the following to decimal fractions (do not reduce): 1 4 Convert the following fractions to decimals and round answer to nearest hundredth: 1 3 1 11. 12. 13. 12 6 8 8 8. .819

9. 16.93

10. .05

✓ Solutions 1. .408 (3 places to right of decimal) 2. Hundredths 4. a. .6 (identified digit 6—digit to right less than 5) 5. a. .7 (identified digit 6—digit to right greater than 5) 6. .0009 (4 places) 8. 10.

819 819 a b 1,000 1 3 zeros

9. 16

93 100

525 525 1 a .01 .0025 .05 .0525b 1 4 zeros 4 10,000

11. .16666 .17

LU 3–1a

3. Thousandths b. .628 (identified digit 7—digit to right greater than 5) b. .683 (identified digit 3—digit to right less than 5) 7. .00014 (5 places)

12. .375 .38

13. 12.125 12.13

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 78)

Write the following as a decimal number: 1. Three hundred nine thousandths Name the place position of the identified digit: 2. 7.9324

3. 8.3682

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71

Round each decimal to place indicated: Tenth Thousandth 4. .84361 a. b. 5. .87938 a. b. Convert the following to decimals: 8 16 6. 7. 10,000 100,000 Convert the following to decimal fractions (do not reduce): 8. .938 9. 17.95 10. .03 41 Convert the following fractions to decimals and round answer to nearest hundredth: 1 4 1 11. 12. 13. 13 8 7 9

Learning Unit 3–2: Adding, Subtracting, Multiplying, and Dividing Decimals People who are contemplating a career move to another city or state usually want to know the cost of living in that city or state. Also, you will hear retirees saying they are moving to another city or state because the cost of living is cheaper in this city or state. The following Wall Street Journal clipping “City by City” gives you some interesting statistics on the costs of various items in selected locations:

Wall Street Journal © 2005

If you frequent cof fee restaurants, you might want to check the cost of a cup of coffee with service in various locations. The “City by City” clipping helps you do this. For example, a cup of cof fee with service costs $1.71 in Atlanta, while a cup of cof fee with service costs $4.76 in Tokyo. The coffee with service in Atlanta saves you $3.05 per cup. If you drink 1 cup of cof fee per day for a year in Atlanta, you would save $1,1 13.25. $4.76 Tokyo: Atlanta: 1.71 $3.05 365 days $1,113.25 This learning unit shows you how to add, subtract, multiply , and divide decimals. You also make calculations involving decimals, including decimals used in foreign currency .

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Addition and Subtraction of Decimals Since you know how to add and subtract whole numbers, to add and subtract decimal numbers you have only to learn about the placement of the decimals. The following steps will help you: ADDING AND SUBTRACTING DECIMALS Step 1.

Vertically write the numbers so that the decimal points align. You can place additional zeros to the right of the decimal point if needed without changing the value of the number.

Step 2.

Add or subtract the digits starting with the right column and moving to the left.

Step 3.

Align the decimal point in the answer with the above decimal points.

EXAMPLES

Add 4 7.3 36.139 .0007 8.22.

Whole number to the right of the last digit is assumed to have a decimal.

Extra zeros have been added to make calculation easier.

4.0000 7.3000 36.1390 .0007 8.2200 55.6597

Subtract 45.3 15.273.

Subtract 7 6.9. 6 10

2 9 10

45.30 0 15.273 30.027

7.0 – 6.9 .1

Multiplication of Decimals The multiplication of decimal numbers is similar to the multiplication of whole numbers except for the additional step of placing the decimal in the answer (product). The steps that follow simplify this procedure. MULTIPLYING DECIMALS Step 1.

Multiply the numbers as whole numbers ignoring the decimal points.

Step 2.

Count and total the number of decimal places in the multiplier and multiplicand.

Step 3.

Starting at the right in the product, count to the left the number of decimal places totaled in Step 2. Place the decimal point so that the product has the same number of decimal places as totaled in Step 2. If the total number of places is greater than the places in the product, insert zeros in front of the product.

EXAMPLES

Step 1

8.52 (2 decimal places) 6.7 (1 decimal place) 5 964 Step 2 51 12 57.084

2.36 (2 places) .016 (3 places) 1416 236 .03776 Need to add zero

Step 3

Division of Decimals If the divisor in your decimal division problem is a whole number , first place the decimal point in the quotient directly above the decimal point in the dividend. Then divide as usual. If the divisor has a decimal point, complete the steps that follow .

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73

DIVIDING DECIMALS Step 1.

Make the divisor a whole number by moving the decimal point to the right.

Step 2.

Move the decimal point in the dividend to the right the same number of places that you moved the decimal point in the divisor (Step 1). If there are not enough places, add zeros to the right of the dividend.

Step 3.

Place the decimal point in the quotient above the new decimal point in the dividend. Divide as usual.

Step 3

EXAMPLE

1 3.12 2.5冄 32.8.00 25 78 75 30 25 50 50

Step 1

Step 2

Stop a moment and study the above example. Note that the quotient does not change when we multiply the divisor and the dividend by the same number . This is why we can move the decimal point in division problems and always divide by a whole number . Decimal Applications in Foreign Currency The Wall Street Journal clipping “Bar gain Hunting” showed the cost of an Apple iPod in New York City at $299. Using the updated currency table that follows, what would be the cost of the iPod in pounds? Check your answer.

Wall Street Journal © 2004

Wall Street Journal © 2006

EXAMPLE

$299 .53230 159.1577 pounds Check

159.1577 pounds 1.8785 $298.98 (cost of iPod in New York City)

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Multiplication and Division Shortcuts for Decimals The shortcut steps that follow show how to solve multiplication and division problems quickly involving multiples of 10 (10,100, 1,000, 10,000, etc.). SHORTCUTS FOR MULTIPLES OF 10 Multiplication Step 1.

Count the zeros in the multiplier.

Step 2.

Move the decimal point in the multiplicand the same number of places to the right as you have zeros in the multiplier.

Division Step 1.

Count the zeros in the divisor.

Step 2.

Move the decimal point in the dividend the same number of places to the left as you have zeros in the divisor.

In multiplication, the answers are larger than the original number . If Toyota spends $60,000 for magazine advertising, what is the total value if it spends this same amount for 10 years? What would be the total cost?

EXAMPLE

OTHER EXAMPLES

Ric Francis/AP Wide World

$60,000 10 $600,000

(1 place to the right)

6.89 10 68.9

(1 place to the right)

6.89 100 689.

(2 places to the right)

6.89 1,000 6,890.

(3 places to the right)

In division, the answers are smaller than the original number. EXAMPLES

6.89 10 .689

(1 place to the left)

6.89 100 .0689

(2 places to the left)

6.89 1,000 .00689

(3 places to the left)

6.89 10,000 .000689

(4 places to the left)

Next, let’s dissect and solve a word problem.

How to Dissect and Solve a Word Problem The Word Problem May O’Mally went to Sears to buy wall-to-wall carpet. She needs

101.3 square yards for downstairs, 16.3 square yards for the upstairs bedrooms, and 6.2 square yards for the halls. The carpet cost $14.55 per square yard. The padding cost $3.25 per square yard. Sears quoted an installation char ge of $6.25 per square yard. What was May O’Mally’s total cost? By completing the following blueprint aid, we will slowly dissect this word problem. Note that before solving the problem, we gather the facts, identify what we are solving for , and list the steps that must be completed before finding the final answer , along with any key points we should remember . Let’s go to it! The facts

Solving for?

Steps to take

Key points

Carpet needed: 101.3 sq. yd.; 16.3 sq. yd.; 6.2 sq. yd.

Total cost of carpet

Total square yards Cost per square yard Total cost.

Align decimals. Round answer to nearest cent.

Costs: Carpet, $14.55 per sq. yd.; padding, $3.25 per sq. yd.; installation, $6.25 per sq. yd.

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Steps to solving problem 1. Calculate the total number of square yards.

101.3 16.3 6.2 123.8 square yards $14.55 3.25 6.25 $24.05

2. Calculate the total cost per square yard.

123.8 $24.05 $2,977.39

3. Calculate the total cost of carpet.

It’s time to check your progress.

LU 3–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

1. Rearrange vertically and add: 2. Rearrange and subtract: 14, .642, 9.34, 15.87321 28.1549 .885 3. Multiply and round the answer to the nearest tenth: 28.53 17.4 4. Divide and round to the nearest hundredth: 2,182 2.83 Complete by the shortcut method: 5. 14.28 100 6. 9,680 1,000 7. 9,812 10,000 8. Could you help Mel decide which product is the “better buy”? Dog food A: $9.01 for 64 ounces Dog food B: $7.95 for 50 ounces Round to the nearest cent as needed. 9. At Avis Rent-A-Car, the cost per day to rent a medium-size car is $39.99 plus 29 cents per mile. What will it cost to rent this car for 2 days if you drive 602.3 miles? Since the solution shows a completed blueprint, you might use a blueprint also. 10. A trip to Mexico cost 6,000 pesos. What would this be in U.S. dollars? Check your answer.

✓ Solutions 1. 14.00000 .64200 9.34000 15.87321 39.85521

3.

28.53 17.4 11 412 199 71 285 3 496.4 496.422

7 10 1414

2. 28.1549 .8850 27.2699

4.

771.024 771.02

2.83冄218200.000 1981 2010 1981 290 283 7 00 5 66 1 340 1 132

5. 14.28 1,428

6. 9.680 9.680

8. A: $9.01 64 $.14

B: $7.95 50 $.16

7. .9812 .9812 Buy A.

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9. Avis Rent-A-Car total rental charge: The facts

Solving for?

Steps to take

Key points

Cost per day, $39.99. 29 cents per mile. Drove 602.3 miles. 2-day rental.

Total rental charge.

Total cost for 2 days’ rental Total cost of driving Total rental charge.

In multiplication, count the number of decimal places. Starting from right to left in the product, insert decimal in appropriate place. Round to nearest cent.

Steps to solving problem 1. Calculate total costs for 2 days’ rental.

$39.99 2 $79.98

2. Calculate the total cost of driving.

$.29 602.3 $174.667 $174.67

3. Calculate the total rental charge.

$ 79.98 174.67 $254.65

10. 6,000 $.09302 $558.12 Check $558.12 10.7504 6.000.01 pesos due to rounding

LU 3–2a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 78)

1. Rearrange vertically and add: 2. Rearrange and subtract: 16, .831, 9.85, 17.8321 29.5832 .998 3. Multiply and round the answer to the nearest tenth: 29.64 18.2 4. Divide and round to the nearest hundredth: 3,824 4.94 Complete by the shortcut method: 5. 17.48 100 6. 8,432 1,000 7. 9,643 10,000 8. Could you help Mel decide which product is the “better buy”? Dog food A: $8.88 for 64 ounces Dog food B: $7.25 for 50 ounces Round to the nearest cent as needed: 9. At Avis Rent-A-Car, the cost per day to rent a medium-size car is $29.99 plus 22 cents per mile. What will it cost to rent this car for 2 days if you drive 709.8 miles? 10. A trip to Mexico costs 7,000 pesos. What would this be in U.S. dollars? Check your answer.

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic Identifying place value, p. 66 Rounding decimals, p. 67

Key point, procedure, formula 10, 1,

1 1 1 , , , etc. 10 100 1,000

1. Identify place value of digit you want to round. 2. If digit to right of identified digit in Step 1 is 5 or more, increase identified digit by 1; if less than 5, do not change identified digit. 3. Drop all digits to right of identified digit.

Example(s) to illustrate situation .439 in thousandths place value .875 rounded to nearest tenth .9 Identified digit

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

77

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (continued) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Converting decimal fractions to decimals, p. 68

1. Decimal fraction has a denominator with multiples of 10. Count number of zeros in denominator. 2. Zeros show how many places are in the decimal.

8 .008 1,000

Converting proper fractions to decimals, p. 68

1. Divide numerator of fraction by its denominator. 2. Round as necessary.

1 (to nearest tenth) .3 3

Converting mixed numbers to decimals, p. 69

1. Convert fractional part of the mixed number to a decimal. 2. Add converted fractional part to whole number.

1 6 4

Converting pure and mixed decimals to decimal fractions, p. 69

1. Place digits to right of decimal point in numerator of fraction. 2. Put 1 in denominator. 3. Add zeros to denominator, depending on decimal places of original number. For mixed decimals, add fraction to whole number.

.984 (3 places)

1. Vertically write and align numbers on decimal points. 2. Add or subtract digits, starting with right column and moving to the left. 3. Align decimal point in answer with above decimal points.

Add 1.3 2 .4 1.3 2.0 .4 3.7

Adding and subtracting decimals, p. 71

6 .0006 10,000

1. 3.

1 .25 6 6.25 4

984

2.

984 1

984 1,000

Subtract 5 3.9

Multiplying decimals, p. 72

1. Multiply numbers, ignoring decimal points. 2. Count and total number of decimal places in multiplier and multiplicand. 3. Starting at right in the product, count to the left the number of decimal places totaled in Step 2. Insert decimal point. If number of places greater than space in answer, add zeros.

2.48 (2 places) .018 (3 places) 1 984 2 48 .04464

Dividing a decimal by a whole number, p. 73

1. Place decimal point in quotient directly above the decimal point in dividend. 2. Divide as usual.

1.1 42冄 46.2 42 42 42

Dividing if the divisor is a decimal, p. 73

1. Make divisor a whole number by moving decimal point to the right. 2. Move decimal point in dividend to the right the same number of places as in Step 1. 3. Place decimal point in quotient above decimal point in dividend. Divide as usual.

14.2 2.9冄 41.39 29 123 116 79 58 21

4 10

5.0 3.9 1.1

(continues)

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Chapter 3 Decimals

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Shortcuts on multiplication and division of decimals, p. 74

When multiplying by 10, 100, 1,000, and so on, move decimal point in multiplicand the same number of places to the right as you have zeros in multiplier. For division, move decimal point to the left.

4.85 100 485

KEY TERMS

Decimal, p. 65 Decimal fraction, p. 67 Decimal point, p. 65

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

Mixed decimal, p. 69 Pure decimal, p. 69 Repeating decimal, p. 67

LU 3–1a (p. 70) .309 938 8. 1,000 Hundredths 95 Ten-thousandths 9. 17 100 A. .8 325 B. .844 10. 10,000 5. A. .9 11. .13 B. .879 12. .57 6. .0008 13. 13.11 7. .00016 1. 2. 3. 4.

4.85 100 .0485

1. 2. 3. 4. 5.

44.5131 28.5852 539.4 774.09 1,748

Rounding decimals, p. 67

LU 3–2a (p. 76) 6. 8.432 7. .9643 8. Buy A $.14 9. $216.14 10. $651.14

Note: For how to dissect and solve a word problem, see page 74.

Critical Thinking Discussion Questions 1. What are the steps for rounding decimals? Federal income tax forms allow the taxpayer to round each amount to the nearest dollar. Do you agree with this? 2. Explain how to convert fractions to decimals. If 1 out of 20 people buys a Land Rover, how could you write an advertisement in decimals? 3. Explain why .07, .70, and .700 are not equal. Assume you take a family trip to Disney World that covers 500 miles. Show that 108 of the trip, or .8 of the trip, represents 400 miles.

4. Explain the steps in the addition or subtraction of decimals. Visit a car dealership and find the dif ference between two sticker prices. Be sure to check each sticker price for accuracy. Should you always pay the sticker price?

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Identify the place value for the following 3–1. 8.56932

3–2.

293.9438

Round the following as indicated: Tenth 3–3.

.7582

3–4.

4.9832

3–5.

5.8312

3–6.

6.8415

3–7.

6.5555

Hundredth

Thousandth

3–8. 75.9913 Round the following to the nearest cent: 3–9. $4,822.775 3–10. $4,892.046 Convert the following types of decimal fractions to decimals (round to nearest hundredth as needed): 3–11.

9 100

3–12.

3 10

3–13.

91 1,000

3–14.

910 1,000

3–15.

64 100

3–16.

979 1,000

91 100

3–17. 14

Convert the following decimals to fractions. Do not reduce to lowest terms. 3–18. .3

3–19.

.62

3–20.

.006

3–21. .0125

3–22.

.609

3–23.

.825

3–24. .9999

3–25.

.7065

3–28.

6.025

Convert the following to mixed numbers. Do not reduce to the lowest terms. 3–26. 7.4

3–27.

28.48

Write the decimal equivalent of the following: 3–29. Four thousandths

3–30.

Three hundred three and two hundredths

3–31. Eighty-five ten thousandths

3–32.

Seven hundred seventy-five thousandths

3–34.

.005, 2,002.181, 795.41, 14.0, .184

Rearrange the following and add: 3–33. .115, 10.8318, 4.7, 802.481

1

Rearrange the following and subtract: 3–35. 9.2 5.8

3–36.

7 2.0815

3–37.

3.4 1.08

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Estimate by rounding all the way and multiply the following (do not round final answer): 3–38. 6.24 3.9

3–39.

Estimate

.413 3.07 Estimate

3–40. 675 1.92

3–41.

Estimate

4.9 .825 Estimate

Divide the following and round to the nearest hundredth: 3–42. .8931 3

3–43.

29.432 .0012

3–44. .0065 .07

3–45.

7,742.1 48

3–46. 8.95 1.18

3–47.

2,600 .381

Convert the following to decimals and round to the nearest hundredth: 3–48.

1 8

3–49.

1 25

3–50.

5 6

3–51.

5 8

Complete these multiplications and divisions by the shortcut method (do not do any written calculations): 3–52. 96.7 10

3–53.

258.5 100

3–54.

8.51 1,000

3–55. .86 100

3–56.

9.015 100

3–57.

48.6 10

3–58. 750 10

3–59.

3,950 1,000

3–60.

8.45 10

3–61. 7.9132 1,000

WORD PROBLEMS As needed, round answers to the nearest cent. 3–62. A Ford Explorer costs $ 30,000. What would be the cost in pounds in London? Use the currency table and check your answer.

3–63. Ken Griffey, Jr. got 7 hits out of 12 at bats. What was his batting average to the nearest thousandths place?

3–64. An article in The Boston Globe dated January 11, 2007 reported ticket prices for Rod Stewart’s February 3rd concert at the TD Banknorth Garden at $125 per ticket. In addition to the price of a ticket, there is a $14.80 convenience char ge, a $2.50 facility fee, and a $2.50 electronic delivery fee. Richard Evans purchased 4 tickets to the concert. What was Richard’s total cost for the tickets?

3–65. At Wal-Mart, Alice Rose purchased 19.10 yards of ribbon. Each yard costs 89 cents. What was the total cost of the ribbon? 3–66. Douglas Noel went to Home Depot and bought 4 doors at $42.99 each and 6 bags of fertilizer at $8.99 per bag. What was the total cost to Douglas? If Douglas had $300 in his pocket, what does he have left to spend?

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3–67. The stock of Intel has a high of $30.25 today. It closed at $28.85. How much did the stock drop from its high?

3–68. Ed Weld is traveling by car to a comic convention in San Diego. His company will reimburse him $.39 per mile. If Ed travels 906.5 miles, how much will Ed receive from his company? 3–69. Mark Ogara rented a truck from Avis Rent-A-Car for the weekend (2 days). The base rental price was $29.95 per day plus 1412 cents per mile. Mark drove 410.85 miles. How much does Mark owe?

3–70. The Houston Chronicle on January 13, 2007 reported on Texans’ ticket prices to be charged Texas football fans for the 2007 season. The average ticket price will be $60.63, an increase of $2.88 from last year . Before the 2006 season, 22 teams increased ticket prices. The average ticket price was $62.38 with the New England Patriots having the highest average ticket at $90.90 per game. (a) What was the price of ticket to a Texan game last year? (b) How much below the average is the Texans’ ticket. (c) How much above the average are the Patriots tickets? (d) What is the average price between Texans’ tickets and the Patriots’ tickets? Round to the nearest hundredth.

3–71. Pete Allan bought a scooter on the Web for $99.99. He saw the same scooter in the mall for $108.96. How much did Pete save by buying on the Web?

3–72. Russell is preparing the daily bank deposit for his cof fee shop. Before the deposit, the cof fee shop had a checking account balance of $3,185.66. The deposit contains the following checks: No. 1 No. 2

$

99.50

No. 3

$8.75

110.35

No. 4

6.83

Russell included $820.55 in currency with the deposit. What is the coffee shop’s new balance, assuming Russell writes no new checks?

3–73. The Chattanooga Times /Free Press ran a story on US Airways offering lower fares for Chattanooga, Tennessee–New York City flights. US Airways Express is offering a $190 round-trip fare for those who buy tickets in the next couple of weeks. Ticket prices had been running between $230 and $330 round-trip. Mark VanLoh, Airport Authority president, said the new fare is lower than the $219 ticket price of fered by Southwest Airlines. How much would a family of four save using US Airways versus Southwest Airlines?

81

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3–74. Randi went to Lowes to buy wall-to-wall carpeting. She needs 110.8 square yards for downstairs, 31.8 square yards for the halls, and 161.9 square yards for the bedrooms upstairs. Randi chose a shag carpet that costs $14.99 per square yard. She ordered foam padding at $3.10 per square yard. The carpet installers quoted Randi a labor char ge of $3.75 per square yard. What will the total job cost Randi?

3–75. Art Norton bought 4 new Aquatred tires at Goodyear for $89.99 per tire. Goodyear char ged $3.05 per tire for mounting, $2.95 per tire for valve stems, and $3.80 per tire for balancing. IfArt paid no sales tax, what was his total cost for the 4 ires? t

3–76. Shelly is shopping for laundry detergent, mustard, and canned tuna. She is trying to decide which of two products is the better buy. Using the following information, can you help Shelly? Laundry detergent A

Mustard A

Canned tuna A

$2.00 for 37 ounces

$.88 for 6 ounces

$1.09 for 6 ounces

Laundry detergent B

Mustard B

Canned tuna B

$2.37 for 38 ounces

$1.61 for 1221 ounces

$1.29 for 834 ounces

3–77. Roger bought season tickets for weekend games to professional basketball games. The cost was $945.60. The season package included 36 home games. What is the average price of the tickets per game? Round to the nearest cent. Marcelo, Roger ’s friend, offered to buy 4 of the tickets from Roger. What is the total amount Roger should receive?

3–78. A nurse was to give each of her patients a 1.32-unit dosage of a prescribed drug. The total remaining units of the drug at the hospital pharmacy were 53.12. The nurse has 38 patients. Will there be enough dosages for all her patients?

3–79. Audrey Long went to Japan and bought an animation cel of Mickey Mouse. The price was 25,000 yen. What is the price in U.S. dollars? Check your answer.

ADDITIONAL SET OF WORD PROBLEMS 3–80. On Monday, the stock of IBM closed at $88.95. At the end of trading on Tuesday, IBM closed at $94.65. How much did the price of stock increase from Monday to Tuesday?

3–81. Tie Yang bought season tickets to the Boston Pops for $698.55. The season package included 38 performances. What is the average price of the tickets per performance? Round to nearest cent. Sam, Tie’s friend, offered to buy 4 of the tickets from Tie. What is the total amount Tie should receive? 3–82. Morris Katz bought 4 new tires at Goodyear for $95.49 per tire. Goodyear also charged Morris $2.50 per tire for mounting, $2.40 per tire for valve stems, and $3.95 per tire for balancing. Assume no tax. What was Morris’s total cost for the 4 tires?

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3–83. The Denver Post reported that Xcel Energy is revising customer charges for monthly residential electric bills and gas bills. Electric bills will increase $3.32. Gas bills will decrease $1.74 a month. (a) What is the resulting new monthly increase for the entire bill? (b) If Xcel serves 2,350 homes, how much additional revenue would Excel receive each month?

3–84. Steven is traveling to a computer convention by car. His company will reimburse him $.29 per mile. If Steven travels 890.5 miles, how much will he receive from his company? 3–85. Gracie went to Home Depot to buy wall-to-wall carpeting for her house. She needs 104.8 square yards for downstairs, 17.4 square yards for halls, and 165.8 square yards for the upstairs bedrooms. Gracie chose a shag carpet that costs $13.95 per square yard. She ordered foam padding at $2.75 per square yard.The installers quoted Gracie a labor cost of $5.75 per square yard in installation. What will the total job cost Gracie?

3–86. On February 1, 2007 The Kansas City Star, reported the Dow Jones Industrial Average rose 98.38 points from the previous day, and closed at 12,621.69. The blue-chip index set a trading high, at 12,657.02 and just missed the record of 12,621.77 points. (a) What were closing points on January 31, 2007? (b) This closing on February 1 was how many points from the record? (c) What were the average points from January 31, 2007’s lowest to February 1, 2007’s highest? Round to the nearest hundredth.

CHALLENGE PROBLEMS 3–87. The Miami Herald ran a story on Carnival Cruise’s profit per share. For the third quarter, Carnival earned $734.3 million with 815.9 million shares of stock outstanding. Last year, earnings were $500.8 million, or 85 cents a share. (a) How much were the earnings per shareholder for the third quarter? Round to the nearest cent.b) ( How many shareholders did Carnival Cruise have last year? Round to the nearest hundred thousands. Check your answers.

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3–88. Jill and Frank decided to take a long weekend in NewYork. City Hotel has a special getaway weekend for $79.95.The price is per person per night, based on double occupancy. The hotel has a minimum two-night stay. For this price, Jill and Frank will receive $50 credit toward their dinners at City’s Skylight Restaurant. Also included in the package is a $3.99 credit per person toward breakfast for two each morning. Since Jill and Frank do not own a car , they plan to rent a car . The car rental agency charges $19.95 a day with an additional charge of $.22 a mile and $1.19 per gallon of gas used. The gas tank holds 24 gallons. From the following facts, calculate the total expenses of Jill and Frank (round all answers to nearest hundredth or cent as appropriate). Assume no taxes. Car rental (2 days):

Dinner cost at Skylight

Beginning odometer reading

4,820

Ending odometer reading

4,940

Beginning gas tank: Gas tank on return:

3 4 1 2

$182.12

Breakfast for two:

full.

Morning No. 1

24.17

Morning No. 2

26.88

full.

Tank holds 24 gallons.

DVD SUMMARY PRACTICE TEST 1. Add the following by translating the verbal form to the decimal equivalent. (p. 71) Three hundred thirty-eight and seven hundred five thousandths Nineteen and fifty-nine hundredths Five and four thousandths Seventy-five hundredths Four hundred three and eight tenths Convert the following decimal fractions to decimals. (p. 68) 2.

7 10

3.

7 100

4.

7 1,000

Convert the following to proper fractions or mixed numbers. Do not reduce to the lowest terms. (p. 68) 5. .9

6. 6.97

7. .685

Convert the following fractions to decimals (or mixed decimals) and round to the nearest hundredth as needed. (p. 68) 8.

84

2 7

9.

1 8

4 7

10. 4

11.

1 13

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12. Rearrange the following decimals and add. (p. 71) 5.93,

11.862,

284.0382,

88.44

13. Subtract the following and round to the nearest tenth. (p. 71) 13.111 3.872 14. Multiply the following and round to the nearest hundredth. (p. 72) 7.4821 15.861 15. Divide the following and round to the nearest hundredth. (p. 73) 203,942 5.88 Complete the following by the shortcut method. (p. 74) 16. 62.94 1,000 17. 8,322,249.821 100 18. The average pay of employees is $795.88 per week. Lee earns $820.44 per week. How much is Lee’ s pay over the average? (p. 71) 19. Lowes reimburses Ron $.49 per mile. Ron submitted a travel log for a total of 1,910.81 miles. How much will Lowes reimburse Ron? Round to the nearest cent. (p. 72) 20. Lee Chin bought 2 new car tires from Michelin for $182.1 1 per tire. Michelin also char ged Lee $3.99 per tire for mounting, $2.50 per tire for valve stems, and $4.10 per tire for balancing. What is Lee’s final bill? (p. 72)

21. Could you help Judy decide which of the following products is cheaper per ounce? (p. 73) Canned fruit A

Canned fruit B

$.37 for 3 ounces

$.58 for 334 ounces

22. Paula Smith bought an iPod for 350 euros. What is this price in U.S. dollars? (p. 73) 23. Google stock traded at a high of $438.22 and closed at $410.12. How much did the stock fall from its high? (p. 71)

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Personal Finance A KIPLINGER APPROACH flying cross-country to New York City. How can I book a cheap ticket today that will let me change my flight dates later without paying hefty fees? —CHARLES KUTTNER, Portland, Ore.

T R AV E L S L E U T H

| You don’t have to break the

bank to phone home from the Alps. By Sean O’Neill

Call U.S. for less uring a recent trip to Germany and Austria, I called the States five times, using pay phones and charging my Visa card. I returned home to find a $105 bill for a 20-minute call and a $192 bill for all my other calls, which each lasted fewer than ten minutes. What might I have done differently? —RAY TAYLOR, Bend, Ore.

JULIETTE BORDA

D

Ouch. Your credit card is one of many that charge sky-high rates for placing international phone calls. For your next trip, try this: If you own a cell phone, ask your wireless carrier if you can use it overseas for a low rate. Cingular (the nation’s largest provider) and T-Mobile have adopted GSM technology, the standard for much of the world. If you own a relatively new phone and use one of these providers, you can usually place calls from overseas. Check first

with your provider, and ask to buy a temporary plan that will allow you to call home cheaply while traveling abroad. Cingular, for instance, lets many of its customers call home from several European countries at rates of about $1 a minute, plus a fee of about $6 for each month of travel. Or buy a prepaid phone card that lets you call for rates of about 10 cents to 15 cents per minute. We like the $20 cards from Nobel (www.nobel .com) and $10 cards from OneSuite (www.onesuite.com). You can use any phone overseas, but note that a hotel phone may come with a high fee.

REBOOK FOR LESS

o get the lowest fares, I usually need to book air tickets well in advance. But my next trip will be to celebrate the birth of my granddaughter, and I can’t know for sure the date I’ll be

T

Congratulations on becoming a grandparent. If you can make an educated guess about when you’re most likely to travel, you’ll save by booking early on a discount airline. In the best case, you’ll have a cheap ticket and you’ll arrive at the right time. In the worst case, you’ll need to rebook, your new flight date will come with a higher round-trip fare—which is typical—and you’ll have to pay the difference plus a rebooking fee. But here’s why booking with a discounter is a good idea, even if a discounter and a major airline offer similar advance ticket prices: You’ll face lower 11th-hour fares and the rebooking fee will be smaller. Major carriers typically charge $100 to rebook, while top discounters charge less. Rebooking fees for ATA are $50 and for JetBlue, $30. Southwest doesn’t charge a fee. Given that you know the due date of your granddaughter, we suggest you book your trip from Portland to New York City now (more than a month ahead). You’ll pay $403 before taxes and fees of $50 on JetBlue. In the past year on this route, fares booked just days before departure weren’t much higher than $400 before taxes, says FareCompare.com, a site that gives you the lowest average fares available on most routes. We estimate you’ll save between $75 and $300 by rebooking on JetBlue instead of a major airline. What if no discounter serves your route? If you have to go with a major airline and you need to rebook, ask the agent to waive the rebooking fee when you call to change flight dates. Agents often have discretion to waive fees— but you’ll always pay the difference between the old and new fare. Have a money-related travel question? Write us at [email protected]

BUSINESS MATH ISSUE Kiplinger’s © 2006

Prepaid phone cards do not really save you money. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A Calculate market total.

2006 Wall Street Journal ©

PROJECT B Show how to calculate A. $914.14 C. $13,318.97 B. $332.24 D. $353.84

b site text We he e e S : s t T t Projec /slater9e) and e. Interne m ce Guid r o u .c o e s h e h R t .m e (www Intern ss Math Busine

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CHAPTER

4

Banking

LEARNING UNIT OBJECTIVES LU 4–1: The Checking Account • Define and state the purpose of signature cards, checks, deposit slips, check stubs, check registers, and endorsements (pp. 89–91). • Correctly prepare deposit slips and write checks (pp. 91–92).

LU 4–2: Bank Statement and Reconciliation Process; Trends in Online Banking • Define and state the purpose of the bank statement (pp. 93–94). • Complete a check register and a bank reconciliation (pp. 96–98). • Explain the trends in online banking (pp. 98–100).

Wall Street Jo urnal © 2005

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Learning Unit 4–1

Mark Lennihan/AP Wide World

89

Too often people think their bank is their best friend. You should remember that your bank is a business. The banking industry is very competitive. Note in the Wall Street Journal clipping “Bank of America to Pay $2.5 Billion for China Foothold” how quickly the banking sectors are changing all over the world to be more competitive. An important fixture in today’s banking is the automatic teller machine (ATM). The ability to get instant cash is a convenience many bank customers enjoy. However , more than half of the ATM customers do not like to deposit Wall Street Journal © 2005 checks because they are afraid the checks will not be correctly deposited to their account. Bank of America, Bank One, and Wells Far go are testing new ATMs that accept a check, scan the check, and print a receipt with a photographic image of the check. When these machines are widely available, they will eliminate the fear of depositing checks. The effect of using an ATM card is the same as using a debit card—both transactions result in money being immediately deducted from your checking account balance. As a result, debit cards have been called enhanced ATM cards or check cards. Often banks charge fees for these card transactions. The frequent complaints of bank customers have made many banks offer their ATMs as a free service, especially if customers use an ATM in the same network as their bank. Some banks char ge fees for using another bank’ s ATM. Remember that the use of debit cards involves planning. As check cards, you must be aware of your bank balance every time you use a debit card. Also, if you use a credit card instead of a debit card, you can only be held responsible for $50 of illegal char ges; and during the time the credit card company investigates the illegal char ges, they are removed from your account. However , with a debit card, this legal limit only applies if you report your card lost or stolen within two business days. We should add that debit cards are profitable for banks. When shopping, if you use a debit card that does not require a personal identification number , the store pays a fee to the bank that issued the card—usually from 1.4 to 2 cents on the dollar . This chapter begins with a discussion of the checking account. You will follow Molly Kate as she opens a checking account for Gracie’ s Natural Superstore and performs her banking transactions. Pay special attention to the procedure used by Gracie’ s to reconcile its checking account and bank statement. This information will help you reconcile your checkbook records with the bank’ s record of your account. The chapter concludes by discussing how the trends in online banking may af fect your banking procedures.

Learning Unit 4–1: The Checking Account A check or draft is a written order instructing a bank, credit union, or savings and loan institution to pay a designated amount of your money on deposit to a person or an or ganization. Checking accounts are of fered to individuals and businesses. Businesses may be charged $.39 per check received for a business transaction. Note that the business checking account usually receives more services than the personal checking account. Most small businesses depend on a checking account for ef ficient record keeping. In this learning unit you will follow the checking account procedures of a newly or ganized small business. You can use many of these procedures in your personal check writing. You will also learn about e-checks—a new trend.

Opening the Checking Account Molly Kate, treasurer of Gracie’ s Natural Superstore, went to Ipswich Bank to open a business checking account. The bank manager gave Molly a signature card. The signature card contained space for the company’ s name and address, references, type of account, and the signature(s) of the person(s) authorized to sign checks. If necessary , the bank will use the signature card to verify that Molly signed the checks. Some companies authorize more than one person to sign checks or require more than one signature on a check.

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FIGURE

4.1

Deposit slip

DEPOSIT TICKET

CASH

➞

LIST CHECK SINGLY

53-7058 53-7058

Gracie's Natural Superstore 80 Garfield St. Bartlet, NH 01835 Preprinted numbers in magnetic ink identify bank number, routing and sorting of the check, and Gracie's Natural Superstore account number

DATE

March 4

20

1,800 00 200 00

09

DEPOSITS MAY NOT BE AVAILABLE FOR IMMEDIATE WITHDRAWAL USE OTHER SIDE FOR ADDITIONAL LISTING ENTER TOTAL HERE

TOTAL FROM OTHER SIDE TOTAL ITEMS

TOTAL

2,000 00

BE SURE EACH ITEM IS PROPERLY ENDORSED.

IPSWICH BANK ipswichbank.com 211370587 88190662 CHECKS AND OTHER ITEMS ARE RECIEVED FOR DEPOSIT SUBJECT TO THE PROVISIONS OF THE UNIFORM COMMERCIAL CODE OR ANY APPLICABLE COLLECTION AGREEMENT.

The 53-7058 is taken from the upper right corner of the check from the top part of the fraction. This number is known as the American Bankers Association transit number. The 53 identifies the city or state where the bank is located and the 7058 identifies the bank.

Molly then lists on a deposit slip (or deposit ticket) the checks and/or cash she is depositing in her company’ s business account. The bank gave Molly a temporary checkbook to use until the company’ s printed checks arrived. Molly also will receive preprinted checking account deposit slips like the one shown in Figure 4.1. Since the deposit slips are in duplicate, Molly can keep a record of her deposit. Note that the increased use of making deposits at ATM machines has made it more convenient for people to make their deposits. Writing business checks is similar to writing personal checks. Before writing any checks, however, you must understand the structure of a check and know how to write a check. Carefully study Figure 4.2. Note that the verbal amount written in the check should FIGURE

4.2

The structure of a check

Date check is written To whom check is payable or payee

Verbal form of amount of check. Note spacing and use of "and" to represent the decimal

Bank ordered to pay is drawee

Bank number and customer's number 53-7058/2113

Gracie's Natural Superstore 80 Garfield St. Bartlet, NH 01835

88190662 DATE

xx

$ 6,000 100

xx

Six thousand and 100

DOLLARS

Amount of check

Security features detailed on back

IPSWICH BANK ipswichbank.com MEMO

Molly Kate

Office Equipment

211370587 88190662 2 Bank number printed with magnetic ink for computer processing matches printed number at upper right-hand corner of check above the date of the check. Customer account number below bank number.

Gracie's Natural Superstore's account number

Preprinted check number

March 8, 2009

Staples Corporation PAY TO THE ORDER OF

633

633 Preprinted check number

MP

0000006000 0000006000 When bank processes check, the 6000 is imprinted here. Note that this should match what is written for amount of check.

Signature–this is the same as on the signature card One who writes check is called the drawer

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FIGURE

A. Blank Endorsement

4.3

Types of common endorsements

Gracie's Natural Superstore 88190662

91

The company stamp or a signature alone on the back left side of a check legally makes the check payable to anyone holding the check. It can be further endorsed. This is not a safe type of endorsement.

B. Full Endorsement Pay to the order of Ipswich Bank Gracie's Natural Superstore 88190662

Safer type of endorsement since Gracie's Natural Superstore indicates the name of the company or person to whom the check is to be payable to. Only the person or company named in the endorsement can transfer the check to someone else.

C. Restrictive Endorsement Pay to the order of Ipswich Bank For deposit only Gracie's Natural Superstore 88190662

Check Stub It should be completed before the check is written. xx

633 6000 100 09 March 8 Staples Corp. Other Furniture 14,416 24

14,416 24 6,000 00 8,416 24

Safest endorsement for businesses. Gracie's stamps the back of the check so that this check must be deposited in the firm's bank account. This limits any further negotiation of the check.

match the figure amount. If these two amounts are dif ferent, by law the bank uses the verbal amount. Also, note the bank imprint on the bottom right section of the check. When processing the check, the bank imprints the check’ s amount. This makes it easy to detect bank errors.

Using the Checking Account Once the check is written, the writer must keep a record of the check. Knowing the amount of your written checks and the amount in the bank should help you avoid writing a bad check. Business checkbooks usually include attached check stubs to keep track of written checks. The sample check stub in the mar gin shows the information that the check writer will want to record. Some companies use a check register to keep their check records instead of check stubs. Figure 4.6 (p. 96) shows a check register with a ✔ column that is often used in balancing the checkbook with the bank statement (Learning Unit 4–2). Gracie’s Natural Superstore has had a busy week, and Molly must deposit its checks in the company’ s checking account. However , before she can do this, Molly must endorse, or sign, the back left side of the checks. Figure 4.3 explains the three types of check endorsements: blank endorsement, full endorsement, and restrictive endorsement. These endorsements transfer Gracie’ s ownership to the bank, which collects the money from the person or company issuing the check. Federal Reserve regulation limits all endorsements to the top 1 12 inches of the trailing edge on the back left side of the check. After the bank receives Molly’ s deposit slip, shown in Figure 4.1 (p. 90), it increases (or credits) Gracie’ s account by $2,000. Often Molly leaves the deposit in a locked bag in a night depository. Then the bank credits (increases) Gracie’ s account when it processes the deposit on the next working day .

E-Checks—A New Trend Before concluding this unit, let’ s look at a new trend using e-checks. In the Wall Street Journal clipping “Taking Rain Check on ‘E-Checks,’” p. 92, we see that retailers are now trying to get your bank account number and “routing” number at the bottom of your checks so bills can be paid directly from your bank account when your bills are due.

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Wall Street Journal © 2005

Let’s check your understanding of the first unit in this chapter .

LU 4–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

113

Complete the following check and check stub for Long Company . Note the $9,500.60 balance brought forward on check stub No. 1 13. You must make a $690.60 deposit on May 3. Sign the check for Roland Small. Date

Check no.

Amount

Payable to

For

June 5, 2009

113

$83.76

Angel Corporation

Rent

Long Company 22 Aster Rd. Salem, MA 01970

113

9,500 60

IPSWICH BANK ipswichbank.com 011000138

14 0380 113

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Learning Unit 4–2

✓

Solution with page reference to check your progress

83.76 113 09 June 5 Angel Corp. Rent

113

Long Company 22 Aster Rd. Salem, MA 01970

June 5 Angel Corporation

9,500 60 690 60

Eighty-three and

09 83

76 100

76 100

IPSWICH BANK ipswichbank.com

10,191 20 83 76 10,107 44

LU 4–1a

93

Roland Small

Rent

011000138

14 0380 113

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 101)

Complete the following check and stub for Long Company . Note the $10,800.80 balance brought forward on check stub No. 1 13. You must make an $812.88 deposit on May 3. Sign the check for Roland Small. Date July 8, 2009

113

Check No. 113

Amount $79.88

Long Company 22 Aster Rd. Salem, MA 01970

Payable to Lowe Corp

For Advertising

113

10,800 80

IPSWICH BANK ipswichbank.com 011000138

14 0380 113

Learning Unit 4–2: Bank Statement and Reconciliation Process; Trends in Online Banking This learning unit is divided into two sections: (1) bank statement and reconciliation process, and (2) trends in online banking . The bank statement discussion will teach you why it was important for Gracie’ s Natural Superstore to reconcile its checkbook balance with the balance reported on its bank balance. Note that you can also use this reconciliation process in reconciling your personal checking account and avoiding the expensive error of an overdrawn account. To introduce you to the “T rends in Online Banking” section, we have included the following Wall Street Journal clipping “Financial Institutions Give Cash to Induce Customers to Use Web-Based Services,” p. 94. As you probably know , financial institutions favor online banking because it is less expensive for them.

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Wall Street Journal © 2004

Would you bank online if you were given a cash incentive? Many customers are still concerned about security issues when banking online. In 2006, more than 40 million households were banking online, which leaves more than 60 million customers who do not bank online.

Bank Statement and Reconciliation Process Each month, Ipswich Bank sends Gracie’ s Natural Superstore a bank statement (Figure 4.4, p. 95). We are interested in the following: 1. 2. 3. 4.

Beginning bank balance. Total of all the account increases. Each time the bank increases the account amount, it credits the account. Total of all account decreases. Each time the bank decreases the account amount, it debits the account. Final ending balance.

Due to dif ferences in timing, the bank balance on the bank statement frequently does not match the customer ’s checkbook balance. Also, the bank statement can show transactions that have not been entered in the customer ’s checkbook. Figure 4.5, p. 95, tells you what to look for when comparing a checkbook balance with a bank balance.

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Learning Unit 4–2

FIGURE

95

4.4

Bank statement Ipswich Bank 1 Pleasant St. Bartlett, NH 01835

Account Statement

Gracie's Natural Superstore 80 Garfield St. Bartlett, NH 01835

Checking Account: 881900662 Checking Account Summary as of 3/31/09

Beginning Balance

Total Deposits

Total Withdrawals

Service Charge

Ending Balance

$13,112.24

$8,705.28

$9,926.00

$28.50

$11,863.02

Checking Accounts Transactions Deposits

Date

Amount

Deposit Deposit Deposit EFT leasing: Bakery dept. EFT leasing: Meat dept. Interest

3/05 3/05 3/09 3/18 3/27 3/31

2,000.00 224.00 389.20 1,808.06 4,228.00 56.02

Charges

Date

Amount

Service charge: Check printing EFT: Health insurance NSF

3/31 3/21 3/21

28.50 722.00 104.00

Checks Number 301 633 634 635 636 637

FIGURE

Daily Balance Date 3/07 3/13 3/13 3/11 3/18 3/31

4.5

Reconciling checkbook with bank statement

Amount 200.00 6,000.00 300.00 200.00 200.00 2,200.00

Date 2/28 3/05 3/07 3/09 3/11 3/13

Balance 13,112.24 15,232.24 14,832.24 15,221.44 15,021.44 8,721.44

Checkbook balance

Date 3/18 3/21 3/28 3/31

Balance 10,529.50 9,807.50 14,035.50 11,863.02

Bank balance

⫹ EFT (electronic funds transfer)

⫺ NSF check

⫹ Deposits in transit

⫹ Interest earned

⫺ Online fees

⫺ Outstanding checks

⫹ Notes collected

⫺ Automatic payments*

⫾ Bank errors

⫹ Direct deposits

⫺ Overdrafts

⫺ ATM withdrawals

⫺ Service charges

⫺ Automatic withdrawals

⫺ Stop payments‡

†

⫾ Book errors§ *Preauthorized payments for utility bills, mortgage payments, insurance, etc. † Overdrafts occur when the customer has no overdraft protection and a check bounces back to the company or person who received the check because the customer has written a check without enough money in the bank to pay for it. ‡ A stop payment is issued when the writer of check does not want the receiver to cash the check. § If a $60 check is recorded at $50, the checkbook balance must be decreased by $10.

Gracie’s Natural Superstore is planning to of fer to its employees the option of depositing their checks directly into each employee’ s checking account. This is accomplished through the electronic funds transfer (EFT)—a computerized operation that electronically transfers funds among parties without the use of paper checks. Gracie’ s, who sublets space in the store, receives rental payments by EFT . Gracie’s also has the bank pay the store’ s health insurance premiums by EFT .

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To reconcile the difference between the amount on the bank statement and in the checkbook, the customer should complete a bank reconciliation. Today, many companies and home computer owners are using software such as Quicken and QuickBooks to complete their bank reconciliation. However , you should understand the following steps for manually reconciling a bank statement. RECONCILING A BANK STATEMENT Step 1. Identify the outstanding checks (checks written but not yet processed by the bank). You can use the ✓ column in the check register (Figure 4.6) to check the canceled checks listed in the bank statement against the checks you wrote in the check register. The unchecked checks are the outstanding checks. Step 2. Identify the deposits in transit (deposits made but not yet processed by the bank), using the same method in Step 1. Step 3. Analyze the bank statement for transactions not recorded in the check stubs or check registers (like EFT). Step 4. Check for recording errors in checks written, in deposits made, or in subtraction and addition. Step 5. Compare the adjusted balances of the checkbook and the bank statement. If the balances are not the same, repeat Steps 1–4.

Molly uses a check register (Figure 4.6) to keep a record of Gracie’ s checks and deposits. By looking at Gracie’ s check register , you can see how to complete Steps 1 and 2 above. The explanation that follows for the first four bank statement reconciliation steps will help you understand the procedure. FIGURE

4.6

Gracie’s Natural Superstore check register

RECORD ALL CHARGES OR CREDITS THAT AFFECT YOUR ACCOUNT NUMBER

DATE 2007

DESCRIPTION OF TRANSACTION

3/04

Deposit

3/04

Deposit

PAYMENT/DEBIT (–)

冑

$

BALANCE DEPOSIT/CREDIT $ (+) $

2,000 00 224 00

Staples Company

6,000 00 ✓

634 3/09

Health Foods Inc.

1,020 00 ✓

Deposit

389 20

635 3/ 10

Liberty Insurance

200 00 ✓

636 3/18

Ryan Press

200 00 ✓

637 3/29

Logan Advertising

3/30

(–) $

633 3/08

3/09

FEE (IF ANY)

2,200 00 ✓

Deposit

3,383 26

638 3/31

Sears Roebuck

572 00

639 3/31

Flynn Company

638 94

640 3/31

Lynn s Farm

166 00

641 3/31

Ron s Wholesale

406 28

642 3/31

Grocery Natural, Inc.

917 06

REMEMBER TO RECORD AUTOMATIC PAYMENTS/DEPOSITS ON DATE AUTHORIZED.

12,912 24 00 14,912 24 + 224 00 15,136 24 – 6,000 00 9,136 24 – 1,020 00 8,116 24 + 389 20 8,505 44 – 200 00 8,305 44 – 200 00 8,105 44 – 2,200 00 5,905 44 + 3,383 26 9,288 70 – 572 00 8,716 70 – 638 94 8,077 76 – 166 00 7,911 76 – 406 28 7,505 48 – 917 06 $6,588 42 + 2,000

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Learning Unit 4–2

97

Step 1. Identify Outstanding Checks Outstanding checks are checks that Gracie’ s Natural Superstore has written but Ipswich Bank has not yet recorded for payment when it sends out the bank statement. Gracie’ s treasurer identifies the following checks written on 3/31 as outstanding: No. 638 No. 639 No. 640 No. 641 No. 642

$572.00 638.94 166.00 406.28 917.06

Step 2. Identify Deposits in Transit Deposits in transit are deposits that did not reach Ipswich Bank by the time the bank prepared the bank statement. The March 30 deposit of $3,383.26 did not reach Ipswich Bank by the bank statement date. You can see this by comparing the company’ s bank statement with its check register . Step 3. Analyze Bank Statement for Transactions Not Recorded in Check Stubs or Check Register The bank statement of Gracie’ s Natural Superstore (Figure 4.4, p. 95) begins with the deposits, or increases, made to Gracie’ s bank account. Increases to accounts are known as credits. These are the result of a credit memo (CM). Gracie’s received the following increases or credits in March: 1. 2.

EFT leasing: $1,808.06 and $4,228.00. Interest credited: $56.02.

Each month the bakery and meat departments pay for space they lease in the store. Gracie’s has a checking account that pays interest; the account has earned $56.02.

When Gracie’ s has char ges against her bank account, the bank decreases, or debits, Gracie’s account for these char ges. Banks usually inform customers of a debit transaction by a debit memo (DM). The following items will result in debits to Gracie’ s account: 1. 2. 3.

Service charge: $28.50 EFT payment: $722. NSF check: $104.

The bank charged $28.50 for printing Gracie’s checks. The bank made a health insurance payment for Gracie’ s. One of Gracie’s customers wrote Gracie’s a check for $104. Gracie’s deposited the check, but the check bounced for nonsufficient funds (NSF). Thus, Gracie’s has $104 less than it figured.

Step 4. Check for Recording Errors The treasurer of Gracie’ s Natural Superstore, Molly Kate, recorded check No. 634 for the wrong amount—$1,020 (see the check register). The bank statement showed that check No. 634 cleared for $300. To reconcile Gracie’s checkbook balance with the bank balance, Gracie’s must add $720 to its checkbook balance. Neglecting to record a deposit also results in an error in the company’ s checkbook balance. As you can see, reconciling the bank’s balance with a checkbook balance is a necessary part of business and personal finance. Step 5. Completing the Bank Reconciliation Now we can complete the bank reconciliation on the back side of the bank statement as shown in Figure 4.7 (p. 98). This form is usually on the back of a bank statement. If necessary, however, the person reconciling the bank statement can construct a bank reconciliation form similar to Figure 4.8 (p. 98).

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FIGURE

4.7

Reconciliation process

$572 638 166 406 917

638 639 640 641 642

11,863.02

00 94 00 28 06

3,383.26

15,246.28 2,700.28 12,546.00 $

12,546.00

$2,700 28 $6,588.42

Checkbook balance

$6,036.06 56.02 720.00 6,812.08 $ 28.50 722.00 104.00 854.50 Ending checkbook balance $12,546.00 + EFT leasing + Interest + Checkbook error – Service charge – EFT: health insurance – NSF

FIGURE

4.8

GRACIE’S NATURAL SUPERSTORE Bank Reconciliation as of March 31, 2009

Bank reconciliation

Bank balance

Checkbook balance Gracie’s checkbook balance

$6,588.42

Bank balance

$11,863.02

Add:

Add:

Deposit in transit, 3/30

EFT leasing: Bakery dept. $1,808.06

3,383.26 $15,246.28

EFT leasing: Meat dept. 4,228.00 Interest Error: Overstated check No. 634

56.02 720.00

$ 6,812.08 $13,400.50 Deduct:

Deduct: Service charge NSF check

104.00

EFT health insurance payment 722.00

Reconciled balance

Outstanding checks:

$ 28.50

854.50

$12,546.00

No. 638

$572.00

No. 639

638.94

No. 640

166.00

No. 641

406.28

No. 642

917.06

Reconciled balance

2,700.28 $12,546.00

Trends in Online Banking In the introduction to this learning unit, you learned that financial institutions are of fering cash incentives to induce customers to use web-based services. We also said that many customers are still concerned about security issues when banking online.

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Learning Unit 4–2

99

To combat fraud online, the following Wall Street Journal clipping “‘V irtual Debit Card’ Aims to Combat Online Fraud,” discusses a new checking account with a “virtual debit card.” As you can see from the clipping, this “eSpend” card is in addition to the regular debit card. With the eSpend card, you can get a daily limit for purchases online, over the phone, or by mail order .

Wall Street Journal © 2005

Today, with the increased use of computers, the trends in online banking are changing. In this section, you will learn about three of the changes. Changes are occurring in the use of the Internet—there is an increase in Internet banking. Customers are also seeing that new bank legislation has resulted in an increase in the speed of check clearing. Finally , the role of middlemen in the Internet is changing. Increased Use of Internet As time passes, people are realizing the convenience of using online banking in the Internet. An enormous amount of time is saved when people can avoid most of their trips to the bank by using the Internet. Also, Internet banking does not keep bank hours. The most popular reason to use Internet banking is the ability to quickly pay your bills. The Internet method of bill paying has several advantages. You do not have to write checks, save the envelopes that come with bills, check to see if you have stamps to put on the envelopes, or be concerned that payments for bills will not reach their destination in time to make a deadline. Online banking also has other advantages. You can transfer money between your accounts, and you can check your transactions and balances. The broad objective of banks is to make your online banking experience the same as your physical bank experience. This objective, of course, cannot be completely fulfilled. If you want to make deposits or withdraw funds, you must do this by wire, mail, ATM, or a physical appearance at your bank. On some online bank websites, however , you can apply for loans and get information on other bank financing. New Bank Legislation Although banks are doing everything they can to get people to avoid writing checks, many people do not want to give up their check writing. In a recent Kansas City Bank presentation, it was emphasized that today more checks than ever are processed. To reduce the costs

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of paper checks, some banks no longer return canceled checks. Instead, these banks use a safekeeping procedure involving holding the checks for a period of time, keeping microfilm copies of checks for at least a year , and returning a check or a photocopy for a small fee. In 2003, a new piece of legislation known as Check 21 was signed into law . This legislation means that canceled checks can now be transferred electronically to customers rather than bundling them up and sending them through the mail. The electronic transfer of their canceled checks gives customers time to study all their canceled checks without the concern for safekeeping.

Reproduced with permission of PayPal, Inc. Copyright © 2007 PAYPAL, INC. ALL RIGHTS RESERVED.

LU 4–2

Role of Middlemen on the Internet Have you ever made a purchase on eBay? eBay is a popular website for many people who use the company to buy and sell items at auction and also to buy and sell items outright. After you make a purchase on eBay , you have the option to pay by check, credit card, or use PayPal. Many people use PayPal because they believe it is safer than their check or credit card. Here, you can see a partial Web screen of eBay with PayPal. PayPal acts like a third person operating between eBay and the seller . Since PayPal performs a service, it char ges customers a fee. Obviously , banks and credit unions object to letting PayPal dominate the field and handle these transactions for a fee. As a result, PayPal recently agreed to be purchased by eBay . In the next five years, banks hope to eliminate middlemen like PayPal. At that time you might be able to e-mail cash to Internetenabled ATM machines or use a cell phone to send your money . The Practice Quiz that follows will test your knowledge of the bank reconciliation process.

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Rosa Garcia received her February 3, 2009, bank statement showing a balance of $212.80. Rosa’s checkbook has a balance of $929.15. The bank statement showed that Rosa had an ATM fee of $12.00 and a deposited check returned fee of $20.00. Rosa earned interest of $1.05. She had three outstanding checks: No. 300, $18.20; No. 302, $38.40; and No. 303, $68.12. A deposit for $810.12 was not on her bank statement. Prepare Rosa Garcia’ s bank reconciliation. ROSA GARCIA Bank Reconciliation as of February 3, 2009

Checkbook balance Rosa’s checkbook balance

Bank balance $929.15

Add:

Bank balance

$ 212.80

Add:

Interest

1.05

Deposit in transit

810.12

$930.20 Deduct: Deposited check returned fee ATM

Reconciled balance

LU 4–2a

$1,022.92 Deduct: Outstanding checks:

$20.00 12.00

32.00

$898.20

No. 300

$18.20

No. 302

38.40

No. 303

68.12

Reconciled balance

124.72 $ 898.20

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 101)

Earl Miller received his March 8, 2009, bank statement, which had a $300.10 balance. Earl’s checkbook has a $1,200.10 balance. The bank statement showed a $15.00 ATM fee and a $30.00 deposited check returned fee. Earl earned $24.06 interest. He had three outstanding checks: No. 300, $22.88; No. 302, $15.90; and No. 303, $282.66. A deposit for $1,200.50 was not on his bank statement. Prepare Earl’ s bank reconciliation.

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

101

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Types of endorsements, p. 91

Blank: Not safe; can be further endorsed.

Jones Co. 21-333-9

Full: Only person or company named in endorsement can transfer check to someone else.

Pay to the order of Regan Bank Jones Co. 21-333-9

Restrictive: Check must be deposited. Limits any further negotiation of the check.

Pay to the order of Regan Bank. For deposit only. Jones Co. 21-333-9

Checkbook balance ⫹ EFT (electronic funds transfer) ⫹ Interest earned ⫹ Notes collected ⫹ Direct deposits ⫺ ATM withdrawals ⫺ NSF check ⫺ Online fees ⫺ Automatic withdrawals ⫺ Overdrafts ⫺ Service charges ⫺ Stop payments ⫾ Book errors* CM—adds to balance DM—deducts from balance

Checkbook balance Balance ⫺ NSF

Bank reconciliation, p. 94

$800 40 $760 4

⫺ Service charge

$756 Bank balance Balance ⫹ Deposits in transit ⫺ Outstanding checks

$ 632 416 $1,048 292 $ 756

Bank balance ⫹ Deposits in transit ⫺ Outstanding checks ⫾ Bank errors *If a $60 check is recorded as $50, we must decrease checkbook balance by $10.

KEY TERMS

Automatic teller machine (ATM), p. 89 Bank reconciliation, p. 96 Bank statement, p. 95 Blank endorsement, p. 91 Check, p. 89 Check register, p. 96 Check stub, p. 91 Credit memo (CM), p. 97 Debit card, p. 89

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 4–1a (p. 93) Ending Balance Forward $11,533.80

Debit memo (DM), p. 97 Deposit slip, p. 90 Deposits in transit, p. 97 Draft, p. 89 Drawee, p. 90 Drawer, p. 90 Electronic funds transfer (EFT), p. 95 Endorse, p. 91 Full endorsement, p. 91

Nonsufficient funds (NSF), p. 97 Outstanding checks, p. 97 Overdrafts, p. 95 Payee, p. 90 Restrictive endorsement, p. 91 Safekeeping, p. 100 Signature card, p. 89

LU 4–2a (p. 100) Reconciled Balance $1,179.16

Critical Thinking Discussion Questions 1. Explain the structure of a check. The trend in bank statements is not to return the canceled checks. Do you think this is fair? 2. List the three types of endorsements. Endorsements are limited to the top 1 12 inches of the trailing edge on the back left side of your check. Why do you think the Federal Reserve made this regulation?

3. List the steps in reconciling a bank statement. Today, many banks char ge a monthly fee for certain types of checking accounts. Do you think all checking accounts should be free? Please explain. 4. What are some of the trends in online banking? Will we become a cashless society in which all transactions are made with some type of credit card?

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Classroom Notes

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS 4–1. Fill out the check register that follows with this information:

2009 July 7 15 19 20 24 29

Check No. 482 Check No. 483 Deposit Check No. 484 Check No. 485 Deposit

Google Microsoft

$133.50 55.10 700.00 451.88 319.24 400.30

Sprint Krispy Kreme

RECORD ALL CHARGES OR CREDITS THAT AFFECT YOUR ACCOUNT NUMBER

DATE 2009

DESCRIPTION OF TRANSACTION

PAYMENT/DEBIT (–) $

冑

FEE (IF ANY)

(–) $

BALANCE DEPOSIT/CREDIT $ (+)

4,500 75

$

4–2. November 1, 2009, Payroll.com, an Internet company, has a $10,481.88 checkbook balance. Record the following transactions for Payroll.com by completing the two checks and check stubs provided. Sign the checks Garth Scholten, controller . a. November 8, 2009, deposited $688.10 b. November 8, check No. 190 payable to Wal-Mart Corporation for office supplies—$766.88 c. November 15, check No. 191 payable to Compaq Corporation for computer equipment—$3,815.99.

PAYROLL.COM 1 LEDGER RD. ST. PAUL, MN 55113

190

IPSWICH BANK ipswichbank.com 011000138

25 11103 190

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PAYROLL.COM 1 LEDGER RD. ST. PAUL, MN 55113

191

IPSWICH BANK ipswichbank.com 011000138

25 11103 191

4–3. Using the check register in Problem 4–1 and the following bank statement, prepare a bank reconciliation for Lee.com. BANK STATEMENT Date

Checks

Deposits

7/1 balance 7/18

$4,500.75 $133.50

7/19 7/26

Balance

4,367.25 $ 700.00

319.24

7/30

15.00 SC

5,067.25 4,748.01 4,733.01

WORD PROBLEMS 4–4. Bank fees are squeezing customers according to an article in The Record (Hackensack, NJ) dated April 20, 2006. Banks are having a hard time making money lending, so many are char ging more and higher fees. Meanwhile, interest paid on interest-bearing checking accounts remains low . Kayla Siska received her bank statement from the Commerce Bank which increased its overdraft fee from $33 to $35. To avoid future overdraft fees and bouncedchecks (NSF) fees, Kayla wants to make sure her checkbook is in balance. The following checks have not cleared the bank: No. 634, $58.30; No. 635, $108.75; and No. 637, $112.68. Her checkbook balance shows $695.23. She received $1.75 in interest. She also was char ged a $35.00 overdraft fee. The bank shows a balance of $320.10. A $621.61 deposit was not recorded. Prepare Kayla’s bank reconciliation.

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4–5. In the February 2007 issue of Consumer Reports Money Adviser an article appeared, stating ATM surcharges and bounced-check fees have reached record highs. The report found the average bounced-check (NSF) fee was $27.40 up from $27.04 last spring. ATM fees rose 10 cents, to $1.64 since last fall’s survey. Norman Rand uses his ATM several times a month. Norman received his June 2007 bank statement showing a balance of $835.38, the statement did not show a $178.79 deposit he had made. His checkbook balance shows $838.40. Check No. 234 for $88.70 and check No. 236 for $124.75 were outstanding. He had an ATM surcharge of $11.48 and had a bounced-check fee of $27.40. He received $1.20 in interest. Prepare Norman’s bank reconciliation.

4–6. A local bank began char ging $2.50 each month for returning canceled checks. The bank also has an $8.00 “maintenance” fee if a checking account slips below $750. Donna Sands likes to have copies of her canceled checks for preparing her income tax. She has received her bank statement with a balance of $535.85. Donna received $2.68 in interest and has been char ged for the canceled checks and the maintenance fee. The following checks were outstanding: No. 94, $121.16; No. 96, $106.30; No. 98, $210.12; and No. 99, $64.84. A deposit of $765.69 was not recorded on Donna’ s bank statement. Her checkbook shows a balance of $806.94. Prepare Donna’s bank reconciliation.

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4–7. USA Today reported on April 18, 2006 that checking account charges are getting nastier. Even as more banks introduce checking accounts with no monthly service fees, they’re raising other char ges and making it harder for customers to avoid them. The average cost of bouncing a check is at a near -record $27.04. ATM fees for non-customers have risen to a record average of $1.60 per withdrawal. Ben Luna received his bank statement with a $27.04 fee for a bouncedcheck (NSF). He has an $815.75 monthly mortgage payment paid through his bank. There was also a $3.00 teller fee and a check printing fee of $3.50. His ATM card fee was $6.40. There was also a $530.50 deposit in transit. The bank shows a balance of $1 19.17. The bank paid Ben $1.23 in interest. Ben’ s checkbook shows a balance of $1,395.28. Check No. 234 for $80.30 and check No. 235 for $28.55 were outstanding. Prepare Ben’ s bank reconciliation.

4–8. John D. Hawks, Jr., controller of the currency, delivered an address titled “Banks—Fees! Fees! Fees!” He points out that consumers who were unable to meet minimum balance requirements paid an average of $217 a year , or $18 a month, to maintain a checking account. Kameron Gibson has a hard time maintaining the minimum balance. He was having difficulty balancing his checkbook because he did not notice this fee on his bank statement. His bank statement showed a balance of $717.72. Kameron’s checkbook had a balance of $209.50. Check No. 104 for $110.07 and check No. 105 for $15.55 were outstanding. A $620.50 deposit was not on the statement. He has his payroll check electronically deposited to his checking account—the payroll check was for $1,025.10. There was also a $4 teller fee and an $18 service charge. Prepare Kameron Gibson’s bank reconciliation.

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4–9. Banks are finding more ways to char ge fees, such as a $25 overdraft fee. Sue McV ickers has an account in Fayetteville; she has received her bank statement with this $25 char ge. Also, she was charged a $6.50 service fee; however, the good news is she earned $5.15 interest. Her bank statement’s balance was $315.65, but it did not show the $1,215.15 deposit she had made. Sue’ s checkbook balance shows $604.30. The following checks have not cleared: No. 250, $603.15; No. 253, $218.90; and No. 254, $130.80. Prepare Sue’ s bank reconciliation.

4–10. Carol Stokke receives her April 6 bank statement showing a balance of $859.75; her checkbook balance is $954.25. The bank statement shows an ATM charge of $25.00, NSF fee of $27.00, earned interest of $2.75, and Carol’s $630.15 refund check, which was processed by the IRS and deposited to her account. Carol has two checks that have not cleared—No. 115 for $521.15 and No. 1 16 for $205.50. There is also a deposit in transit for $1,402.05. Prepare Carol’s bank reconciliation.

4–11. Lowell Bank reported the following checking account fees: $2 to see a real-live teller,$20 to process a bounced check, and $1 to $3 if you need an original check to prove you paid a bill or made a charitable contribution. This past month you had to transact business through a teller 6 times—a total $12 cost to you. Your bank statement shows a $305.33 balance; your checkbook shows a $1,009.76 balance. You received $1.10 in interest. An $801.15 deposit was not recorded on your statement. The following checks were outstanding: No. 413, $28.30; No. 414, $18.60; and No. 418, $60.72. Prepare your bank reconciliation.

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CHALLENGE PROBLEMS 4–12. Carolyn Crosswell, who banks in New Jersey, wants to balance her checkbook, which shows a balance of $985.20. The bank shows a balance of $1,430.33. The following transactions occurred: $135.20 automatic withdrawal to the gas company, $6.50 ATM fee, $8.00 service fee, and $1,030.05 direct deposit from the IRS. Carolyn used her debit card 5 times and was char ged 45 cents for each transaction; she was also char ged $3.50 for check printing. A $931.08 deposit was not shown on her bank statement. The following checks were outstanding: No. 235, $158.20; No. 237, $184.13; No. 238, $118.12; and No. 239, $38.83. Carolyn received $2.33 interest. Prepare Carolyn’s bank reconciliation.

4–13. Melissa Jackson, bookkeeper for Kinko Company, cannot prepare a bank reconciliation. From the following facts, can you help her complete the June 30, 2009, reconciliation? The bank statement showed a $2,955.82 balance. Melissa’s checkbook showed a $3,301.82 balance. Melissa placed a $510.19 deposit in the bank’ s night depository on June 30. The deposit did not appear on the bank statement. The bank included two DMs and one CM with the returned checks: $690.65 DM for NSF check, $8.50 DM for service char ges, and $400.00 CM (less $10 collection fee) for collecting a $400.00 non-interestbearing note. Check No. 81 1 for $1 10.94 and check No. 912 for $82.50, both written and recorded on June 28, were not with the returned checks. The bookkeeper had correctly written check No. 884, $1,000, for a new cash register, but she recorded the check as $1,069. The May bank reconciliation showed check No. 748 for $210.90 and check No. 710 for $195.80 outstanding on April 30. The June bank statement included check No. 710 but not check No. 748.

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DVD SUMMARY PRACTICE TEST 1. Walgreens has a $12,925.55 beginning checkbook balance. Record the following transactions in the check stubs provided. (p. 91) a. November 4, 2009, check No. 180 payable to Johnson and Johnson Corporation, $1,700.88 for drugs. b. $5,250 deposit—November 24. c. November 24, 2009, check No. 181 payable to Gillette Corporation, $825.55 merchandise.

2. On April 1, 2009, Lester Company received a bank statement that showed a $8,950 balance. Lester showed an $8,000 checking account balance. The bank did not return check No. 1 15 for $750 or check No. 1 18 for $370. A $900 deposit made on March 31 was in transit. The bank char ged Lester $20 for printing and $250 for NSF checks. The bank also collected a $1,400 note for Lester . Lester for got to record a $400 withdrawal at the ATM. Prepare a bank reconciliation. (p. 95)

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3. Felix Babic banks at Role Federal Bank. Today he received his March 31, 2009, bank statement showing a $762.80 balance. Felix’s checkbook shows a balance of $799.80. The following checks have not cleared the bank: No. 140, $130.55; No. 149, $66.80; and No. 161, $102.90. Felix made a $820.15 deposit that is not shown on the bank statement. He has his $617.30 monthly mortgage payment paid through the bank. His $1,100.20 IRS refund check was mailed to his bank. Prepare Felix Babic’ s bank reconciliation. (p. 95)

4. On June 30, 2009, Wally Company’s bank statement showed a $7,500.10 bank balance. Wally has a beginning checkbook balance of $9,800.00. The bank statement also showed that it collected a $1,200.50 note for the company . A $4,500.10 June 30 deposit was in transit. Check No. 1 19 for $650.20 and check No. 130 for $381.50 are outstanding. Wally’s bank char ges $.40 cents per check. This month, 80 checks were processed. Prepare a reconciled statement. ( p. 95)

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Personal Finance A KIPLINGER APPROACH CREDIT

| Online banking is convenient

about writing checks or running out of but not foolproof. By Joan Goldwasser stamps thanks to automatic bill-paying services, you still need to monitor your bills. After you sign up with a vendor or your bank, review your account h e n e mily statements on a regular basis, advises and Greg MarMike Herd of Nacha, the electronictinez moved payments association. And make sure from Philadelthat your payee mailing addresses are phia to Los up-to-date. Angeles earlier this year, they thought When you sign up for automatic that transferring their online bank bill-paying with an individual vendor account would be a snap. They were or with your bank, you can generally wrong. Because they didn’t notify decide whether you want the funds Sprint in time, their monthly celldebited from your checking account phone bill was or charged to a paid twice— credit card. You once from their can change the Philadelphia acamount, or even count and again cancel the payfrom their new ment, sometimes Los Angeles bank as late as the day account. before your bill Other online is due. bill-payers have If you have a had bills that problem with narrowly escaped unauthorized paybeing paid late, or ments—which weren’t paid at all. happens to about One couple failed 25 out of every to notify Check100,000 transacfree, their bill-paytions, according G Because of an online glitch, Emily Martinez ing service, that to Nacha—be ended up overpaying her cell-phone bill. their mortgage sure to notify your lender had changbank or credit-card ed addresses. Their mortgage check was company immediately. Nacha rules forwarded and ended up arriving on require the financial institution to retime. However, a young teacher wasn’t imburse your account if a transaction so fortunate. He assumed that his car is unauthorized. Neither Visa nor Maspayments would be automatically terCard holds consumers liable for debited from his checking account. unintended charges to their accounts. But he neglected to sign the necessary In a rare instance, automatic billdocuments, and after several missed paying can be too efficient. Take the payments, he had to scramble to fix case of the Canadian man who died in the problem—after his car had been his Winnipeg apartment but wasn’t repossessed in the middle of the night. discovered for nearly two years. No If you’re among the two-thirds of one noticed because all of his monthly U.S. consumers who no longer worry bills had been paid on time.

Electronic billpaying SNAFUS

D A N C H AV K I N

W

BUSINESS MATH ISSUE Kiplinger’s © 2006

In ten years everyone will pay bills online. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

111

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A Visit the Toyota Motor Corp. Web site and check out the banking options they offer. Do you think retailers should be in the banking business?

2004 Wall Street Journal ©

112

b site text We he e e S : s t T t Projec /slater9e) and e. Interne ce Guid m r o u .c o e s h e h R t .m e (www Intern ss Math Busine

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CHAPTER

5

Solving for the Unknown: A How-to Approach for Solving Equations

LEARNING UNIT OBJECTIVES LU 5–1: Solving Equations for the Unknown • Explain the basic procedures used to solve equations for the unknown (pp. 114–116 ). • List the five rules and the mechanical steps used to solve for the unknown in seven situations; know how to check the answers (pp. 117–119 ). urnal © 2006 Wall Street Jo

LU 5–2: Solving Word Problems for the Unknown • List the steps for solving word problems (p. 121 ). • Complete blueprint aids to solve word problems; check the solutions (pp. 121–123 ).

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Chapter 5 Solving for the Unknown: A How-to Approach for Solving Equations

When you shop at Home Depot, have you noticed that Home Depot employs many older employees? Often you are greeted by an older employee. Older , experienced employees frequently are ready to answer customers’ questions. Traditionally, many employers have avoided hiring older people. Now this has changed. The following Wall Street Journal clipping “Gray Is Good: Employers Make Ef forts to Retain Older, Experienced Workers” gives interesting facts about the hiring of older , experienced employees. Some companies are seeking employees 55 and over . Home Depot and 1 Stanley Consultants are two examples. At Stanley Consultants about 4 of the 1,100 employees are over 50. This means that 275 employees are over 50: 1 1,100 275 4

Wall Street Journal © 2005

Learning Unit 5–1 explains how to solve for unknowns in equations. In Learning Unit 5–2 you learn how to solve for unknowns in word problems. When you complete these learning units, you will not have to memorize as many formulas to solve business and personal math applications. Also, with the increasing use of computer software, a basic working knowledge of solving for the unknown has become necessary .

Learning Unit 5–1: Solving Equations for the Unknown The Rose Smith letter at the top of the following page is based on a true story . Note how Rose states that the blueprint aids, the lesson on repetition, and the chapter or ganizers were important factors in the successful completion of her business math course.

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Learning Unit 5–1

Rose Smith 15 Locust Street Lynn, MA 01915 Dear Professor Slater,

115

net

Decorating Service

Thank you for helping me get through your Business Math class. When I first started, my math anxiety level was real high. I felt I had no head for numbers. When you told us we would be covering the chapter on solving equations, I’ll never forget how I started to shake. I started to panic. I felt I could never solve a word problem. I thought I was having an algebra attack. Now that it’s over (90 on the chapter on unknowns), I’d like to tell you what worked for me so you might pass this on to other students. It was your blueprint aids. Drawing boxes helped me to think things out. They were a tool that helped me more clearly understand how to dissect each word problem. They didn’t solve the problem for me, but gave me the direction I needed. Repetition was the key to my success. At first I got them all wrong but after the third time, things started to click. I felt more confident. Your chapter organizers at the end of the chapter were great. Thanks for your patience – your repetition breeds success – now students are asking me to help them solve a word problem. Can you believe it! Best,

Rose Rose Smith

Many of you are familiar with the terms variables and constants. If you are planning to prepare for your retirement by saving only what you can af ford each year , your saving is a variable; if you plan to save the same amount each year , your saving is a constant. Now you can also say that you cannot buy clothes by size because of the many variables involved. This unit explains the importance of mathematical variables and constants when solving equations.

Basic Equation-Solving Procedures Do you wait for the after -Christmas sales to make your purchases? What happens when retailers have fewer inventories to sell after Christmas because they had a good Christmas season and discounted merchandise deeply before Christmas? This means it will be harder for customers to find bar gains. The best bar gains will be found in computers and clothes.

Wall Street Journal © 2005

From the Wall Street Journal heading “Navigating the New World of Post-Christmas Sales,” you can see that stores had a strong Christmas season. To have merchandise to sell, retailers offered large discounts to gift-card recipients with the hope that retailers could sell the gift-card recipients new, full-priced items. The heading also stated that gift cards change the equation. But no explanation is given on how the equation is changed or what the equation was before the change. The definition of an equation given in the next paragraph may suggest to you what is meant by “the equation.”

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Chapter 5 Solving for the Unknown: A How-to Approach for Solving Equations

Do you know the dif ference between a mathematical expression, equation, and formula? A mathematical expression is a meaningful combination of numbers and letters called terms. Operational signs (such as or ) within the expression connect the terms to show a relationship between them. For example, 6 2 or 6 A 4A are mathematical expressions. An equation is a mathematical statement with an equal sign showing that a mathematical expression on the left equals the mathematical expression on the right. An equation has an equal sign; an expression does not have an equal sign. A formula is an equation that expresses in symbols a general fact, rule, or principle. Formulas are shortcuts for expressing a word concept. For example, in Chapter 10 you will learn that the formula for simple interest is Interest ( I) Principal ( P) Rate ( R) Time ( T). This means that when you see I P R T, you recognize the simple interest formula. Now let’ s study basic equations. As a mathematical statement of equality , equations show that two numbers or groups of numbers are equal. For example, 6 4 10 shows the equality of an equation. Equations also use letters as symbols that represent one or more numbers. These symbols, usually a letter of the alphabet, are variables that stand for a number . We can use a variable even though we may not know what it represents. For example, A 2 6. The variable A represents the number or unknown (4 in this example) for which we are solving. We distinguish variables from numbers, which have a fixed value. Numbers such as 3 or 7 are constants or knowns, whereas A and 3A (this means 3 times the variable A) are variables. So we can now say that variables and constants are terms of mathematical expressions. Usually in solving for the unknown, we place variable(s) on the left side of the equation and constants on the right. The following rules for variables and constants are important. VARIABLES AND CONSTANTS RULES 1. If no number is in front of a letter, it is a 1: B 1B; C 1C. 2. If no sign is in front of a letter or number, it is a : C C; 4 4.

You should be aware that in solving equations, the meaning of the symbols , , , and has not changed. However , some variations occur . For example, you can also write A B (A times B) as A ⴢ B, A(B), or AB. Also, A divided by B is the same as A/B. Remember that to solve an equation, you must find a number that can replace the unknown in the equation and make it a true statement. Now let’ s take a moment to look at how we can change verbal statements into variables. Assume Dick Hersh, an employee of Nike, is 50 years old. Let’s assign Dick Hersh’ s changing age to the symbol A. The symbol A is a variable. Verbal statement

FIGURE

5.1

Equality in equations

A+8

58

Left side of equation

Right side of equation

Dick's age in 8 years will equal 58.

Variable A (age)

Dick’s age 8 years ago

A8

Dick’s age 8 years from today

A8

Four times Dick’s age

4A

One-fifth Dick’s age

A/5

To visualize how equations work, think of the old-fashioned balancing scale shown in Figure 5.1. The pole of the scale is the equals sign. The two sides of the equation are the two pans of the scale. In the left pan or left side of the equation, we have A 8; in the right pan or right side of the equation, we have 58. To solve for the unknown (Dick’ s present age), we isolate or place the unknown (variable) on the left side and the numbers on the right. We will do this soon. For now , remember that to keep an equation (or scale) in balance, we must perform mathematical operations (addition, subtraction, multiplication, and division) to both sides of the equation. SOLVING FOR THE UNKNOWN RULE Whatever you do to one side of an equation, you must do to the other side.

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Learning Unit 5–1

117

How to Solve for Unknowns in Equations This section presents seven drill situations and the rules that will guide you in solving for unknowns in these situations. We begin with two basic rules—the opposite process rule and the equation equality rule. OPPOSITE PROCESS RULE If an equation indicates a process such as addition, subtraction, multiplication, or division, solve for the unknown or variable by using the opposite process. For example, if the equation process is addition, solve for the unknown by using subtraction.

EQUATION EQUALITY RULE You can add the same quantity or number to both sides of the equation and subtract the same quantity or number from both sides of the equation without affecting the equality of the equation. You can also divide or multiply both sides of the equation by the same quantity or number (except zero) without affecting the equality of the equation. To check your answer(s), substitute your answer(s) for the letter(s) in the equation. The sum of the left side should equal the sum of the right side.

Drill Situation 1: Subtracting Same Number from Both Sides of Equation Example A 8 58 Dick’s age A plus 8 equals 58.

Mechanical steps A 8 58 8 8 A 50

Explanation 8 is subtracted from both sides of equation to isolate variable A on the left. Check 50 8 58 58 58

Note: Since the equation process used addition, we use the opposite process rule and solve for variable A with subtraction. We also use the equation equality rule when we subtract the same quantity from both sides of the equation. Drill Situation 2: Adding Same Number to Both Sides of Equation Example B 50 80 Some number B less 50 equals 80.

Mechanical steps B 50 80 50 50 B 130

Explanation 50 is added to both sides to isolate variable B on the left. Check 130 50 80 80 80

Note: Since the equation process used subtraction, we use the opposite process rule and solve for variable B with addition. We also use the equation equality rule when we add the same quantity to both sides of the equation. Drill Situation 3: Dividing Both Sides of Equation by Same Number Example

7G 35 Some number G times 7 equals 35.

Mechanical steps 7G 35 7G 35 7 7 G5

Explanation By dividing both sides by 7, G equals 5. Check 7(5) 35 35 35

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Chapter 5 Solving for the Unknown: A How-to Approach for Solving Equations

Note: Since the equation process used multiplication, we use the opposite process rule and solve for variable G with division. We also use the equation equality rule when we divide both sides of the equation by the same quantity . Drill Situation 4: Multiplying Both Sides of Equation by Same Number Example Mechanical steps Explanation V 70 5

V 70 5 Some number V divided by 5 equals 70.

By multiplying both sides by 5, V is equal to 350.

V 5a b 70(5) 5 V 350

Check 350 70 5 70 70

Note: Since the equation process used division, we use the opposite process rule and solve for variable V with multiplication. We also use the equation equality rule when we multiply both sides of the equation by the same quantity . Drill Situation 5: Equation That Uses Subtraction and Multiplication to Solve Unknown MULTIPLE PROCESSES RULE When solving for an unknown that involves more than one process, do the addition and subtraction before the multiplication and division.

Example

Mechanical steps

Explanation

H 25 4

H 2 4

5

When we divide unknown H by 4 and add the result to 2, the answer is 5.

H 2 4

5

1. Move constant to right side by subtracting 2 from both sides. 2. To isolate H, which is divided by 4, we do the opposite process and multiply 4 times both sides of the equation.

2 H 4

2

3

H 4a b 4(3) 4 H

12

Check 12 25 4 325 55

Drill Situation 6: Using Parentheses in Solving for Unknown PARENTHESES RULE When equations contain parentheses (which indicate grouping together), you solve for the unknown by first multiplying each item inside the parentheses by the number or letter just outside the parentheses. Then you continue to solve for the unknown with the opposite process used in the equation. Do the additions and subtractions first; then the multiplications and divisions.

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Learning Unit 5–1

119

Example

Mechanical steps

Explanation

5(P 4) 20

5(P 4)

The unknown P less 4, multiplied by 5 equals 20.

5P 20 20 20 20 5P 40 5 5 P 8

1. Parentheses tell us that everything inside parentheses is multiplied by 5. Multiply 5 by P and 5 by 4. 2. Add 20 to both sides to isolate 5P on left. 3. To remove 5 in front of P, divide both sides by 5 to result in P equals 8.

20

Check 5(8 4) 20 5(4) 20 20 20 Drill Situation 7: Combining Like Unknowns LIKE UNKNOWNS RULE To solve equations with like unknowns, you first combine the unknowns and then solve with the opposite process used in the equation.

Example

4A A 20

Mechanical steps 4A A 20 5A 20 5 5 A 4

Explanation To solve this equation: 4A 1A 5A. Thus, 5A 20. To solve for A, divide both sides by 5, leaving A equals 4.

Before you go to Learning Unit 5–2, let’ s check your understanding of this unit.

LU 5–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1.

DVD

2.

✓ 1.

2.

Write equations for the following (use the letter Q as the variable). Do not solve for the unknown. a. Nine less than one-half a number is fourteen. b. Eight times the sum of a number and thirty-one is fifty . c. Ten decreased by twice a number is two. d. Eight times a number less two equals twenty-one. e. The sum of four times a number and two is fifteen. f. If twice a number is decreased by eight, the dif ference is four. Solve the following: a. B 24 60 b. D 3D 240 c. 12B 144 B B d. e. f. 3(B 8) 18 50 4 16 6 4

Solutions 1 Q 9 14 2 d. 8Q 2 21

8(Q 31) 50

a.

b.

a. B 24 60 24 24 B 36

e. 4Q 2 15 b. 4D 240 4 4 D 60

c.

10 2Q 2

f. 2Q 8 4 c. 12B 144 12 12 B 12

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Chapter 5 Solving for the Unknown: A How-to Approach for Solving Equations

d.

LU 5–1a

B 6a b 50(6) 6 B 300

e. B 4 4 4 B 4 B 4a b 4 B

16 4 12 12(4)

f.

3(B 8) 18 3B 24 18 24 24 3B 42 3 3 B 14

48

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 127)

1.

2.

Write equations for the following (use the letter Q as the variable). Do not solve for the unknown. a. Eight less than one-half a number is sixteen. b. Twelve times the sum of a number and forty-one is 1,200. c. Seven decreased by twice a number is one. d. Four times a number less two equals twenty-four . e. The sum of three times a number and three is nineteen. f. If twice a number is decreased by six, the dif ference is five. Solve the following: a. B 14 70 b. D 4D 250 c. 11B 121 B B d. e. f. 3(B 6) 18 90 2 16 8 2

Learning Unit 5–2: Solving Word Problems for the Unknown When you buy a candy bar such as a Snickers, you should turn the candy bar over and carefully read the ingredients and calories contained on the back of the candy bar wrapper . For example, on the back of the Snickers wrapper you will read that there are “170 calories per piece.” You could misread this to mean that the entire Snickers bar has 170 calories. However, look closer and you will see that the Snickers bar is divided into three pieces, so if you eat the entire bar , instead of consuming 170 calories, you will consume 510 calories. Making errors like this could result in a weight gain that you cannot explain. 1 S 170 calories 3 1 3 a Sb 170 3 3 S 510 calories per bar In this unit, we use blueprint aids in six dif ferent situations to help you solve for unknowns. Be patient and persistent. Remember that the more problems you work, the easier the process becomes. Do not panic! Repetition is the key . Study the five steps that follow. They will help you solve for unknowns in word problems.

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Learning Unit 5–2

121

SOLVING WORD PROBLEMS FOR UNKNOWNS Step 1.

Carefully read the entire problem. You may have to read it several times.

Step 2.

Ask yourself: “What is the problem looking for?”

Step 3.

When you are sure what the problem is asking, let a variable represent the unknown. If the problem has more than one unknown, represent the second unknown in terms of the same variable. For example, if the problem has two unknowns, Y is one unknown. The second unknown is 4Y—4 times the first unknown.

Step 4.

Visualize the relationship between unknowns and variables. Then set up an equation to solve for unknown(s).

Step 5.

Check your result to see if it is accurate.

Word Problem Situation 1: Number Problems From the Wall Street Journal clipping “The Flagging Division,” you can determine that Disney Stores reduced its product of ferings by 1,600. Disney now has 1,800 product of ferings. What was the original number of product offerings?

Reprinted by permission of The WallStreet Journal, © 2000 Dow Jones & Company, Inc. All Rights Reserved Worldwide. Bill Aron/PhotoEdit

Blueprint aid

Mechanical steps

Unknown(s)

Variable(s)

Relationship*

Original number of product offerings

P

P 1,600 New offerings

P 1,600 1,800 1,600 1,600 P 3,400

New offerings 1,800

*This column will help you visualize the equation before setting up the actual equation.

Explanation

Check

The original offerings less 1,600 1,800. Note that we added 1,600 to both sides to isolate P on the left. Remember, 1P P.

3,400 1,600 1,800 1,800 1,800

Word Problem Situation 2: Finding the Whole When Part Is Known A local Burger

King budgets 18 of its monthly profits on salaries. Salaries for the month were $12,000. What were Burger King’s monthly profits?

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Chapter 5 Solving for the Unknown: A How-to Approach for Solving Equations

oMechanical steps

Blueprint aid

Unknown(s)

Variable(s)

Monthly profits

P

Relationship 1 P 8 Salaries $12,000

Explanation 1 8P

represents Burger King’s monthly salaries. Since the equation used division, we solve for P by multiplying both sides by 8.

1 P $12,000 8 P 8a b $12,000(8) 8 P $96,000

Check 1 ($96,000) $12,000 8 $12,000 $12,000

Word Problem Situation 3: Difference Problems ICM Company sold 4 times as many computers as Ring Company. The difference in their sales is 27. How many computers of each company were sold? Blueprint aid

Mechanical steps

Unknown(s)

Variable(s)

Relationship

ICM

4C

4C

Ring

C

C 27

Note: If problem has two unknowns, assign the variable to smaller item or one who sells less. Then assign the other unknown using the same variable. Use the same letter.

Explanation

4C C 27 3C 27 3 3 C 9 Ring 9 computers ICM 4(9) 36 computers

Check

The variables replace the names ICM and Ring. We assigned Ring the variable C, since it sold fewer computers. We assigned ICM 4C, since it sold 4 times as many computers.

36 computers 9 27 computers

Word Problem Situation 4: Calculating Unit Sales Together Barry Sullivan and Mitch

Ryan sold a total of 300 homes for Regis Realty. Barry sold 9 times as many homes as Mitch. How many did each sell? Mechanical steps

Blueprint aid

Unknown(s)

Variable(s)

Relationship

Homes sold: B. Sullivan M. Ryan

9H

H*

9H H 300 homes

9H H 300 10H 300 10 10 H 30 Ryan: 30 homes Sullivan: 9(30) 270 homes

*

Assign H to Ryan since he sold less.

Explanation

Check

We assigned Mitch H, since he sold fewer homes. We assigned Barry 9H, since he sold 9 times as many homes. Together Barry and Mitch sold 300 homes.

30 270 300

Word Problem Situation 5: Calculating Unit and Dollar Sales (Cost per Unit) When Total Units Are Not Given Andy sold watches ($9) and alarm clocks ($5) at a flea

market. Total sales were $287. People bought 4 times as many watches as alarm clocks. How many of each did Andy sell? What were the total dollar sales of each?

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Learning Unit 5–2

Mechanical steps

Blueprint aid

Unknown(s)

Variable(s)

Price

Relationship

4C

$9

36C

C

5

5C

Unit sales: Watches Clocks

123

36C 5C 287 41C 287 41 41 C 7 7 clocks 4(7) 28 watches

$287 total sales Explanation

Check

Number of watches times $9 sales price plus number of alarm clocks times $5 equals $287 total sales.

7($5) 28($9) $287 $35 $252 $287 $287 $287

Word Problem Situation 6: Calculating Unit and Dollar Sales (Cost per Unit) When Total Units Are Given Andy sold watches ($9) and alarm clocks ($5) at a flea market. Total

sales for 35 watches and alarm clocks were $287. How many of each did were the total dollar sales of each?

Mechanical steps

Blueprint aid

Unknown(s)

Andy sell? What

Variable(s)

Price

Watches

W*

$9

Clocks

35 W

Relationship

Unit sales: 5

9W 5(35 W )

9W 5(35 W) 9W 175 5W 4W 175 175 4W 4

W 28

$287 total sales *

The more expensive item is assigned to the variable first only for this situation to make the mechanical steps easier to complete.

287 287 287 175 112 4

Watches 28 Clocks 35 28 7

Explanation

Check

Number of watches (W ) times price per watch plus number of alarm clocks times price per alarm clock equals $287. Total units given was 35.

28($9) 7($5) $287 $252 $35 $287 $287 $287

Why did we use 35 W? Assume we had 35 pizzas (some cheese, others meatball). If I said that I ate all the meatball pizzas (5), how many cheese pizzas are left? Thirty? Right, you subtract 5 from 35. Think of 35 W as meaning one number . Note in Word Problem Situations 5 and 6 that the situation is the same. In Word Problem Situation 5, we were not given total units sold (but we were told which sold better). In Word Problem Situation 6, we were given total units sold, but we did not know which sold better. Now try these six types of word problems in the Practice Quiz. Be sure to complete blueprint aids and the mechanical steps for solving the unknown(s).

LU 5–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Situations 1. An L. L. Bean sweater was reduced $30. The sale price was $90. What was the original price? 2. Kelly Doyle budgets 18 of her yearly salary for entertainment. Kelly’ s total entertainment bill for the year is $6,500. What is Kelly’ s yearly salary? 3. Micro Knowledge sells 5 times as many computers as Morse Electronics. The difference in sales between the two stores is 20 computers. How many computers did each store sell? 4. Susie and Cara sell stoves at Elliott’ s Appliances. Together they sold 180 stoves in January. Susie sold 5 times as many stoves as Cara. How many stoves did each sell? 5. Pasquale’s Pizza sells meatball pizzas ($6) and cheese pizzas ($5). In March, Pasquale’s total sales were $1,600. People bought 2 times as many cheese pizzas as meatball pizzas. How many of each did Pasquale sell? What were the total dollar sales of each?

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6.

✓

Pasquale’s Pizza sells meatball pizzas ($6) and cheese pizzas ($5). In March, Pasquale’s sold 300 pizzas for $1,600. How many of each did Pasquale’s sell? What was the dollar sales price of each?

Solutions

1.

Unknown(s)

Variable(s)

Relationship

Original price

P*

P $30 Sale price Sale price $90

Mechanical steps P $30 $90 30 30 P $120

*P Orignal price.

2.

Unknown(s) Yearly salary

Variable(s)

Relationship

S*

1 8S

Entertainment $6,500

1 S $6,500 8 S 8a b $6,500(8) 8 S $52,000

*S Salary.

3.

Mechanical steps

Unknown(s)

Variable(s)

Relationship

Micro

5C*

5C

Morse

C

C

Mechanical steps 5C C 20 4C 20 4 4

20 computers *C Computers.

4.

Unknown(s)

C 5 (Morse) 5C 25 (Micro)

Variable(s)

Relationship

6S 180 6 6

Stoves sold: Susie

5S*

5S

Cara

S

S

Mechanical steps 5S S 20

S 30 (Cara)

180 stoves

5S 150 (Susie)

S Stoves.

*

5.

Unknown(s)

Variable(s)

Price

Relationship

Mechanical steps 6M 10M 1,600

Meatball Cheese

M

$6

6M

2M

5

10M $1,600 total sales

16M 1,600 16 16 M 100 (meatball) 2M 200 (cheese)

Check (100 $6) (200 $5) $1,600 $600 $1,000 $1,600 $1,600 $1,600

6.

Unknown(s)

Variable(s)

Price

Relationship

Unit sales: Meatball Cheese

M* 300 M

$6 5

6M 5(300 M)

Mechanical steps 6M 5(300 M) 6M 1,500 5M M 1,500 1,500 M

1,600 1,600 1,600 1,500 100

$1,600 total sales *We assign the variable to the most expensive to make the mechanical steps easier to complete.

Check 100($6) 200($5) $600 $1,000 $1,600

Meatball 100 Cheese 300 100 200

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

LU 5–2a

125

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 127)

Situations 1. 2. 3.

4.

5.

6.

An L. L. Bean sweater was reduced $50. The sale price was $140. What was the original price? 1 Kelly Doyle budgets 7 of her yearly salary for entertainment. Kelly’ s total entertainment bill for the year is $7,000. What is Kelly’ s yearly salary? Micro Knowledge sells 8 times as many computers as Morse Electronics. The difference in sales between the two stores is 49 computers. How many computers did each store sell? Susie and Cara sell stoves at Elliott’ s Appliances. Together they sold 360 stoves in January. Susie sold 2 times as many stoves as Cara. How many stoves did each sell? Pasquale’s Pizza sells meatball pizzas ($7) and cheese pizzas ($6). In March, Pasquale’s total sales were $1,800. People bought 3 times as many cheese pizzas as meatball pizzas. How many of each did Pasquale sell? What were the total dollar sales of each? Pasquale’s Pizza sells meatball pizzas ($7) and cheese pizzas ($6). In March, Pasquale sold 288 pizzas for $1,800. What was the dollar sales price of each?

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Solving for unknowns from basic equations

Mechanical steps to solve unknowns

Key point(s)

Situation 1: Subtracting same number from both sides of equation, p. 117

D 10 12 10 10 D 2

Subtract 10 from both sides of equation to isolate variable D on the left. Since equation used addition, we solve by using opposite process—subtraction.

Situation 2: Adding same number to both sides of equation, p. 117

L 24 40 24 24 L 64

Add 24 to both sides to isolate unknown L on left. We solve by using opposite process of subtraction—addition.

Situation 3: Dividing both sides of equation by same number, p. 117

6B 24 6B 24 6 6 B4

To isolate B by itself on the left, divide both sides of the equation by 6. Thus, the 6 on the left cancels—leaving B equal to 4. Since equation used multiplication, we solve unknown by using opposite process—division.

Situation 4: Multiplying both sides of equation by same number, p. 118

R 15 3 R 3 a b 15(3) 3 R 45

To remove denominator, multiply both sides of the equation by 3—the 3 on the left side cancels, leaving R equal to 45. Since equation used division, we solve unknown by using opposite process—multiplication.

Situation 5: Equation that uses subtraction and multiplication to solve for unknown, p. 118

B 6 13 3 6 6 B 7 3 B 3 a b 7(3) 3 B 21

1. Move constant 6 to right side by subtracting 6 from both sides. 2. Isolate B by itself on left by multiplying both sides by 3.

(continues)

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Chapter 5 Solving for the Unknown: A How-to Approach for Solving Equations

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (continued) Solving for unknowns from basic equations

Mechanical steps to solve unknowns

Key point(s)

Situation 6: Using parentheses in solving for unknown, p. 118

6(A 5) 12 6A 30 12 30 30 6A 42 6 6 A7

Parentheses indicate multiplication. Multiply 6 times A and 6 times 5. Result is 6A 30 on left side of the equation. Now add 30 to both sides to isolate 6A on left. To remove 6 in front of A, divide both sides by 6, to result in A equal to 7. Note that when deleting parentheses, we did not have to multiply the right side.

Situation 7: Combining like unknowns, p. 119

6A 2A 64 8A 64 8 8 A 8

6A 2A combine to 8A. To solve for A, we divide both sides by 8.

Solving for unknowns from word problems Situation 1: Number problems, p. 121 U.S. Air reduced its airfare to California by $60. The sale price was $95. What was the original price? Situation 2: Finding the whole when part is known, p. 122 K. McCarthy spends 1⁄8 of her budget for school. What is the total budget if school costs $5,000?

Mechanical steps to solve unknown with check

Blueprint aid Unknown(s)

Variable(s)

Relationship

Original price

P

P $60 Sale price Sale price $95

Unknown(s)

Variable(s)

Relationship

Total budget

B

1

/8B School $5,000

P $60 $ 95 60 60 P $155 Check $155 $60 $95 $95 $95 1 B $5,000 8 B 8 a b $5,000(8) 8 B $40,000 Check 1 ($40,000) $5,000 8 $5,000 $5,000

Situation 3: Difference problems, p. 122 Moe sold 8 times as many suitcases as Bill. The difference in their sales is 280 suitcases. How many suitcases did each sell?

Situation 4: Calculating unit sales, p. 122 Moe sold 8 times as many suitcases as Bill. Together they sold a total of 360. How many did each sell?

Unknown(s)

Variable(s)

Relationship

Suitcases sold: Moe Bill

8S S

8S S 280 suitcases

8S S 280 (Bill) 7S 280 7 7 S 40 (Bill) 8(40) 320 (Moe) Check 320 40 280 280 280

Unknown(s)

Variable(s)

Relationship

Suitcases sold: Moe Bill

8S S

8S S 360 suitcases

8S S 280 9S 360 9 9 S 40 (Bill) 8(40) 320 (Moe) Check 320 40 360 360 360

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

127

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Solving for unknowns from word problems Situation 5: Calculating unit and dollar sales (cost per unit) when total units not given, p. 123 Blue Furniture Company ordered sleepers ($300) and nonsleepers ($200) that cost $8,000. Blue expects sleepers to outsell nonsleepers 2 to 1. How many units of each were ordered? What were dollar costs of each? Situation 6: Calculating unit and dollar sales (cost per unit) when total units given, p. 123 Blue Furniture Company ordered 30 sofas (sleepers and nonsleepers) that cost $8,000. The wholesale unit cost was $300 for the sleepers and $200 for the nonsleepers. How many units of each were ordered? What were dollar costs of each?

Mechanical steps to solve unknown with check

Blueprint aid Unknown(s)

Variable(s)

Price

Relationship

Sleepers Nonsleepers

2N N

$300 200

600N 200N $8,000 total cost

600N 200N 8,000 800N 8,000 800 800 N 10 (nonsleepers) 2N 20 (sleepers) Check 10 $200 $2,000 20 $300 6,000 $8,000

Unknown(s)

Variable(s)

Price

Relationship

Unit costs Sleepers Nonsleepers

S 30 S

$300 200

300S 200(30 S) $ 8,000 total cost

300S 200(30 S) 8,000 300S 6,000 200S 8,000 100S 6,000 8,000 6,000 6,000 100S 2,000 100 100 S 20 Nonsleepers 30 20 10 Check 20($300) 10($200) $8,000 $6,000 $2,000 $8,000 $8,000 $8,000

Note: When the total units are given, the higherpriced item (sleepers) is assigned to the variable first. This makes the mechanical steps easier to complete.

KEY TERMS

Constants, p. 116 Equation, p. 116 Expression, p. 116

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 5–1a (p. 120) 1. A. Q /2 8 16 B. 12 (Q 41) 1,200 C. 7 2Q 1 D. 4Q 2 24 E. 3Q 3 19 F. 2Q 6 5 2. A. 56 B. 50 C. 11 D. 720 E. 28 F. 12

Formula, p. 116 Knowns, p. 116 Unknown, p. 116

Variables, p. 116

LU 5–2a (p. 125) P $190 S $49,000 Morse 7; Micro 56 Cara 120; Susie 240 Meatball 72; cheese 216; Meatball $504; cheese $1,296 6. Meatball $504; cheese $1,296

1. 2. 3. 4. 5.

Critical Thinking Discussion Questions 1. Explain the dif ference between a variable and a constant. What would you consider your monthly car payment—a variable or a constant? 2. How does the opposite process rule help solve for the variable in an equation? If a Mercedes costs 3 times as much as a Saab, how could the opposite process rule be used? The selling price of the Mercedes is $60,000.

3. What is the dif ference between Word Problem Situations 5 and 6 in Learning Unit 5–2? Show why the more expensive item in Word Problem Situation 6 is assigned to the variable first.

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Classroom Notes

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS (First Of Three Sets) Solve the unknown from the following equations: 5–1. D ⫹ 19 ⫽ 100

5–2.

E ⫹ 90 ⫽ 200

5–3.

Q ⫹ 100 ⫽

5–5. 5Y ⫽ 75

5–6.

P ⫽ 92 6

5–7.

8Y ⫽ 96

5–9. 4(P ⫺ 9) ⫽ 64

5–10. 3(P ⫺ 3) ⫽ 27

400

5–4.

Q ⫺ 60 ⫽ 850

5–8.

N ⫽5 16

WORD PROBLEMS (First of Three Sets) 5–11. On February 14, 2007, The Fresno Bee reported Yosemite Fitness Center recently opened its second indoor climbing gym. The new gym, which is 6,975 square feet, is 3 times larger than the former gym. What was the size of the old gym?

5–12. In 1955 only 435 Kaiser-Darrins were built, because Kaiser-Frazer bailed out of the car business. Only 435 of these fantastic cars were ever built, they sold for $3,668 according to an article in the Chicago Sun-Times March 5, 2007 edition. The Kaiser-Darrin ended up being the most prized of Henry J. Kaiser ’s cars. It’s valued today at $62,125 if in excellent condition, which is 134 times as much as a car in very nice condition—if you can find an owner willing to part with one for any price. What would be the value of the car in very nice condition?

5–13. Joe Sullivan and Hugh Kee sell cars for a Ford dealer . Over the past year, they sold 300 cars. Joe sells 5 times as many cars as Hugh. How many cars did each sell?

5–14. Nanda Yueh and Lane Zuriff sell homes for ERARealty. Over the past 6 months they sold 120 homes. Nanda sold 3 times as many homes as Lane. How many homes did each sell?

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5–15. Dots sells T-shirts ($2) and shorts ($4). InApril, total sales were $600. People bought 4 times as manyT-shirts as shorts. How many T-shirts and shorts did Dots sell? Check your answer.

5–16. Dots sells 250 T-shirts ($2) and shorts ($4). In April, total sales were $600. How many T-shirts and shorts did Dots sell? Check your answer. Hint: Let .S ⫽ Shorts

DRILL PROBLEMS (Second of Three Sets) 5–17. 8D ⫽ 640

5–18. 7(A ⫺ 5) ⫽ 63

5–20. 18(C ⫺ 3) ⫽ 162

5–21.

9Y ⫺ 10 ⫽

53

5–19.

N ⫽7 9

5–22.

7B ⫹ 5 ⫽

26

WORD PROBLEMS (Second of Three Sets) 5–23. On a flight from New York to Portland, Delta reduced its Internet price by $170.00. The sale price was $315.99. What was the original price?

5–24. Jill, an employee at Old Navy, budgets 15 of her yearly salary for clothing. Jill’s total clothing bill for the year is $8,000.What is her yearly salary?

5–25. Bill’s Roast Beef sells 5 times as many sandwiches as Pete’ s Deli. The difference between their sales is 360 sandwiches. How many sandwiches did each sell?

130

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5–26. Some job seekers who have dif ficulty finding new employment are described as discouraged workers. The September 6, 2003, issue of The New York Times reported that job losses were mounting. In August 2003, the count of discouraged workers rose to 503,000, 212 times as many as in August 2002. How many discouraged workers were there in August 2002?

5–27. Computer City sells batteries ($3) and small boxes of pens ($5). InAugust, total sales were $960. Customers bought 5 times as many batteries as boxes of pens. How many of each did Computer City sell? Check your answer.

5–28. Staples sells cartons of pens ($10) and rubber bands ($4). Leona ordered a total of 24 cartons for $210. How many cartons of each did Leona order? Check your answer. Hint: Let .P ⫽ Pens

DRILL PROBLEMS (Third of Three Sets) 5–29. A ⫹ 90 ⫺ 15 ⫽ 210

5–30.

5Y ⫹ 15(Y ⫹ 1) ⫽

5–31. 3M ⫹ 20 ⫽

5–32.

20(C ⫺ 50) ⫽ 19,000

2M ⫹ 80

35

WORD PROBLEMS (Third of Three Sets) 5–33. The St. Louis Post-Dispatch, on October 25, 2006 reported on ticket scalpers. Cardinals World Series tickets were selling at 15 times more than the highest ticket at face value—others were about 8 times the lowest face value. Pete Moran paid $400 for the lowest ticket at face value. Dennis Spivey paid 938 times the amount paid by Pete. How much did Dennis pay for his ticket?

131

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5–34. At General Electric, shift 1 produced 4 times as much as shift 2. General Electric’ s total production for July was 5,500 jet engines. What was the output for each shift?

5–35. Ivy Corporation gave 84 people a bonus. If Ivy had given 2 more people bonuses, Ivy would have rewarded force. How large is Ivy’s workforce?

2 3

of the work-

5–36. Jim Murray and Phyllis Lowe received a total of $50,000 from a deceased relative’s estate. They decided to put $10,000 in a trust for their nephew and divide the remainder. Phyllis received 34 of the remainder; Jim received 14. How much did Jim and Phyllis receive?

5–37. The first shift of GME Corporation produced112 times as many lanterns as the second shift. GME produced 5,600 lanterns in November. How many lanterns did GME produce on each shift?

5–38. Wal-Mart sells thermometers ($2) and hot-water bottles ($6). In December, Wal-Mart’s total sales were $1,200. Customers bought 7 times as many thermometers as hot-water bottles. How many of each did Wal-Mart sell? Check your answer.

5–39. Ace Hardware sells cartons of wrenches ($100) and hammers ($300). Howard ordered 40 cartons of wrenches and hammers for $8,400. How many cartons of each are in the order? Check your answer.

132

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CHALLENGE PROBLEMS 5–40. The Omaha World-Herald reported the number of homeless people counted during an August census. In homeless shelters, 572 men were counted. This number was 234 times the number of children and 212 times the number of women. (a) How many children were homeless? (b) How many women were homeless?(c) What was the total number of homeless? Round answers to the nearest whole number.

5–41. Bessy has 6 times as much money as Bob, but when each earns $6, Bessy will have 3 times as much money as Bob. How much does each have before and after earning the $6?

nknown(s)

Variable(s)

Relationship

DVD SUMMARY PRACTICE TEST 1.

Delta reduced its round-trip ticket price from Portland to Boston by $140. The sale price was $401.90. What was the original price? (p. 121)

2.

David Role is an employee of Google. He budgets 17 of his salary for clothing. If Dave’s total clothing for the year is $12,000, what is his yearly salary? (p. 122)

3.

A local Best Buy sells 8 times as many iPods as Sears. The difference between their sales is 490 iPods. How many iPods did each sell? (p. 122)

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4.

Working at Staples, Jill Reese andAbby Lee sold a total of 1,200 calculators. Jill sold 5 times as many calculators as Abby. How many did each sell? (p. 122)

5.

Target sells sets of pots ($30) and dishes ($20) at the local store. On the July 4 weekend, Target’s total sales were $2,600. People bought 6 times as many pots as dishes. How many of each did Target sell? Check your answer. (p. 123)

6.

A local Dominos sold a total of 1,600 small pizzas ($9) and pasta dinners ($13) during the Super Bowl. How many of each did Dominos sell if total sales were $15,600? Check your answer. (p. 123)

134

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Personal Finance A KIPLINGER APPROACH R E T I R E A M I L L I O N A I R E Time is on your side (and so is Uncle Sam)

T

he road to $1 million starts early, but if you’re a late bloomer, help is at hand. The table below shows how much you need to save each month to accumulate $1 million by age 65, along with strategies for achieving that goal. At age 25, you’re starting from scratch. At ages 35, 45

IF YOU’RE

IF YOU’RE

25

IF YOU’RE

IF YOU’RE

45

35

55

YOU’VE SAVED

YOU’VE SAVED

YOU’VE SAVED

YOU’VE SAVED

WHAT YOU NEED TO SAVE PER MONTH

WHAT YOU NEED TO SAVE PER MONTH

WHAT YOU NEED TO SAVE PER MONTH

WHAT YOU NEED TO SAVE PER MONTH

$0

$286

GET HELP FROM UNCLE S A M You may qualify for

F R O M L E F T: V E E R , C O R B I S /J U P I T E R I M A G E S , V E E R , B A N A N A S TO C K /J U P I T E R I M A G E S

and 55, we assume you already have money in savings, on which you’re earning 8% annually. If you’re setting your goal lower or higher than $1 million, go to kiplinger.com/ links/whatyouneed to see how much you need to save if you’re aiming to stockpile $500,000 or $2 million.

a retirement-savings tax credit of 10% to 50% of the amount you contribute to an IRA, 401(k) or other retirement account. The credit can reduce your tax bill by up to $1,000. To qualify, your income must be $25,000 or less if you’re single, $37,500 or less if you’re a head of household or $50,000 or less if you’re married.

$0

$671

YOU’VE SAVED

$0

$1,698 YOU’VE SAVED

$0

$5,466 YOU’VE SAVED

$50,000

$50,000

$50,000

WHAT YOU NEED TO SAVE PER MONTH

WHAT YOU NEED TO SAVE PER MONTH

WHAT YOU NEED TO SAVE PER MONTH

$304

G E T H E L P F R O M YO U R B O S S If your employer

offers a matching contribution, contribute at least enough to your 401(k) to capture the full match. Otherwise, you’re walking away from free money. Try to save 15% of your gross income for retirement, including your employer match.

$1,298 YOU’VE SAVED

$4,859 YOU’VE SAVED

$100,000

$100,000

WHAT YOU NEED TO SAVE PER MONTH

WHAT YOU NEED TO SAVE PER MONTH

$861

P L AY C AT C H - U P Aim

to contribute the maximum $15,500 to your 401(k) this year or $4,000 to your traditional or Roth IRA. Once you turn 50, you can contribute an additional $5,000 in catch-up contributions to your 401(k) and an extra $1,000 to your IRA.

$4,253 YOU’VE SAVED

$200,000 WHAT YOU NEED TO SAVE PER MONTH

$3,040

S TAY O N T H E J O B

Working a few years longer can boost your savings. SOURCE : Nuveen Investments

IS YOUR RETIREMENT SAVING ON COURSE? | Go to kiplinger.com/tools

BUSINESS MATH ISSUE Kiplinger’s © 2007

Saving at a young age is not realistic. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A Calculate the total price tag for each item (do not include maintenance fee).

A.

B.

C.

D.

2005 Wall Street Journal ©

136

b site text We he e e S : s t T t Projec /slater9e) and e. Interne m ce Guid r o u .c o e s h e h R t .m e (www Intern ss Math Busine

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CHAPTER

6

Percents and Their Applications

LEARNING UNIT OBJECTIVES LU 6–1: Conversions • Convert decimals to percents (including rounding percents), percents to decimals, and fractions to percents (pp. 139–141). • Convert percents to fractions (p. 142).

LU 6–2: Application of Percents—Portion Formula • List and define the key elements of the portion formula (pp. 144–145). • Solve for one unknown of the portion formula when the other two key elements are given (pp. 145–148 ). • Calculate the rate of percent decreases and increases (pp. 148–151).

urnal © 2005 Wall Street Jo

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Chapter 6 Percents and Their Applications

Wall Street Journal © 2005

Wall Street Journal © 2005

Did you know that 60% of Internet users in China are men and that Ford plans to cut 10% of its workforce of salaried jobs? These facts are from the two Wall Street Journal clippings “Buying Potential” and “Ford to Cut 4,000 U.S. Salaried Jobs in Retooling Ef fort.” Note in these Wall Street Journal clippings how companies frequently use percents to express various decreases and increases between two or more numbers, or to determine a decrease or increase. To understand percents, you should first understand the conversion relationship between decimals, percents, and fractions as explained in Learning Unit 6–1. Then, in Learning Unit 6–2, you will be ready to apply percents to personal and business events.

Learning Unit 6–1: Conversions When we described parts of a whole in previous chapters, we used fractions and decimals. Percents also describe parts of a whole. The word percent means per 100. The percent symbol (%) indicates hundredths (division by 100). Percents are the result of expressing numbers as part of 100. Thus, Ford’s 10% cut in its workforce of salaried jobs represents 10 out of 100.

Percents can provide some revealing information. The Wall Street Journal clipping “Outsourcing Ratio for World’s Top Laptop PC Brands, 2004” shows that Dell, Apple, Gateway, and Acer outsource 100% of their top laptop PC brands.

Wall Street Journal © 2005

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Learning Unit 6–1

TABLE

6.1 Color

Analyzing a bag of M&M’s® Yellow Red Blue Orange Brown Green Total

Information adapted from http://us.mms.com/ us/about/products/milkchocolate/

Fraction 18 55 10 55 9 55 7 55 6 55 5 55 55 1 55

Decimal (hundredth)

Percent (hundredth)

.33

32.73%

.18

18.18

.16

16.36

.13

12.73

.11

10.91

.09

9.09

1.00

139

100.00%

Let’s return to the M&M’ s® example from earlier chapters. In Table 6.1, we use our bag of 55 M&M’ s® to show how fractions, decimals, and percents can refer to the same parts of a whole. For example, the bag of 55 M&M’ s® contains 18 yellow M&M’ s®. As you can see in Table 6.1, the 18 candies in the bag of 55 can be expressed as a fraction (18 s® website, you will see 55 ), decimal (.33), and percent (32.73%). If you visit the M&M’ ® that the standard is 1 1 yellow M&M’ s . The clipping “What Colors Come in Your Bag?” shows an M&M’ s® Milk Chocolate Candies Color Chart. In this unit we discuss converting decimals to percents (including rounding percents), percents to decimals, fractions to percents, and percents to fractions. You will see when you study converting fractions to percents why you should first learn how to convert decimals to percents.

Converting Decimals to Percents The following Wall Street Journal clipping “Getting in the Door: More Online” shows that 2% or 2 out of 100 vendors are able to get a foot in the door of Wal-Mart. If the clipping had stated the 2% as a decimal (.02), could you give its equivalent in percent? The 2 . As you know , percents are the result of expressing decimal .02 in decimal fraction is 100 2 2 . You can now conclude that .02 100 numbers as part of 100, so 2% 100 2%.

Wall Street Journal © 2005

The steps for converting decimals to percents are as follows: CONVERTING DECIMALS TO PERCENTS Step 1.

Move the decimal point two places to the right. You are multiplying by 100. If necessary, add zeros. This rule is also used for whole numbers and mixed decimals.

Step 2.

Add a percent symbol at the end of the number.

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Chapter 6 Percents and Their Applications

EXAMPLES

.66 .66. 66%

.425 .42.5 42.5%

.8 .80. 80%

8 8.00. 800%

Add 1 zero to make two places.

Add 2 zeros to make two places.

.007 .00.7 .7%

2.51 2.51. 251%

7 Caution: One percent means 1 out of every 100. Since .7% is less than 1%, it means 10 of 1%—a very small amount. Less than 1% is less than .01. To show a number less than 1%, you must use more than two decimal places and add 2 zeros. Example: .7% .007.

Rounding Percents When necessary , percents should be rounded. Rounding percents is similar to rounding whole numbers. Use the following steps to round percents: ROUNDING PERCENTS Step 1.

When you convert from a fraction or decimal, be sure your answer is in percent before rounding.

Step 2.

Identify the specific digit. If the digit to the right of the identified digit is 5 or greater, round up the identified digit.

Step 3.

Delete digits to right of the identified digit.

For example, Table 6.1 (p. 139) shows that the 18 yellow M&M’ s® rounded to the nearest hundredth percent is 32.73% of the bag of 55 M&M’ s®. Let’s look at how we arrived at this figure. When using a calculator, you press 18 55 % . This allows you to go right to percent, avoiding the decimal step.

Step 1.

18 .3272727 32.72727% 55

Step 2.

32.73727%

Step 3.

32.73%

Note that the number is in percent! Identify the hundredth percent digit. Digit to the right of the identified digit is greater than 5, so the identified digit is increased by 1. Delete digits to the right of the identified digit.

Converting Percents to Decimals Note that in the following Barron’s clipping “Kellogg by the Numbers,” 54.5% of Kellogg’ s revenue came from cereal sales.

Barron’s © 2006

Battle Creek Enquirer/Scott Erskine/ AP Wide World

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Learning Unit 6–1

141

In the paragraph and steps that follow , you will learn how to convert percents to decimals. The example below the steps using 2% comes from the clipping “Getting in the Door” (p. 139). As previously indicated, the example using 54.5% comes from the clipping “Kellogg by the Numbers.” To convert percents to decimals, you reverse the process used to convert decimals to percents. In our earlier discussion on converting decimals to percents (p. 139), we asked if the 2% in the “Getting in the Door” clipping had been in decimals and not percent, could you convert the decimals to the 2%? Once again, the definition of percent states that 2% 2/100. The fraction 2/100 can be written in decimal form as .02. You can conclude that 2% 2/100 .02. Now you can see this procedure in the following conversion steps: CONVERTING PERCENTS TO DECIMALS Step 1.

Drop the percent symbol.

Step 2.

Move the decimal point two places to the left. You are dividing by 100. If necessary, add zeros.

EXAMPLES Note that when a percent is less than 1%, the decimal conversion has at least two leading zeros before the number .0095.

.95% .00.95 .0095 Add 2 zeros to make two places. 54.5% .54.5 .545

2% .02. .02

66% .66. .66

Add 1 zero to make two places. 824.4% 8.24.4 8.244

1 Now we must explain how to change fractional percents such as 5% to a decimal. 1 1 Remember that fractional percents are values less than 1%. For example, 5% is 5 of 1%. Fractional percents can appear singly or in combination with whole numbers. To convert them to decimals, use the following steps:

CONVERTING FRACTIONAL PERCENTS TO DECIMALS Step 1.

Convert a single fractional percent to its decimal equivalent by dividing the numerator by the denominator. If necessary, round the answer.

Step 2.

If a fractional percent is combined with a whole number (mixed fractional percent), convert the fractional percent first. Then combine the whole number and the fractional percent.

Step 3.

Drop the percent symbol; move the decimal point two places to the left (this divides the number by 100).

EXAMPLES

1 % .20% .00.20 .0020 5 1 % .25% .00.25 .0025 4 3 7 % 7.75% .07.75 .0775 4 1 6 % 6.5% .06.5 .065 2

3 Think of 7 % as 4 7%

.07

3 % .0075 4 3 7 % 4

.0775

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Chapter 6 Percents and Their Applications

Converting Fractions to Percents When fractions have denominators of 100, the numerator becomes the percent. Other fractions must be first converted to decimals; then the decimals are converted to percents. CONVERTING FRACTIONS TO PERCENTS Step 1.

Divide the numerator by the denominator to convert the fraction to a decimal.

Step 2.

Move the decimal point two places to the right; add the percent symbol.

EXAMPLES

3 .75 .75. 75% 4

1 .20 .20. 20% 5

1 .05 .05. 5% 20

Converting Percents to Fractions Using the definition of percent, you can write any percent as a fraction whose denominator is 100. Thus, when we convert a percent to a fraction, we drop the percent symbol and 1 write the number over 100, which is the same as multiplying the number by 100 . This method 1 of multiplying by 100 is also used for fractional percents. CONVERTING A WHOLE PERCENT (OR A FRACTIONAL PERCENT) TO A FRACTION Step 1.

Drop the percent symbol.

Step 2.

Multiply the number by

Step 3.

Reduce to lowest terms.

1 100 .

EXAMPLES

76% 76

1 76 19 100 100 25

156% 156

1 1 1 1 % 8 8 100 800

1 156 56 14 1 1 100 100 100 25

Sometimes a percent contains a whole number and a fraction such as 1212% or 22.5%. Extra steps are needed to write a mixed or decimal percent as a simplified fraction. CONVERTING A MIXED OR DECIMAL PERCENT TO A FRACTION Step 1.

Drop the percent symbol.

Step 2.

Change the mixed percent to an improper fraction.

Step 3.

Multiply the number by

Step 4.

Reduce to lowest terms.

1 100 .

Note: If you have a mixed or decimal percent, change the decimal portion to fractional equivalent and continue with Steps 1 to 4.

EXAMPLES

1 25 1 25 1 12 % 2 2 100 200 8

1 25 1 25 1 12.5% 12 % 2 2 100 200 8 1 45 1 45 9 22.5% 22 % 2 2 100 200 40 It’s time to check your understanding of Learning Unit 6–1.

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Learning Unit 6–1

LU 6–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Convert to percents (round to the nearest tenth percent as needed): 1. .6666 2. .832 3. .004 4. 8.94444 Convert to decimals (remember , decimals representing less than 1% will have at least 2 leading zeros before the number): 1 3 % 5. 6. 6 % 4 4 7. 87% 8. 810.9% Convert to percents (round to the nearest hundredth percent): 1 2 9. 10. 7 9 Convert to fractions (remember , if it is a mixed number , first convert to an improper fraction): 1 11. 19% 12. 71 % 13. 130% 2 1 % 14. 15. 19.9% 2

✓

LU 6–1a

143

Solutions

1.

.66.66 66.7%

2.

.83.2 83.2%

3.

.00.4 .4%

4.

8.94.444 894.4%

5.

1 % .25% .0025 4

6.

3 6 % 6.75% .0675 4

7.

87% .87. .87

8.

810.9% 8.10.9 8.109

9.

1 .14.285 14.29% 7 1 19 100 100

11.

19% 19

13.

130% 130

15.

19

1 130 30 3 1 1 100 100 100 10

10.

2 .22.22 22.22% 9

12.

1 143 1 143 71 % 2 2 100 200

14.

1 1 1 1 % 2 2 100 200

199 1 199 9 % 10 10 100 1,000

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 156)

Convert to percents (round to the nearest tenth percent as needed): 1. .4444 2. .782 3. .006 4. 7.93333 Convert to decimals (remember , decimals representing less than 1% will have at least 2 leading zeros before the number): 1 4 6. 7 % % 5 5 7. 92% 8. 765.8% Convert to percents (round to the nearest hundredth percent): 3 1 9. 10. 3 7 5.

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Chapter 6 Percents and Their Applications

Convert to fractions (remember , if it is a mixed number fraction): 1 11. 17% 12. 82 % 4 14.

1 % 4

15.

, first convert to an improper 13.

150%

17.8%

Learning Unit 6–2: Application of Percents—Portion Formula The bag of M&M’ s® we have been studying contains Milk Chocolate M&M’ s®. ® ® M&M/Mars also makes Peanut M&M’ s and some other types of M&M’ s . To study the application of percents to problems involving M&M’ s®, we make two key assumptions:

1. Total sales of Milk Chocolate M&M’ s®, Peanut M&M’ s®, and other M&M’ s® chocolate candies are $400,000. 2. Eighty percent of M&M’ s® sales are Milk Chocolate M&M’ s®. This leaves the Peanut and other M&M’ s® chocolate candies with 20% of sales (100% 80%). 80% M&M’s® 20% M&M’s® 100% Milk Chocolate Peanut and other Total sales M&M’s® chocolate candies ($400,000)

Before we begin, you must understand the meaning of three terms— portion. These terms are the key elements in solving percent problems. •

•

•

base, rate, and

Base (B). The base is the beginning whole quantity or value (100%) with which you will compare some other quantity or value. Often the problems give the base after the word of. For example, the whole (total) sales of M&M’ s®—Milk Chocolate M&M’ s, Peanut, and other M&M’ s® chocolate candies—are $400,000. Rate (R). The rate is a percent, decimal, or fraction that indicates the part of the base that you must calculate. The percent symbol often helps you identify the rate. For example, Milk Chocolate M&M’ s® currently account for 80% of sales. So the rate is 80%. Remember that 80% is also 45, or .80. Portion (P). The portion is the amount or part that results from the base multiplied by the rate. For example, total sales of M&M’ s® are $400,000 (base); $400,000 times .80 (rate) equals $320,000 (portion), or the sales of Milk Chocolate M&M’ s®. A key point to remember is that portion is a number and not a percent. In fact, the portion can be larger than the base if the rate is greater than 100%.

Solving Percents with the Portion Formula In problems involving portion, base, and rate, we give two of these elements. You must find the third element. Remember the following key formula: Portion (P) Base (B) Rate (R) is

Portion

This line is called the dividing line

Base X Rate of

%

To help you solve for the portion, base, and rate, this unit shows pie charts. The shaded area in each pie chart indicates the element that you must solve for . For example, since we shaded portion in the pie chart at the left, you must solve for portion. To use the pie charts, put your finger on the shaded area (in this case portion). The formula that remains tells you what to do. So in the pie chart at the left, you solve the problem by multiplying base by the rate. Note the circle around the pie chart is broken since we want to emphasize that portion can be lar ger than base if rate is greater than 100%. The horizontal line in the pie chart is called the dividing line, and we will use it when we solve for base or rate. The following example summarizes the concept of base, rate, and portion. Assume that you received a small bonus check of $100. This is a gross amount—your company did not withhold any taxes. You will have to pay 20% in taxes.

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Learning Unit 6–2

Base: 100%—whole. Usually given after the word of—but not always.

Rate: Usually expressed as a percent but could also be a decimal or fraction.

$100 bonus check

20% taxes

145

Portion: A number—not a percent and not the whole. $20 taxes

First decide what you are looking for . You want to know how much you must pay in taxes—the portion. How do you get the portion? From the portion formula Portion ( P) ⫽ Base (B) ⫻ Rate (R), you know that you must multiply the base ($100) by the rate (20%). When you do this, you get $100 ⫻ .20 ⫽ $20. So you must pay $20 in taxes. Let’s try our first word problem by taking a closer look at the M&M’ s® example to ® see how we arrived at the $320,000 sales of Milk Chocolate M&M’ s given earlier . We will be using blueprint aids to help dissect and solve each word problem. Solving for Portion The Word Problem Sales of Milk Chocolate M&M’s® are 80% of the total M&M’s® sales.

Total M&M’s® sales are $400,000. What are the sales of Milk Chocolate M&M’ s®? The facts

Solving for?

Steps to take

Milk Chocolate M&M’s® sales: 80%.

Sales of Milk Chocolate M&M’s®.

Identify key elements.

Total M&M’s® sales: $400,000.

Key points Amount or part of beginning

Base: $400,000. Rate: .80.

Portion (?)

Portion: ? Portion ⫽ Base ⫻ Rate.

Base ⫻ Rate ($400,000) (.80) Beginning whole quantity (often after “of”)

Percent symbol or word (here we put into decimal)

Portion and rate must relate to same piece of base.

Steps to solving problem 1. Set up the formula. 2. Calculate portion (sales of Milk Chocolate M&M’s®).

Portion ⫽ Base ⫻ Rate

P ⫽ $400,000 ⫻ .80 P ⫽ $320,000

In the first column of the blueprint aid, we gather the facts. In the second column, we state that we are looking for sales of Milk Chocolate M&M’ s®. In the third column, we identify each key element and the formula needed to solve the problem. Review the pie chart in the fourth column. Note that the portion and rate must relate to the same piece of the base. In this word problem, we can see from the solution below the blueprint aid that sales of Milk Chocolate M&M’s® are $320,000. The $320,000 does indeed represent 80% of the base. Note here that the portion ($320,000) is less than the base of $400,000 since the rate is less than 100%. Now let’s work another word problem that solves for the portion. The Word Problem Sales of Milk Chocolate M&M’s® are 80% of the total M&M’s® sales.

Total M&M’s® sales are $400,000. What are the sales of Peanut and other M&M’ s® chocolate candies?

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Chapter 6 Percents and Their Applications

The facts

Solving for?

Milk Chocolate M&M’s® sales: 80%.

Sales of Peanut and other M&M’s® chocolate candies.

Total M&M’s® sales: $400,000.

Steps to take

Key points

Identify key elements.

If 80% of sales are Milk Chocolate M&M’s, then 20% are Peanut and other M&M’s® chocolate candies.

Base: $400,000. Rate: .20 (100% 80%). Portion: ? Portion Base Rate.

Portion (?) Base Rate ($400,000) (.20) Portion and rate must relate to same piece of base.

Steps to solving problem Portion Base Rate

1. Set up the formula.

P $400,000 .20

2. Calculate portion (sale of Peanut and other M&M’s® chocolate candies).

P $80,000

In the previous blueprint aid, note that we must use a rate that agrees with the portion so the portion and rate refer to the same piece of the base. Thus, if 80% of sales are Milk Chocolate M&M’ s®, 20% must be Peanut and other M&M’ s® chocolate candies (100% 80% 20%). So we use a rate of .20. In Step 2, we multiplied $400,000 .20 to get a portion of $80,000. This portion represents the part of the sales that were not Milk Chocolate M&M’ s®. Note that the rate of .20 and the portion of $80,000 relate to the same piece of the base—$80,000 is 20% of $400,000. Also note that the portion ($80,000) is less than the base ($400,000) since the rate is less than 100%. Take a moment to review the two blueprint aids in this section. Be sure you understand why the rate in the first blueprint aid was 80% and the rate in the second blueprint aid was 20%. Solving for Rate The Word Problem Sales of Milk Chocolate M&M’ s® are $320,000. Total M&M’s® sales

s® sales compared to total

are $400,000. What is the percent of Milk Chocolate M&M’ M&M’s® sales? The facts

Solving for?

Steps to take

Key points

Milk Chocolate M&M’s® sales: $320,000.

Percent of Milk Chocolate M&M’s® sales to total M&M’s® sales.

Identify key elements.

Since portion is less than base, the rate must be less than 100%

Total M&M’s® sales: $400,000.

Base: $400,000. Rate: ? Portion: $320,000 Rate

Portion ($320,000)

Portion Base

Base Rate ($400,000) (?)

Portion and rate must relate to the same piece of base. Steps to solving problem 1. Set up the formula. 2. Calculate rate (percent of Milk Chocolate M&M’s® sales).

Portion Base $320,000 R $400,000

Rate

R 80%

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Learning Unit 6–2

147

Note that in this word problem, the rate of 80% and the portion of $320,000 refer to the same piece of the base. Sales of Milk Chocolate M&M’ s® are $320,000. Total sales of Milk Chocolate M&M’ s, Peanut, and other M&M’ s® chocolate candies are $400,000. What percent of Peanut and other M&M’ s® chocolate candies are sold compared to total M&M’s® sales? The Word Problem

The facts

Solving for?

Steps to take

Key points

Milk Chocolate M&M’s® sales: $320,000.

Percent of Peanut and other M&M’s® chocolate candies sales compared to total M&M’s® sales.

Identify key elements.

Represents sales of Peanut and other M&M’s® chocolate candies

Total M&M’s® sales: $400,000.

Base: $400,000. Rate: ? Portion: $80,000 ($400,000 $320,000). Rate

Portion Base

Portion ($80,000) Base Rate ($400,000) (?)

When portion becomes $80,000, the portion and rate now relate to same piece of base.

Steps to solving problem Rate

1. Set up the formula.

R

2. Calculate rate.

Portion Base $80,000 ($400,000 $320,000) $400,000

R 20%

The word problem asks for the rate of candy sales that are not Milk Chocolate M&M’s. Thus, $400,000 of total candy sales less sales of Milk Chocolate M&M’ s® ($320,000) ® allows us to arrive at sales of Peanut and other M&M’ s chocolate candies ($80,000). The $80,000 portion represents 20% of total candy sales. The $80,000 portion and 20% rate refer to the same piece of the $400,000 base. Compare this blueprint aid with the blueprint aid for the previous word problem. Ask yourself why in the previous word problem the rate was 80% and in this word problem the rate is 20%. In both word problems, the portion was less than the base since the rate was less than 100%. Now we go on to calculate the base. Remember to read the word problem carefully so that you match the rate and portion to the same piece of the base. Solving for Base The Word Problem Sales of Peanut and other M&M’ s® chocolate candies are 20% of

total M&M’s® sales. Sales of Milk Chocolate M&M’ s® are $320,000. What are the total sales of all M&M’ s®? The facts

Solving for?

Steps to take

Peanut and other M&M’s® chocolate candies sales: 20%.

Total M&M’s® sales.

Identify key elements.

Milk Chocolate M&M’s® sales: $320,000.

Base: ? Rate: .80 (100% 20%)

Portion: $320,000 Portion Base Rate

Key points Portion ($320,000) Base Rate (?) (.80) (100% – 20%) Portion ($320,000) and rate (.80) do relate to the same piece of base.

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Steps to solving problem Base

1. Set up the formula.

Portion Rate

$320,000 .80 B $400,000 B

2. Calculate the base.

$320,000 is 80% of base

Note that we could not use 20% for the rate. The $320,000 of Milk Chocolate M&M’s® represents 80% (100% 20%) of the total sales of M&M’ s®. We use 80% so that the portion and rate refer to same piece of the base. Remember that the portion ($320,000) is less than the base ($400,000) since the rate is less than 100%.

Calculating Percent Decreases and Increases In the following Wall Street Journal clipping “Wal-Mart’s Entry Likely to Reshape Warranty Game,” we see a product’ s warranty as a percentage of the price of a 42-inch plasma TV. If you buy a 42-inch plasma TV, would you buy an extended warranty? Did you realize how much extended warranties can cost? Using this clipping, let’ s look at how to calculate percent decreases and increases.

Wall Street Journal © 2005

Rate of Percent Decrease Using Sears Assume: Sears drops its 42-inch plasma TV price to $900 from $1,500. Difference between old and new TV price Portion Old TV amount Base $600 ($1,500 $900) R $1,500 R 40%

Rate

Let’s prove the 40% with a pie chart. Decrease in TV price Portion ($600) Base Rate ($1,500) (?) Original TV price

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149

The formula for calculating Sears’ percent decrease is as follows: Percent decrease Amount of decrease (P) Percent of ($600) decrease (R) Original TV price (B) (40%) ($1,500)

Now let’s look at how to calculate Best Buy’ s percent increase in plasma TVs using the portion formula for solving the rate. Rate of Percent Increase Using Best Buy Assume: Best Buy increases its 42-inch plasma TV price to $1,200 from $1,000. Difference between old and new TV price Portion Old TV amount Base $200 ($1,200 $1,000) R $1,000

Rate

R 20% Let’s prove the 20% with a pie chart. Increase in TV price Portion ($200) Base Rate ($1,000) (?) Original TV price

The formula for calculating Best Buy’ s percent increase is as follows: Percent increase Amount of increase (P) Percent of ($200) increase (R) Original TV price (B) (20%) ($1,000)

In conclusion, the following steps can be used to calculate percent decreases and increases: CALCULATING PERCENT DECREASES AND INCREASES Step 1.

Find the difference between amounts (such as advertising costs).

Step 2.

Divide Step 1 by the original amount (the base): R P B. Be sure to express your answer in percent.

Before concluding this chapter , we will show how to calculate a percent increase and decrease using M&M’ s® (Figure 6.1). FIGURE

6.1

Bag of 18.40-ounce M&M’s®

% 15 EXTRA... FREE 15% MORE THAN REGULAR 1 LB. BAG

NET WT 18.40 oz. (1 LB 2.40 oz) 521.6g

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Additional Examples Using M&M’s The Word Problem Sheila Leary went to her local supermarket and bought the bag of

M&M’s® shown in Figure 6.1 ( p. 149). The bag gave its weight as 18.40 ounces, which was 15% more than a regular 1-pound bag of M&M’ s®. Sheila, who is a careful shopper , wanted to check and see if she was actually getting a 15% increase. Let’ s help Sheila dissect and solve this problem. The facts

Solving for?

Steps to take

Key points

New bag of M&M’s®: 18.40 oz.

Checking percent increase of 15%.

Identify key elements.

Difference between base and new weight

Base: 16 oz.

15% increase in weight.

Rate: ?

Portion (2.40 oz.)

Portion: 2.40 oz.

Original bag of M&M’s®: 16 oz. (1 lb.)

¢

18.40 oz. ≤ 16.00 2.40 oz.

Rate

Portion Base

Base Rate (16 oz.) (?) Original amount sold

Steps to solving problem Portion Base 2.40 oz . R 16.00 oz .

Rate

1. Set up the formula. 2. Calculate the rate.

Difference between base and new weight. Old weight equals 100%.

R 15% increase

The new weight of the bag of M&M’ s® is really 1 15% of the old weight: 16.00 oz. 100% 2.40 15 18.40 oz. 115% 1.15 We can check this by looking at the following pie chart: Portion Base Rate

Portion (18.40 oz.)

18.40 oz. 16 oz. 1.15

Base Rate (16 oz.) (1.15) 100%

Why is the portion greater than the base? Remember that the portion can be lar ger than the base only if the rate is greater than 100%. Note how the portion and rate relate to the same piece of the base—18.40 oz. is 115% of the base (16 oz.). Let’s see what could happen if M&M/Mars has an increase in its price of sugar . This is an additional example to reinforce the concept of percent decrease. The Word Problem The increase in the price of sugar caused the M&M/Mars company to

decrease the weight of each 1-pound bag of M&M’ s® to 12 ounces. What is the rate of percent decrease? The facts

Solving for?

Steps to take

16-oz. bag of M&M’s®: reduced to 12 oz.

Rate of percent decrease.

Identify key elements.

Key points

Portion (4 oz.)

Base: 16 oz. Rate: ? Portion: 4 oz. (16 oz. 12 oz.) Rate

Portion Base

Base Rate (16 oz.) (?) Old base 100%

Amount of decrease

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151

Steps to solving problem Rate

1. Set up the formula.

R

2. Calculate the rate.

Portion Base 4 oz . 16.00 oz .

R 25% decrease ®

The new weight of the bag of M&M’ s is 75% of the old weight: 16 oz. 100% 4 25 12 oz. 75% We can check this by looking at the following pie chart: Portion Base Rate Portion (12 oz.)

12 oz. 16 oz. .75 100%

Base Rate (16 oz.) (.75)

Note that the portion is smaller than the base because the rate is less than 100%. Also note how the portion and rate relate to the same piece of the base—12 ounces is 75% of the base (16 oz.). After your study of Learning Unit 6–2, you should be ready for the Practice Quiz.

LU 6–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Solve for portion: 1. 38% of 900. 2. 60% of $9,000. Solve for rate (round to nearest tenth percent as needed): 3. 430 is % of 5,000. 4. 200 is % of 700. Solve for base (round to the nearest tenth as needed): 5. 55 is 40% of . 6. 900 is 412% of . Solve the following (blueprint aids are shown in the solution; you might want to try some on scrap paper): 7. Five out of 25 students in Professor Ford’ s class received an A grade. What percent of the class did not receive the A grade? 8. Abby Biernet has yet to receive 60% of her lobster order . Abby received 80 lobsters to date. What was her original order? 9. In 2006, Dunkin’ Donuts Company had $300,000 in doughnut sales. In 2007, sales were up 40%. What are Dunkin’ Donuts sales for 2007? 10. The price of an Apple computer dropped from $1,600 to $1,200. What was the percent decrease? 11. In 1982, a ticket to the Boston Celtics cost $14. In 2007, a ticket cost $50. What is the percent increase to the nearest hundredth percent?

✓ 1.

3. 5.

Solutions 342 900 .38 (P) (B) (R) (P)430 .086 8.6% (R) (B)5,000 (P)55 137.5 (B) (R).40

$5,400 $9,000 .60 (P) (B) (R) (P)200 .2857 28.6% (R) 4. (B)700 (P)900 20,000 (B) 6. (R).045

2.

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7. Percent of Professor Ford’ s class that did not receive an A grade: The facts

Solving for?

Steps to take

5 As.

Percent that did not receive A.

Identify key elements.

25 in class.

Base: 25 Rate: ? Portion: 20 (25 5). Rate

Portion Base

Key points Portion (20) Base Rate (25) (?) The whole Portion and rate must relate to same piece of base.

Steps to solving problem Portion Base 20 R 25

Rate

1. Set up the formula. 2. Calculate the rate.

R 80%

8. Abby Biernet’s original order: The facts

Solving for?

Steps to take

60% of the order not in.

Total order of lobsters.

Identify key elements.

80 lobsters received.

Base: ? Rate: .40 (100% 60%)

Key points Portion (80) Base Rate (?) (.40)

Portion: 80. Base

Portion Rate

80 lobsters represent 40% of the order Portion and rate must relate to same piece of base.

Steps to solving problem Base

1. Set up the formula.

B

2. Calculate the rate.

Portion Rate 80 .40

80 lobsters is 40% of base.

B 200 lobsters

9.

Dunkin’ Donuts Company sales for 2007:

The facts

Solving for?

Steps to take

2006: $300,000 sales.

Sales for 2007.

Identify key elements.

2007: Sales up 40% from 2006.

Key points 2007 sales

Base: $300,000. Rate: 1.40. Old year New year

100% 40 140%

Portion: ? Portion Base Rate.

Portion (?) Base Rate ($300,000) (1.40) 2006 sales When rate is greater than 100%, portion will be larger than base.

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Learning Unit 6–2

Steps to solving problem Portion Base Rate

1. Set up the formula. 2. Calculate the portion.

P $300,000 1.40 P $420,000

10.

Percent decrease in Apple computer price:

The facts

Solving for?

Steps to take

Apple computer was $1,600; now, $1,200.

Percent decrease in price.

Identify key elements.

Key points Difference in price

Base: $1,600. Rate: ? Portion: $400 ($1,600 $1,200). Portion Rate Base

Portion ($400) Base Rate ($1,600) (?) Original price

Steps to solving problem Portion Base $400 R $1,600

Rate

1. Set up the formula. 2. Calculate the rate.

R 25%

11.

Percent increase in Boston Celtics ticket:

The facts

Solving for?

Steps to take

$14 ticket (old).

Percent increase in price.

Identify key elements.

$50 ticket (new).

Key points Difference in price

Base: $14 Rate: ? Portion: $36 ($50 $14) Portion Rate Base

Portion ($36) Base Rate ($14) (?) Original price When portion is greater than base, rate will be greater than 100%.

Steps to solving problem 1. Set up the formula. 2. Calculate the rate.

Portion Base $36 R $14

Rate

R 2.5714 257.14%

153

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LU 6–2a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 156)

Solve for portion: 1. 42% of 1,200 2. 7% of $8,000 Solve for rate (round to nearest tenth percent as needed): 3. 510 is ———– % of 6,000. 4. 400 is ———–% of 900. Solve for base (round to the nearest tenth as needed): 5. 30 is 60% of ———–. 6. 1,200 is 3 12% of ———–. 7. Ten out of 25 students in Professor Ford’ s class received an A grade. What percent of the class did not receive the A grade? 8. Abby Biernet has yet to receive 70% of her lobster order . Abby received 90 lobsters to date. What was her original order? 9. In 2006, Dunkin’Donuts Company had $400,000 in doughnut sales. In 2007, sales were up 35%. What are Dunkin’ Donuts sales for 2007?

10. 11.

The price of an Apple computer dropped from $1,800 to $1,000. What was the percent decrease? (Round to the nearest hundredth percent.) In 1982, a ticket to the Boston Celtics cost $14. In 2009, a ticket cost $75. What is the percent increase to the nearest hundredth percent?

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Converting decimals to percents, p. 139

1. Move decimal point two places to right. If necessary, add zeros. This rule is also used for whole numbers and mixed decimals. 2. Add a percent symbol at end of number.

Rounding percents, p. 140

Converting percents to decimals, p. 141

Converting fractions to percents, p. 142

1. Answer must be in percent before rounding. 2. Identify specific digit. If digit to right is 5 or greater, round up. 3. Delete digits to right of identified digit. 1. Drop percent symbol. 2. Move decimal point two places to left. If necessary, add zeros. For fractional percents: 1. Convert to decimal by dividing numerator by denominator. If necessary, round answer. 2. If a mixed fractional percent, convert fractional percent first. Then combine whole number and fractional percent. 3. Drop percent symbol, move decimal point two places to left. 1. Divide numerator by denominator. 2. Move decimal point two places to right; add percent symbol.

Example(s) to illustrate situation .81 .81. 81% .008 .00.8 .8% 4.15 4.15. 415% Round to the nearest hundredth percent. 3 .4285714 42.85714% 42.86% 7

.89% .0089 95% .95 195% 1.95 3 8 % 8.75% .0875 4 1 % .25% .0025 4 1 % .20% .0020 5 4 .80 80% 5

(continues)

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155

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (continued) Topic

Key point, procedure, formula

Converting percents to fractions, p. 142

Whole percent (or fractional percent) to a fraction: 1. Drop percent symbol. 1 2. Multiply number by 100 . 3. Reduce to lowest terms. Mixed or decimal percent to a fraction: 1. Drop percent symbol. 2. Change mixed percent to an improper fraction. 1 3. Multiply number by 100 . 4. Reduce to lowest terms. If you have a mixed or decimal percent, change decimal portion to fractional equivalent and continue with Steps 1 to 4.

Solving for portion, p. 145

“is” Portion (?) Base Rate ($1,000) (.10) “of”

“%”

Solving for rate, p. 146 Portion ($100) Base Rate ($1,000) (?)

Solving for base, p. 147

Portion ($100) Base Rate (?) (.10)

Calculating percent decreases and increases, p. 148

Amount of decrease or increase

Portion Base Rate (?) Original price

KEY TERMS

Base, p. 144 Percent decrease, p. 149

Example(s) to illustrate situation 64

64% 1 % 4

1 64 16 100 100 25

1 1 1 4 100 400 1 119 19 1 100 100 100

119%

119

1 16 % 4

65 1 65 13 4 100 400 80

16.25%

1 65 1 16 % 4 4 100

65 13 100 80

10% of Mel’s paycheck of $1,000 goes for food. What portion is deducted for food? $100 $1,000 .10 Note: If question was what amount does not go for food, the portion would have been: $900 $1,000 .90 (100% 10% 90% Assume Mel spends $100 for food from his $1,000 paycheck. What percent of his paycheck is spent on food? $100 .10 10% $1,000 Note: Portion is less than base since rate is less than 100%. Assume Mel spends $100 for food, which is 10% of his paycheck. What is Mel’s total paycheck? $100 $1,000 .10 Stereo, $2,000 original price. Stereo, $2,500 new price. $500 .25 25% increase $2,000 Check $2,000 1.25 $2,500 Note: Portion is greater Portion than base since rate is ($2,500) greater than 100%. Base Rate ($2,000) (1.25)

Percent increase, p. 149 Percents, p. 138

Portion, p. 144 Rate, p. 144

(continues)

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CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

Key point, procedure, formula 1. 44.4% 2. 78.2% 3. .6% 4. 793.3% 5. .0020 6. .0780 7. .92

LU 6–1a (p. 143) 8. 7.658 9. 33.33% 10. 42.86% 17 11. 100 329 12. 400 1 13. 1 2 1 14. 400 89 15. 500

Example(s) to illustrate situation 1. 2. 3. 4. 5. 6.

504 560 8.5% 44.4% 50 34,285.7

LU 6–2a (p. 154) 7. 60% 8. 300 9. $540,000 10. 44.44% 11. 435.71%

Note: For how to dissect and solve a word problem, see page 145.

Critical Thinking Discussion Questions 1. In converting from a percent to a decimal, when will you have at least 2 leading zeros before the whole number? Explain this concept, assuming you have 100 bills of $1. 2. Explain the steps in rounding percents. Count the number of students who are sitting in the back half of the room as a percent of the total class. Round your answer to the nearest hundredth percent. Could you have rounded to the nearest whole percent without changing the accuracy of the answer? 3. Define portion, rate, and base. Create an example using Walt Disney World to show when the portion could be larger than the base. Why must the rate be greater than 100% for this to happen?

4. How do we solve for portion, rate, and base? Create an example using IBM computer sales to show that the portion and rate do relate to the same piece of the base. 5. Explain how to calculate percent decreases or increases. Many years ago, comic books cost 10 cents a copy . Visit a bookshop or newsstand. Select a new comic book and explain the price increase in percent compared to the 10-cent comic. How important is the rounding process in your final answer?

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Convert the following decimals to percents: 6–1. .74

6–2.

.824

6–3.

.9

6–4. 8.00

6–5.

3.561

6–6.

6.006

6–8.

14%

6–9.

3 64 % 10

6–11.

119%

6–12.

Convert the following percents to decimals: 6–7. 8% 6–10. 75.9%

89%

Convert the following fractions to percents (round to the nearest tenth percent as needed): 6–13.

1 12

6–14.

1 400

6–15.

7 8

6–16.

11 12

Convert the following to fractions and reduce to the lowest terms: 6–17. 4%

6–18.

2 6–19. 31 % 3

6–20.

6–21. 6.75%

6–22.

1 18 % 2 1 61 % 2 182%

Solve for the portion (round to the nearest hundredth as needed): 6–23. 7% of 150

6–24.

125% of 4,320

6–25.

25% of 410

6–26. 119% of 128.9

6–27.

17.4% of 900

6–28.

11.2% of 85

6–29. 12 % of 919

1 2

6–30.

45% of 300

6–31. 18% of 90

6–32.

30% of 2,000

Solve for the base (round to the nearest hundredth as needed): 6–33. 170 is 120% of

6–34. 36 is .75% of

6–35. 50 is .5% of

6–36. 10,800 is 90% of

1 6–37. 800 is 4 % of 2 Solve for rate (round to the nearest tenth percent as needed): 6–38.

of 80 is 50

6–39.

6–40.

of 250 is 65

6–41.

110 is

6–43.

16 is

6–42.

.09 is

of 2.25

of 85 is 92 of 100 of 4 157

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Solve the following problems. Be sure to show your work. Round to the nearest hundredth or hundredth percent as needed: 6–44. What is 180% of 310? 6–45. 66% of 90 is what? 6–46. 40% of what number is 20? 6–47. 770 is 70% of what number? 6–48. 4 is what percent of 90? 6–49. What percent of 150 is 60? Complete the following table: Sales in millions Product

2007

2008

6–50. Digital cameras

$380

$410

6–51. DVD players

$ 50

$ 47

Amount of decrease or increase

Percent change (to nearest hundredth percent as needed)

WORD PROBLEMS (First of Four Sets) 6–52. At a local Wendy’s, a survey showed that out of 12,000 customers eating lunch, 3,000 ordered Diet Pepsi with their meal. What percent of customers ordered Diet Pepsi?

6–53. What percent of customers in Problem 6–52 did not order Diet Pepsi?

6–54. The Rhinelander Daily News March 4, 2007 issue, ran a story on rising gas prices. Last week, gas was selling for $1.99 a gallon and the world looked rosy. Not so now. The price of a gallon of regular unleaded nosed up to $2.24. What was the percent increase? Round to the nearest hundredth percent.

6–55. Wally Chin, the owner of an Exxon-Mobil station, bought a used Ford pickup truck, paying $2,000 as a down payment. He still owes 80% of the selling price. What was the selling price of the truck?

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6–56. Maria Fay bought 4 Aquatread tires at a local Goodyear store. The salesperson told her that her mileage would increase by 6%. Before this purchase, Maria was getting 22 mpg. What should her mileage be with the new tires to nearest hundredth?

6–57. Jeff Rowe went to Best Buy and bought a Cannon digital camera. The purchase price was $400. Jeff made a down payment of 40%. How much was Jeff’s down payment?

6–58. Assume that in the year 2009, 800,000 people attended the Christmas Eve celebration at Walt Disney World. In 2010, attendance for the Christmas Eve celebration is expected to increase by 35%. What is the total number of people expected at Walt Disney World for this event?

6–59. Pete Smith found in his attic a Woody Woodpecker watch in its original box. It had a price tag on it for $4.50. The watch was made in 1949. Pete brought the watch to an antiques dealer and sold it for $35. What was the percent of increase in price? Round to the nearest hundredth percent.

6–60. Fuel inventories were lower than last year according to the San Francisco Chronicle dated February 24, 2007. This year there are 31.7 million barrels of inventory, a 7.32 percent drop. (a) What was the amount of inventory last year to nearest tenth? (b) What was the amount of the decrease?

6–61. Christie’s Auction sold a painting for $24,500. It charges all buyers a 15% premium of the final bid price. How much did the bidder pay Christie’s?

WORD PROBLEMS (Second of Four Sets) 6–62. Out of 9,000 college students surveyed, 540 responded that they do not eat breakfast. What percent of the students do not eat breakfast?

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6–63. What percent of college students in Problem 6–62 eat breakfast?

6–64. Alice Hall made a $3,000 down payment on a new Ford Explorer wagon. She still owes 90% of the selling price. What was the selling price of the wagon?

6–65. The Kansas City Star on February 16, 2007 reported on the lowering prices of natural gas. With lower natural gas prices and unseasonably warm weather early this winter the forecast is that gas-heating customers will pay $825 as compared to $1,019 last year. At Kansas Gas Service, last year, the price of 1,000 cubic feet of gas was $9.95; this has been reduced by 12.1608 percent. (a) What percent was the decrease for customer’s bills? (b) What was the amount of decreased charge for 1,000 cubic feet of gas? Round to nearest cent. (c) What is the new price for 1,000 cubic feet? Round to the nearest hundredth percent.

6–66. Jim and Alice Lange, employees at Wal-Mart, have put themselves on a strict budget. Their goal at year’s end is to buy a boat for $15,000 in cash. Their budget includes the following: 40% food and lodging

20% entertainment

10% educational

Jim earns $1,900 per month and Alice earns $2,400 per month. After one year, will Alice and Jim have enough cash to buy the boat?

6–67. The price of a Fossil watch dropped from $49.95 to $30.00. What was the percent decrease in price? Round to the nearest hundredth percent.

6–68. The Museum of Science in Boston estimated that 64% of all visitors came from within the state. On Saturday, 2,500 people attended the museum. How many attended the museum from out of state?

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6–69. Staples pays George Nagovsky an annual salary of $36,000. Today, George’s boss informs him that he will receive a $4,600 raise. What percent of George’s old salary is the $4,600 raise? Round to the nearest tenth percent.

6–70. In 2009, Dairy Queen had $550,000 in sales. In 2010, Dairy Queen’s sales were up 35%. What were Dairy Queen’s sales in 2010?

6–71. Blue Valley College has 600 female students. This is 60% of the total student body. How many students attend Blue Valley College?

6–72. Dr. Grossman was reviewing his total accounts receivable. This month, credit customers paid $44,000, which represented 20% of all receivables (what customers owe) due. What was Dr. Grossman’s total accounts receivable?

6–73. Massachusetts has a 5% sales tax. Timothy bought a Toro lawn mower and paid $20 sales tax. What was the cost of the lawn mower before the tax?

6–74. The price of an antique doll increased from $600 to $800. What was the percent of increase? Round to the nearest tenth percent.

6–75. Borders bookstore ordered 80 marketing books but received 60 books. What percent of the order was missing?

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WORD PROBLEMS (Third of Four Sets) 6–76. At a Christie’s auction, the auctioneer estimated that 40% of the audience was from within the state. Eight hundred people attended the auction. How many out-of-state people attended?

6–77. Due to increased mailing costs, the new rate will cost publishers $50 million; this is 12.5% more than they paid the previous year. How much did it cost publishers last year? Round to the nearest hundreds.

6–78. In 2010, Jim Goodman, an employee at Walgreens, earned $45,900, an increase of 17.5% over the previous year. What were Jim’s earnings in 2009? Round to the nearest cent.

6–79. If the number of mortgage applications declined by 7% to 1,625,415, what had been the previous year’s number of applications?

6–80. In 2010, the price of a business math text rose to $100. This is 6% more than the 2009 price. What was the old selling price? Round to the nearest cent.

6–81. Web Consultants, Inc., pays Alice Rose an annual salary of $48,000. Today, Alice’s boss informs her that she will receive a $6,400 raise. What percent of Alice’s old salary is the $6,400 raise? Round to nearest tenth percent.

6–82. Earl Miller, a lawyer, charges Lee’s Plumbing, his client, 25% of what he can collect for Lee from customers whose accounts are past due. The attorney also charges, in addition to the 25%, a flat fee of $50 per customer. This month, Earl collected $7,000 from 3 of Lee’s past-due customers. What is the total fee due to Earl?

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6–83. Petco ordered 100 dog calendars but received 60. What percent of the order was missing?

6–84. Blockbuster Video uses MasterCard. MasterCard charges 212% on net deposits (credit slips less returns). Blockbuster made a net deposit of $4,100 for charge sales. How much did MasterCard charge Blockbuster?

6–85. In 2009, Internet Access had $800,000 in sales. In 2010, Internet Access sales were up 45%. What are the sales for 2010?

WORD PROBLEMS (Fourth of Four Sets) 6–86. Saab Corporation raised the base price of its popular 900 series by $1,200 to $33,500. What was the percent increase? Round to the nearest tenth percent.

6–87. The sales tax rate is 8%. If Jim bought a new Buick and paid a sales tax of $1,920, what was the cost of the Buick before the tax?

6–88. Puthina Unge bought a new Compaq computer system on sale for $1,800. It was advertised as 30% off the regular price. What was the original price of the computer? Round to the nearest dollar.

6–89. John O’Sullivan has just completed his first year in business. His records show that he spent the following in advertising: Newspaper

$600

Radio

$650

Yellow Pages

$700

Local flyers

$400

What percent of John’s advertising was spent on the Yellow Pages? Round to the nearest hundredth percent.

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6–90. The Cincinnati Post reported holiday spending predictions. Columbus-based Big Research LLC surveyed nearly 7,700 consumers. The survey found 22% of consumers planned to begin shopping in either September or October, and another 17% had started in August or earlier. (a) How many consumers planned to begin shopping in September or October? (b) How many consumers planned to begin in August or earlier?

6–91. Abby Kaminsky sold her ski house at Attitash Mountain in New Hampshire for $35,000. This sale represented a loss of 15% off the original price. What was the original price Abby paid for the ski house? Round your answer to the nearest dollar.

6–92. Out of 4,000 colleges surveyed, 60% reported that SAT scores were not used as a high consideration in viewing their applications. How many schools view the SAT as important in screening applicants?

6–93. If refinishing your basement at a cost of $45,404 would add $18,270 to the resale value of your home, what percent of your cost is recouped? Round to the nearest percent.

6–94. A major airline laid off 4,000 pilots and flight attendants. If this was a 12.5% reduction in the workforce, what was the size of the workforce after the layoffs?

6–95. Assume 450,000 people line up on the streets to see the Macy’s Thanksgiving Parade in 2008. If attendance is expected to increase 30%, what will be the number of people lined up on the street to see the 2009 parade?

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CHALLENGE PROBLEMS 6–96. Continental Airlines stock climbed 4% from $18.04. Shares of AMR Corporation, American Airlines’ parent company, closed up 7% at $12.55. AirTran Airways went from $17.27 to $17.96. Round answers to the nearest hundredth. (a) What is the new price of Continental Airlines stock? (b) What had been the price of AMR Corporation stock? (c) What percent did AirTran Airways increase? Round to the nearest percent.

6–97. A local Dunkin’ Donuts shop reported that its sales have increased exactly 22% per year for the last 2 years. This year’s sales were $82,500. What were Dunkin’ Donuts sales 2 years ago? Round each year’s sales to the nearest dollar.

DVD SUMMARY PRACTICE TEST Convert the following decimals to percents. (p. 139) 1. .921

2. .4

3. 15.88

4. 8.00

7. 400%

8.

Convert the following percents to decimals. (p. 141) 5. 42%

6. 7.98%

1 % 4

Convert the following fractions to percents. Round to the nearest tenth percent. (p. 142) 9.

1 6

10.

1 3

Convert the following percents to fractions and reduce to the lowest terms as needed. (p. 142) 3 11. 19 % 8

12. 6.2%

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Solve the following problems for portion, base, or rate: 13. An Arby’s franchise has a net income before taxes of $900,000. The company’s treasurer estimates that 40% of the company’s net income will go to federal and state taxes. How much will the Arby’s franchise have left? (p. 145)

14. Domino’s projects a year-end net income of $699,000. The net income represents 30% of its annual sales. What are Domino’s projected annual sales? (p. 147)

15. Target ordered 400 iPods. When Target received the order, 100 iPods were missing. What percent of the order did Target receive? (p. 146)

16. Matthew Song, an employee at Putnam Investments, receives an annual salary of $120,000. Today his boss informed him that he would receive a $3,200 raise. What percent of his old salary is the $3,200 raise? Round to the nearest hundredth percent. (p. 146)

17. The price of a Delta airline ticket from Los Angeles to Boston increased to $440. This is a 15% increase. What was the old fare? Round to the nearest cent. (p. 147)

18. Scupper Grace earns a gross pay of $900 per week at Office Depot. Scupper’s payroll deductions are 29%. What is Scupper’s take-home pay? (p. 145)

19. Mia Wong is reviewing the total accounts receivable of Wong’s department store. Credit customers paid $90,000 this month. This represents 60% of all receivables due. What is Mia’s total accounts receivable? (p. 147)

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Personal Finance A KIPLINGER APPROACH HOME

| Will remodeling pay off when you move? By Patricia Mertz Esswein

Live better and SELL HIGHER

R

PAY B A C K The dollars and sense of a dozen popular projects heck out the national average costs for 12 remodeling projects and estimates of how much that cost will be recouped. Payback can vary dramatically by region. The recovery for new siding, for example, ranges from 80% in the West to 105% in the East.

C

THE PROJECT

THE PRICE

% COST RECOUPED

Minor kitchen remodel

$15,273

93%

Major kitchen remodel, mid-range

$42,660

79%

Major kitchen remodel, upscale

$75,206

80%

Bathroom remodel, mid-range

$9,861

90%

Bathroom remodel, upscale

$25,273

86%

Bathroom addition, mid-range

$21,087

86%

Bathroom addition, upscale

$41,587

81%

Master suite, mid-range

$70,245

80%

Master suite, upscale

$134,364

78%

Window replacement, mid-range

$9,273

85%

Window replacement, upscale

$15,383

84%

Siding replacement

$6,946

93%

SOURCE: Hanley Wood, LLC

F R O M TO P : S U B -Z E R O/ W O L F; A M E R I C A N S TA N D A R D ; R I C H A R D L E O J O H N S O N /G E T T Y I M A G E S ; A N D E R S E N W I N D O W S

emo d eli n g projects are enticing investments. You get to play the Iron Chef in a new, modern kitchen or pamper yourself in a spa-style bathroom, then recoup your money when you sell your house. In fact, anticipating that payback is often a driving force in convincing yourself—or your spouse—that a project is worth the money. But how much return can you count on? The latest report from Remodeling magazine says it’s not uncommon to recover 80% or more. Despite unrelenting new construction, the average U.S. home is 32 years old and in need of lifts, tucks and addons. So, home remodeling has become a national obsession. In 2004, Americans spent $186 billion on remodeling, according to Harvard University’s Joint Center for Housing Studies. The accompanying table shows the average price tag for a dozen popular projects, based on figures provided by HomeTech Information Systems, a company that develops software for estimating remodeling costs. The percentage of cost recouped at resale is based on estimates by members of the National Association of Realtors. The numbers are national averages; the full report (which can be ordered for $37.50 at www.remodelingmaga zine.com) includes estimates by region and for 60 cities. The payback can vary dramatically by region. Sal Alfano, editorial director of Remodeling magazine, notes that in extremely hot markets and those with a lot of new construction, resale values may slip below national averages. That’s because buyers would just as soon purchase a new house with all the amenities than a remodeled house. —Research: KATY MARQUARDT

BUSINESS MATH ISSUE Kiplinger’s © 2005

In today’s real estate market these % recouped numbers are unrealistic. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A What was the original circulation figure of the New York Daily News? Round to nearest hundredth. Check your answer.

2006 Wall Street Journal ©

PROJECT B Assume a package cost $42 to deliver by UPS in 2006. What would it cost assuming a new list rate increase of 4.9%?

Wall Street Journal © 2006

168

b site text We he e e S : s t T t Projec /slater9e) and e. Interne m ce Guid r o u .c o e s h e h R t .m e (www Intern ss Math Busine

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DVD

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Video Case AMERICAN PRESIDENT LINES

American President Lines (APL) has automated its terminal so the average turnaround time for a trucker picking up a 40foot container is only 17 minutes. APL uses an automated wireless system to track containers parked across its recently remodeled 160-acre facility in Seattle. The fast turnaround time gives customers who operate under the just-in-time mode the opportunity to make more trips. Independent truck drivers also benefit. The international freight industry is plagued by red tape and inefficiency. APL has used its website to help clients like Excel Corporation, the country’s second-largest beef packer and processor, speed up its billing time. Excel now wants to ask online for a place on a ship and for a call from APL when room will be available. The shipping market is enormous, estimated anywhere from $100 billion to $1 trillion. Imports in the United States

PROBLEM

1

The $170 billion in international trade volume per year given in the video is expected to increase by 50% in 5 years and expected to double over the next 25 years. (a) What is the expected total dollar amount in 5 years? (b) What is the expected total dollar amount in 25 years? PROBLEM

2

The video stated that thousands of containers arrive each day. Each 40-foot container will hold, for example, 16,500 boxes of running shoes, 132,000 videotapes, or 25,000 blouses. At an average retail price of $49.50 for a pair of running shoes, $14.95 for a videotape, and $26.40 for a blouse, what would be the total retail value of the goods in these three containers (assume different goods in each container)? PROBLEM

3

APL spent $600 million to build a 230-acre shipping terminal in California. The terminal can handle 4 wide-body container ships. Each ship can hold 4,800 20-foot containers, or 2,400 40-foot containers. (a) What was the cost per acre to build the facility? (b) How many 20-foot containers can the terminal handle at one time? (c) How many 40-foot containers can the terminal handle at one time? PROBLEM

4

According to Shanghai Daily, the recent decline in China’s export container prices (which fell by 1.4%) has not taken its toll on the general interest in this sector. China’s foreign trade grew by 35%, reaching $387.1 billion. APL reported that it would increase its services from Asia to Europe to take advantage of China’s growth in exports. What was the dollar amount of China’s foreign trade last year?

alone totaled 10 million containers, while exports totaled 6.5 million containers, together carrying $375 billion worth of goods. One of the most difficult transactions is to source goods from overseas and have them delivered with minimal paperwork all the way through to the end customer. Shipping lines must provide real-time information on the location of ships and goods. Most significant are attempts to automate shipping transactions online. The industry’s administrative inefficiencies, which account for 4% to 10% of international trade costs, are targeted. Industry insiders peg error rates on documents even higher, at 25% to 30%. It’s no secret that startups must overcome the reluctance of hidebound shipping lines, which have deep-seated emotional fears of dot-coms coming between them and their customers. In conclusion, American President Lines needs to get on board by staying online, or it might go down with the ship.

PROBLEM

5

APL has expanded its domestic fleet to 5,100 53-foot containers; it is expanding its global fleet to 253,000 containers. The 5,100 containers represent what percent of APL’s total fleet? Round to the nearest hundredth percent. PROBLEM

6

The cost of owning a shipping vessel is very high. Operating costs for large vessels can run between $75,000 and $80,000 per day. Using an average cost per day, what would be the operating costs for one week? PROBLEM

7

The Port of Los Angeles financed new terminal construction through operating revenues and bonds. They will collect about $30 million a year in rent from APL, who signed a 30-year lease on the property. What is APL’s monthly payment? PROBLEM

8

According to port officials, APL expanded cargo-handling capabilities at the Los Angeles facility that are expected to generate 10,500 jobs, with $335 million in wages and annual industry sales of $1 billion. What would be the average wage received? Round to the nearest dollar. PROBLEM

9

APL has disclosed that it ordered over 34,000 containers from a Chinese container manufacturer. With 253,000 containers in its possession, what will be the percent increase in containers owned by APL? Round to the nearest hundredth percent.

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CHAPTER

7

Discounts: Trade and Cash

LEARNING UNIT OBJECTIVES LU 7–1: Trade Discounts—Single and Chain (Includes Discussion of Freight) • Calculate single trade discounts with formulas and complements (pp. 171–172). • Explain the freight terms FOB shipping point and FOB destination (pp. 172–174). • Find list price when net price and trade discount rate are known (p. 174). • Calculate chain discounts with the net price equivalent rate and single equivalent discount rate (pp. 175–177).

LU 7–2: Cash Discounts, Credit Terms, and Partial Payments • List and explain typical discount periods and credit periods that a business may offer (pp. 179–185). • Calculate outstanding balance for partial payments (p. 186).

Wall Street Jo urnal © 2006

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Learning Unit 7–1

171

Wall Street Journal © 2005

Are you a good online bar gain hunter? The Wall Street Journal clipping “A Deal Seeker ’s Cheat Sheet” shows a variety of strategies customers can use to get the best price online. This chapter discusses two types of discounts taken by retailers—trade and cash. A trade discount is a reduction of f the original selling price (list price) of an item and is not related to early payment. A cash discount is the result of an early payment based on the terms of the sale.

Learning Unit 7–1: Trade Discounts—Single and Chain (Includes Discussion of Freight) Today we see “employee discounts” of fered to nonemployees. The Wall Street Journal clipping “New Deals” shows three examples of these nonemployee discounts offered by companies. Where do companies like Randall Scott Cycle Co. get their merchandise? The merchandise sold by retailers is bought from manufacturers and wholesalers who sell only to retailers and not to customers. These manufacturers and wholesalers of fer retailer discounts so retailers can resell the merchandise at a profit. The discounts are of f the manufacturers’ and wholesalers’ list price (suggested retail price), and the amount of discount that retailers receive of f the list price is the trade discount amount. When you make a purchase, the retailer (seller) Wall Street Journal © 2005 gives you a purchase invoice. Invoices are important business documents that help sellers keep track of sales transactions and buyers keep track of purchase transactions. North Shore Community College Bookstore is a retail seller of textbooks to students. The bookstore usually purchases its textbooks directly from publishers. Figure 7.1 ( p. 172) shows a textbook invoice from McGraw-Hill/Irwin Publishing Company to the North Shore Community College Bookstore. Note that the trade discount amount is given in percent. This is the trade discount rate, which is a percent of f the list price that retailers can deduct. The following formula

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Chapter 7 Discounts: Trade and Cash

FIGURE

7.1

Bookstore invoice showing a trade discount

Invoice No.: 5582 McGraw-Hill/Irwin Publishing Co. 1333 Burr Ridge Parkway Burr Ridge, Illinois 60527 Date: July 8, 2008 Ship: Two-day UPS Terms: 2/10, n/30

Sold to: North Shore Community College Bookstore 1 Ferncroft Road Danvers, MA 01923

Total amount

Unit list price

Description 50 Financial Management—Block/Hirt 10 Introduction to Business—Nichols

$95.66 89.50 Total List Price Less: Trade Discount 25% Net Price Plus: Prepaid Shipping Charge Total Invoice Amount

for calculating a trade discount amount gives the numbers from the Figure 7.1 parentheses:

$4,783.00 895.00 $5,678.00 _ 1,419.50 $4,258.50 125.00 $4,383.50

invoice in

TRADE DISCOUNT AMOUNT FORMULA Trade discount amount List price Trade discount rate ($1,419.50)

($5,678.00)

(25%)

The price that the retailer (bookstore) pays the manufacturer (publisher) or wholesaler is the net price. The following formula for calculating the net price gives the numbers from the Figure 7.1 invoice in parentheses: NET PRICE FORMULA Net price ($4,258.50)

List price ($5,678.00)

Trade discount amount ($1,419.50)

Frequently, manufacturers and wholesalers issue catalogs to retailers containing list prices of the seller ’s merchandise and the available trade discounts. To reduce printing costs when prices change, these sellers usually update the catalogs with new discount sheets. The discount sheet also gives the seller the flexibility of of fering different trade discounts to different classes of retailers. For example, some retailers buy in quantity and service the products. They may receive a lar ger discount than the retailer who wants the manufacturer to service the products. Sellers may also give discounts to meet a competitor ’s price, to attract new retailers, and to reward the retailers who buy product-line products. Sometimes the ability of the retailer to negotiate with the seller determines the trade discount amount. Retailers cannot take trade discounts on freight, returned goods, sales tax, and so on. Trade discounts may be single discounts or a chain of discounts. Before we discuss single trade discounts, let’ s study freight terms.

Freight Terms Do you know how successful the shipping businesses of DHL, UPS, and FedEx are in China? The Wall Street Journal clipping “Faster, Faster . . .” shows that the shipping businesses of these three companies can be quite profitable.

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Learning Unit 7–1

173

Claro Cortes IV/Reuters/Landov

Wall Street Journal © 2004

The most common freight terms are FOB shipping point and FOB destination. These terms determine how the freight will be paid. The key words in the terms are shipping point and destination. FOB shipping point means free on board at shipping point; that is, the buyer pays the freight cost of getting the goods to the place of business. For example, assume that IBM in San Diego bought goods from Argo Suppliers in Boston. Argo ships the goods FOB Boston by plane. IBM takes title to the goods when the aircraft in Boston receives the goods, so IBM pays the freight from Boston to San Diego. Frequently, the seller (Ar go) prepays the freight and adds the amount to the buyer ’s (IBM) invoice. When paying the invoice, the buyer takes the cash discount of f the net price and adds the freight cost. FOB shipping point can be illustrated as follows: FOB shipping point (Boston) Boston

San Diego

Argo Suppliers (IBM takes title here)

IBM (Buyer pays the freight costs) Buyer pays the freight

FOB destination means the seller pays the freight cost until it reaches the buyer ’s place of business. If Argo ships its goods to IBM FOB destination or FOB San Diego, the title to the goods remains with Argo. Then it is Argo’s responsibility to pay the freight from Boston to IBM’ s place of business in San Diego. FOB destination can be illustrated as follows: FOB destination (San Diego) Boston Argo Suppliers (Has title)

IBM (Get title on arrival of goods) Seller pays the freight

The following Wall Street Journal clipping (p. 174) shows the results of a performance test of four companies: Federal Express, DHL, United Parcel Service, and the United States Postal Service. Note the costs and conveniences of these four online delivery services relating to pickup fees, fuel surchar ge/own packaging, and website discounts. From the comments on the clipping, which of the four companies was most impressive?

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Chapter 7 Discounts: Trade and Cash

Wall Street Journal © 2005

Now you are ready for the discussion on single trade discounts.

Single Trade Discount In the introduction to this unit, we showed how to use the trade discount amount formula and the net price formula to calculate the McGraw-Hill/Irwin Publishing Company textbook sale to the North Shore Community College Bookstore. Since McGraw-Hill/Irwin gave the bookstore only one trade discount, it is a single trade discount. In the following word problem, we use the formulas to solve another example of a single trade discount. Again, we will use a blueprint aid to help dissect and solve the word problem. The Word Problem The list price of a Macintosh computer is $2,700. The manufacturer offers dealers a 40% trade discount. What are the trade discount amount and the net price? The facts

Solving for?

Steps to take

List price: $2,700.

Trade discount amount.

Trade discount amount ⫽ List price ⫻ Trade discount rate.

Trade discount rate: 40%.

Net price.

Key points Trade discount amount Portion (?)

Net price ⫽ List price ⫺ Trade discount amount.

Base ⫻ Rate ($2,700) (.40) List price

Trade discount rate

Steps to solving problem 1. Calculate the trade discount amount. 2. Calculate the net price.

$2,700 ⫻ .40 ⫽ $1,080 $2,700 ⫺ $1,080 ⫽ $1,620

Now let’s learn how to check the dealers’ net price of $1,620 with an alternate procedure using a complement. How to Calculate the Net Price Using Complement of Trade Discount Rate The complement of a trade discount rate is the dif ference between the discount rate and 100%. The following steps show you how to use the complement of a trade discount rate: CALCULATING NET PRICE USING COMPLEMENT OF TRADE DISCOUNT RATE Step 1.

To find the complement, subtract the single discount rate from 100%.

Step 2.

Multiply the list price times the complement (from Step 1).

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Learning Unit 7–1

175

Think of a complement of any given percent (decimal) as the result of subtracting the percent from 100%. Step 1.

100% ⫺ 40 Trade discount rate 60% or .60

Portion (?) Base ⫻ Rate ($2,700) (.60) List price

The complement means that we are spending 60 cents per dollar because we save 40 cents per dollar. Since we planned to spend $2,700, we multiply .60 by $2,700 to get a net price of $1,620. Step 2.

$1,620 ⫽ $2,700 ⫻ .60

Note how the portion ($1,620) and rate (.60) relate to the same piece of the base ($2,700). The portion ($1,620) is smaller than the base, since the rate is less than 100%. Be aware that some people prefer to use the trade discount amount formula and the net price formula to find the net price. Other people prefer to use the complement of the trade discount rate to find the net price. The result is always the same. Finding List Price When You Know Net Price and Trade Discount Rate The following formula has many useful applications: CALCULATING LIST PRICE WHEN NET PRICE AND TRADE DISCOUNT RATE ARE KNOWN List price ⫽

Net price Complement of trade discount rate

Next, let’s see how to dissect and solve a word problem calculating list price. The Word Problem A Macintosh computer has a $1,620 net price and a 40% trade dis-

count. What is its list price? The facts

Solving for?

Steps to take

Net price: $1,620.

List price.

List price ⫽ Net price Complement of trade discount rate

Trade discount rate: 40%.

Key points Net price Portion ($1,620) Base ⫻ Rate (?) (.60) List price

100% – 40%

Steps to solving problem 1. Calculate the complement of the trade discount.

2. Calculate the list price.

100% ⫺ 40 60% ⫽ .60 $1,620 ⫽ $2,700 .60

Note that the portion ($1,620) and rate (.60) relate to the same piece of the base. Let’s return to the McGraw-Hill/Irwin invoice in Figure 7.1 (p. 172) and calculate the list price using the formula for finding list price when net price and trade discount rate are known. The net price of the textbooks is $4,258.50. The complement of the trade discount rate is 100% ⫺ 25% ⫽ 75% ⫽ .75. Dividing the net price $4,258.50 by the complement .75

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Chapter 7 Discounts: Trade and Cash

equals $5,678.00, the list price shown in the McGraw-Hill/Irwin invoice. We can show this as follows: $4,258.50 $5,678.00, the list price .75

Chain Discounts Frequently, manufacturers want greater flexibility in setting trade discounts for dif ferent classes of customers, seasonal trends, promotional activities, and so on. To gain this flexibility, some sellers give chain or series discounts—trade discounts in a series of two or more successive discounts. Sellers list chain discounts as a group, for example, 20/15/10. Let’ s look at how Mick Company arrives at the net price of of fice equipment with a 20/15/10 chain discount. The list price of the office equipment is $15,000. The chain discount is 20/15/10. The long way to calculate the net price is as follows:

EXAMPLE

Never add the 20/15/10 together.

Step 1

Step 2

Step 3

Step 4

$15,000 .20 $ 3,000

$15,000 3,000 $12,000 .15 $ 1,800

$12,000 1,800 $10,200 .10 $ 1,020

$10,200 1,020 $ 9,180 net price

Note how we multiply the percent (in decimal) times the new balance after we subtract the previous trade discount amount. For example, in Step 3, we change the last discount, 10%, to decimal form and multiply times $10,200. Remember that each percent is multiplied by a successively smaller base. You could write the 20/15/10 discount rate in any order and still arrive at the same net price. Thus, you would get the $9,180 net price if the discount were 10/15/20 or 15/20/10. However , sellers usually give the lar ger discounts first. Never try to shorten this step process by adding the discounts. Your net price will be incorrect because, when done properly , each percent is calculated on a dif ferent base. Net Price Equivalent Rate In the example above, you could also find the $9,180 net price with the net price equivalent rate—a shortcut method. Let’ s see how to use this rate to calculate net price. CALCULATING NET PRICE USING NET PRICE EQUIVALENT RATE Step 1.

Subtract each chain discount rate from 100% (find the complement) and convert each percent to a decimal.

Step 2.

Multiply the decimals. Do not round off decimals, since this number is the net price equivalent rate.

Step 3.

Multiply the list price times the net price equivalent rate (Step 2).

The following word problem with its blueprint aid illustrates how to use the net price equivalent rate method. The Word Problem The list price of of fice equipment is $15,000. The chain discount is

20/15/10. What is the net price? The facts

Solving for?

Steps to take

Key points

List price: $15,000.

Net price.

Net price equivalent rate.

Do not round net price equivalent rate.

Chain discount: 20/15/10

Net price List price Net price equivalent rate.

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Learning Unit 7–1

177

Steps to solving problem 1. Calculate the complement of each rate and convert each percent to a decimal.

100% 20 80%

100% 15 85%

100% 10 90%

.8

.85

.9

2. Calculate the net price equivalent rate. (Do not round.)

.8 .85 .9 .612

3. Calculate the net price (actual cost to buyer).

$15,000 .612 $9,180

Net price equivalent rate For each $1, you are spending about 61 cents.

Next we see how to calculate the trade discount amount with a simpler method. In the previous word problem, we could calculate the trade discount amount as follows: $15,000 9,180 $ 5,820

List price Net price Trade discount amount

Single Equivalent Discount Rate You can use another method to find the trade discount by using the discount rate.

single equivalent

CALCULATING TRADE DISCOUNT AMOUNT USING SINGLE EQUIVALENT DISCOUNT RATE Step 1.

Subtract the net price equivalent rate from 1. This is the single equivalent discount rate.

Step 2.

Multiply the list price times the single equivalent discount rate. This is the trade discount amount.

Let’s now do the calculations. Step 1.

1.000 .612 .388

If you are using a calculator, just press 1. This is the single equivalent discount rate.

Step 2. $15,000 .388 $5,820

This is the trade discount amount.

Remember that when we use the net price equivalent rate, the buyer of the of fice equipment pays $.612 on each $1 of list price. Now with the single equivalent discount rate, we can say that the buyer saves $.388 on each $1 of list price. The .388 is the single equivalent discount rate for the 20/15/10 chain discount. Note how we use the .388 single equivalent discount rate as if it were the only discount. It’s time to try the Practice Quiz.

LU 7–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing1

1. 2.

DVD

3.

The list price of a dining room set with a 40% trade discount is $12,000. What are the trade discount amount and net price (use complement method for net price)? The net price of a video system with a 30% trade discount is $1,400. What is the list price? Lamps Outlet bought a shipment of lamps from a wholesaler . The total list price was $12,000 with a 5/10/25 chain discount. Calculate the net price and trade discount amount. (Use the net price equivalent rate and single equivalent discount rate in your calculation.)

For all three problems we will show blueprint aids. You might want to draw them on scrap paper .

1

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Chapter 7 Discounts: Trade and Cash

✓ 1.

Solutions Dining room set trade discount amount and net price:

The facts

Solving for?

Steps to take

List price: $12,000.

Trade discount amount.

Trade discount amount List price Trade discount rate.

Trade discount rate: 40%.

Net price.

Key points Trade discount amount Portion (?)

Net price List price Complement of trade discount rate.

Base Rate ($12,000) (.40) List price

Trade discount rate

Steps to solving problem 1. Calculate the trade discount.

$12,000 .40 $4,800 Trade discount amount

2. Calculate the net price.

$12,000 .60 $7,200 (100% 40% 60%)

2.

Video system list price: The facts

Solving for?

Steps to take

Net price: $1,400.

List price.

List price

Key points Net price

Net price Complement of trade discount

Trade discount rate: 30%.

Portion ($1,400) Base Rate (?) (.70) 100% –30%

List price

Steps to solving problem 1. Calculate the complement of trade discount.

100% 30 70% .70 $1,400 $2,000 .70

2. Calculate the list price.

3.

Lamps Outlet’s net price and trade discount amount: The facts

Solving for?

Steps to take

Key points

List price: $12,000.

Net price.

Chain discount: 5/10/25.

Trade discount amount.

Net price List price Net price equivalent rate.

Do not round off net price equivalent rate or single equivalent discount rate.

Trade discount amount List price Single equivalent discount rate.

Steps to solving problem 1. Calculate the complement of each chain discount.

2. Calculate the net price equivalent rate.

100% 5 95% .95

100% 25 75%

100% 10 90%

.90

.75

.64125

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Learning Unit 7–2

3. Calculate the net price.

$12,000 .64125 $7,695

4. Calculate the single equivalent discount rate.

1.00000 .64125 .35875 $12,000 .35875 $4,305

5. Calculate the trade discount amount.

LU 7–1a

179

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 190)

1. 2. 3.

The list price of a dining room set with a 30% trade discount is $16,000. What are the trade discount amount and net price (use complement method for net price)? The net price of a video system with a 20% trade discount is $400. What is the list price? Lamps Outlet bought a shipment of lamps from a wholesaler . The total list price was $14,000 with a 4/8/20 chain discount. Calculate the net price and trade discount amount. (Use the net price equivalent rate and single equivalent discount rate in your calculation.)

Learning Unit 7–2: Cash Discounts, Credit Terms, and Partial Payments

To introduce this learning unit, we will use the New Hampshire Propane Company invoice that follows. The invoice shows that if you pay your bill early, you will receive a 19-cent discount. Every penny counts.

Sean Clayton/The Image Works

New Hampshire Propane Company Date

Description

Qty.

Price

Total

06/24/08

Previous Balance PROPANE

3.60

$3.40

$0.00 $12.24

Totals this invoice:

$12.24

AMOUNT DUE:

$12.24

Invoice No. 004433L Invoice Date 6/26/08

Prompt Pay Discount: $0.19 Net Amount Due if RECEIVED by 07/10/08: Due Date

$12.05

7/26/08

Now let’s study cash discounts.

Cash Discounts

A cash discount is for prompt payment. A trade discount is not.

In the New Hampshire Propane Company invoice, we receive a cash discount of 19 cents. This amount is determined by the terms of the sale, which can include the credit period, cash discount, discount period, and freight terms. Buyers can often benefit from buying on credit. The time period that sellers give buyers to pay their invoices is the credit period. Frequently, buyers can sell the goods bought during this credit period. Then, at the end of the credit period, buyers can pay sellers with the funds from the sales of the goods. When buyers can do this, they can use the consumer ’s money to pay the invoice instead of their money . Sellers can also of fer a cash discount, or reduction from the invoice price, if buyers pay the invoice within a specified time. This time period is the discount period, which is

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Trade discounts should be taken before cash discounts.

part of the total credit period. Sellers of fer this cash discount because they can use the dollars to better advantage sooner than later . Buyers who are not short of cash like cash discounts because the goods will cost them less and, as a result, provide an opportunity for larger profits. Remember that buyers do not take cash discounts on freight, returned goods, sales tax, and trade discounts. Buyers take cash discounts on the net price of the invoice. Before we discuss how to calculate cash discounts, let’ s look at some aids that will help you calculate credit due dates and end of credit periods. Aids in Calculating Credit Due Dates Sellers usually give credit for 30, 60, or 90 days. Not all months of the year have 30 days. So you must count the credit days from the date of the invoice. The trick is to remember the number of days in each month. You can choose one of the following three options to help you do this.

Years divisible by 4 are leap years. Leap years occur in 2008 and 2012.

Option 1: Days-in-a-Month Rule You may already know this rule. Remember that every

4 years is a leap year.

Thirty days has September, April, June, and November; all the rest have 31 except February has 28, and 29 in leap years.

Option 2: Knuckle Months Some people like to use the knuckles on their hands to remember which months have 30 or 31 days. Note in the following diagram that each knuckle represents a month with 31 days. The short months are in between the knuckles. Jan. March May

Aug.

July

Feb. Apr. June

Oct.

Sept.

Dec.

Nov.

31 days: Jan., March, May, July, Aug., Oct., Dec.

Option 3: Days-in-a-Year Calendar The days-in-a-year calendar (excluding leap year) is

another tool to help you calculate dates for discount and credit periods (T example, let’s use Table 7.1 to calculate 90 days from August 12.

EXAMPLE

By Table 7.1: August 12

able 7.1). For

224 days 90 314 days

Search for day 314 in Table 7.1. You will find that day 314 is November 10. In this example, we stayed within the same year . Now let’ s try an example in which we overlap from year to year . What date is 80 days after December 5? Table 7.1 shows that December 5 is 339 days from the beginning of the year . Subtracting 339 from 365 (the end of the year) tells us that we have used up 26 days by the end of the year . This leaves 54 days in the new year . Go back in the table and

EXAMPLE

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Learning Unit 7–2

TABLE Day of month

7.1

181

Exact days-in-a-year calendar (excluding leap year)*

31 Jan.

28 Feb.

31 Mar.

30 Apr.

31 May

30 June

31 July

31 Aug.

30 Sept.

31 Oct.

30 Nov.

31 Dec.

1

1

32

60

91

121

152

182

213

244

274

305

335

2

2

33

61

92

122

153

183

214

245

275

306

336

3

3

34

62

93

123

154

184

215

246

276

307

337

4

4

35

63

94

124

155

185

216

247

277

308

338

5

5

36

64

95

125

156

186

217

248

278

309

339

6

6

37

65

96

126

157

187

218

249

279

310

340

7

7

38

66

97

127

158

188

219

250

280

311

341

8

8

39

67

98

128

159

189

220

251

281

312

342

9

9

40

68

99

129

160

190

221

252

282

313

343

10

10

41

69

100

130

161

191

222

253

283

314

344

11

11

42

70

101

131

162

192

223

254

284

315

345

12

12

43

71

102

132

163

193

224

255

285

316

346

13

13

44

72

103

133

164

194

225

256

286

317

347

14

14

45

73

104

134

165

195

226

257

287

318

348

15

15

46

74

105

135

166

196

227

258

288

319

349

16

16

47

75

106

136

167

197

228

259

289

320

350

17

17

48

76

107

137

168

198

229

260

290

321

351

18

18

49

77

108

138

169

199

230

261

291

322

352

19

19

50

78

109

139

170

200

231

262

292

323

353

20

20

51

79

110

140

171

201

232

263

293

324

354

21

21

52

80

111

141

172

202

233

264

294

325

355

22

22

53

81

112

142

173

203

234

265

295

326

356

23

23

54

82

113

143

174

204

235

266

296

327

357

24

24

55

83

114

144

175

205

236

267

297

328

358

25

25

56

84

115

145

176

206

237

268

298

329

359

26

26

57

85

116

146

177

207

238

269

299

330

360

27

27

58

86

117

147

178

208

239

270

300

331

361

28

28

59

87

118

148

179

209

240

271

301

332

362

29

29

—

88

119

149

180

210

241

272

302

333

363

30

30

—

89

120

150

181

211

242

273

303

334

364

31

31

—

90

—

151

—

212

243

—

304

—

365

*Often referred to as a Julian calendar.

start with the beginning of the year and search for 54 February 23.

By table 365 days in year 339 days until December 5 26 days used in year 80 days from December 5 26 days used in year 54 days in new year or February 23

(80 26) days. The 54th day is

Without use of table December 31 December 5 26 31 days in January 57 23 due date (February 23) 80 total days

When you know how to calculate credit due dates, you can understand the common business terms sellers of fer buyers involving discounts and credit periods. Remember that discount and credit terms vary from one seller to another .

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Common Credit Terms Offered by Sellers The common credit terms sellers of fer buyers include ordinary dating, receipt of goods (ROG), and end of month (EOM). In this section we examine these credit terms. To determine the due dates, we used the exact days-in-a-year calendar (T able 7.1, p. 181). Ordinary Dating Today, businesses frequently use the ordinary dating method. It gives the buyer a cash discount period that begins with the invoice date. The credit terms of two common ordinary dating methods are 2/10, n/30 and 2/10, 1/15, n/30. 2/10, n/30 Ordinary Dating Method The 2/10, n/30 is read as “two ten, net thirty.” Buyers can take a 2% cash discount of f the gross amount of the invoice if they pay the bill within 10 days from the invoice date. If buyers miss the discount period, the net amount—without a discount—is due between day 1 1 and day 30. Freight, returned goods, sales tax, and trade discounts must be subtracted from the gross before calculating a cash discount. EXAMPLE

$400 invoice dated July 5: terms 2/10, n/30; no freight; paid on July 1 1.

Step 1. Calculate end of 2% discount period:

July 5 date of invoice ⫹ 10 days July 15 end of 2% discount period Step 2. Calculate end of credit period: July 5 by Table 7.1 186 days ⫹ 30 216 days

Search in Table 7.1 for 216

August 4

end of credit period

Step 3. Calculate payment on July 1 1:

.02 ⫻ $400 ⫽ $8 cash discount $400 ⫺ $8 ⫽ $392 paid Note: A 2% cash discount means that you save 2 cents on the dollar and pay 98 cents on the dollar. Thus, $.98 ⫻ $400 ⫽ $392. The following time line illustrates the 2/10, n/30 ordinary dating method beginning and ending dates of the above example: Date of invoice, July 5

End of credit period, August 4

End of 2% discount period, July 15 10 days

Day 11 to 30

Discount period

Cannot take discount

30-day credit period

2/10, 1/15, n/30 Ordinary Dating Method The 2/10, 1/15, n/30 is read “two ten, one

fifteen, net thirty.” The seller will give buyers a 2% (2 cents on the dollar) cash discount if they pay within 10 days of the invoice date. If buyers pay between day 1 1 and day 15 from the date of the invoice, they can save 1 cent on the dollar. If buyers do not pay on day 15, the net or full amount is due 30 days from the invoice date.

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Learning Unit 7–2

183

$600 invoice dated May 8; $100 of freight included in invoice price; paid on May 22. Terms 2/10, 1/15, n/30.

EXAMPLE Step 1.

Calculate the end of the 2% discount period: May 8 date of invoice 10 days May 18 end of 2% discount period

Step 2. Calculate end of 1% discount period:

May 18 end of 2% discount period 5 days May 23 end of 1% discount period Step 3. Calculate end of credit period:

May 8 by Table 7.1 128 days 30 158 days

Step 4.

Search in Table 7.1 for 158 June 7 end of credit period Calculate payment on May 22 (14 days after date of invoice): $600 invoice 100 freight $500 .01 $5.00 $500 $5.00 $100 freight $595 A 1% discount means we pay $.99 on the dollar or $500 $.99 $495 $100 freight $595. Note: Freight is added back since no cash discount is taken on freight.

The following time line illustrates the 2/10, 1/15, n/30 ordinary dating method beginning and ending dates of the above example: Date of invoice, May 8

End of 2% discount period, May 18 10 days

End of credit period, June 7

End of 1% discount period, May 23

Day 11 to 15

Discount periods

Day 16 to 30 Cannot take discount

30-day credit period

Receipt of Goods (ROG) 3/10, n/30 ROG With the receipt of goods (ROG), the cash discount period begins when buyer receives goods, not the invoice date. Industry often uses the ROG terms when buyers cannot expect delivery until a long time after they place the order . Buyers can take a 3% discount within 10 days after receipt of goods. Full amount is due between day 1 1 and day 30 if cash discount period is missed.

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$900 invoice dated May 9; no freight or returned goods; the goods were received on July 8; terms 3/10, n/30 ROG; payment made on July 20.

EXAMPLE

Step 1. Calculate the end of the 3% discount period:

July 8 date goods arrive 10 days July 18 end of 3% discount period Step 2. Calculate the end of the credit period:

July 8 by Table 7.1 189 days 30 219 days

Step 3.

Search in Table 7.1 for 219 August 7 end of credit period Calculate payment on July 20: Missed discount period and paid net or full amount of $900.

The following time line illustrates 3/10, n/30 ROG beginning and ending dates of the above example: Date goods arrive, July 8

End of credit period, August 7

End of 3% discount period, July 18 10 days

Day 11 to 30

Discount period

Cannot take discount

30-day credit period

End of Month (EOM)2 In this section we look at terms involving end of the month (EOM). If an invoice is dated the 25th or earlier of a month, we follow one set of rules. If an invoice is dated after the 25th of the month, a new set of rules is followed. Let’ s look at each situation. Invoice Dated 25th or Earlier in Month, 1/10 EOM If sellers date an invoice on the 25th

or earlier in the month, buyers can take the cash discount if they pay the invoice by the first 10 days of the month following the sale (next month). If buyers miss the discount period, the full amount is due within 20 days after the end of the discount period.

EXAMPLE

August 8. Step 1.

$600 invoice dated July 6; no freight or returns; terms 1/10 EOM; paid on

Calculate the end of the 1% discount period: August 10

First 10 days of month following sale.

Step 2. Calculate the end of the credit period:

August 10 20 days Credit period is 20 days after discount period. August 30 Step 3. Calculate payment on August 8: .99 $600 $594 Sometimes the Latin term proximo is used. Other variations of EOM exist, but the key point is that the seller guarantees the buyer 15 days’ credit. We assume a 30-day month. 2

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Learning Unit 7–2

185

The following timeline illustrates the beginning and ending dates of the EOM invoice of the above example: Date of invoice, July 6

Next month following sale, August*

End of credit period, August 30

End of 1% discount period, August 10 10 days

20 days

Discount period

Cannot take discount

*Even though the discount period begins with the next month following the sale, if buyers wish, they can pay before the discount period (date of invoice until the discount period).

Invoice Dated after 25th of Month, 2/10 EOM When sellers sell goods after the 25th of the month, buyers gain an additional month. The cash discount period ends on the 10th day of the second month that follows the sale. Why? This occurs because the seller guarantees the 15 days’ credit of the buyer. If a buyer bought goods on August 29, September 10 would be only 12 days. So the buyer gets the extra month. EXAMPLE $800 invoice dated April 29; no freight or returned goods; terms 2/10 EOM; payment made on June 18. Step 1.

Step 2.

Calculate the end of the 2% discount period: June10

First 10 days of second month following sale

Calculate the end of the credit period: June 10 20 days Credit period is 20 days after discount period.

June 30 Step 3. Calculate the payment on June 18:

No discount; $800 paid. The following time line illustrates the beginning and ending dates of the EOM invoice of the above example: Date of invoice, April 29

2nd month following sale, June*

End of credit period, June 30

End of 2% discount period, June 10 10 days

20 days

Discount period

Cannot take discount

*Even though the discount period begins with the second month following the sale, if buyers wish, they can pay before the discount date (date of invoice until the discount period)

Solving a Word Problem with Trade and Cash Discount Now that we have studied trade and cash discounts, let’ s look at a combination that involves both a trade and a cash discount. The Word Problem Hardy Company sent Regan Corporation an invoice for of fice equipment with a $10,000 list price. Hardy dated the invoice July 29 with terms of 2/10 EOM (end of month). Regan receives a 30% trade discount and paid the invoice on September 6. Since terms were FOB destination, Regan paid no freight charge. What was the cost of office equipment for Regan?

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Chapter 7 Discounts: Trade and Cash

The facts

Solving for?

Steps to take

Key points

List price: $10,000.

Cost of office equipment.

Net price List price Complement of trade discount rate.

Trade discounts are deducted before cash discounts are taken.

Trade discount rate: 30%.

After 25th of month for EOM. Discount period is 1st 10 days of second month that follows sale.

Terms: 2/10 EOM. Invoice date: 7/29. Date paid: 9/6.

Cash discounts are not taken on freight or returns.

Steps to solving problem 1. Calculate the net price.

$10,000 .70 $7,000

2. Calculate the discount period.

Sale: 7/29 Month 1: Aug. Month 2: Sept 10

3. Calculate the cost of office equipment.

$7,000 .98 $6,860

100% 30% (trade discount) Paid on Sept. 6—is entitled to 2% off.

If you save 2 cents on a dollar, you are spending 98 cents.

100% 2%

Partial Payments Often buyers cannot pay the entire invoice before the end of the discount period. culate partial payments and outstanding balance, use the following steps:

To cal-

CALCULATING PARTIAL PAYMENTS AND OUTSTANDING BALANCE Step 1.

Calculate the complement of a discount rate.

Step 2.

Divide partial payments by the complement of a discount rate (Step 1). This gives the amount credited.

Step 3.

Subtract Step 2 from the total owed. This is the outstanding balance.

EXAMPLE Molly McGrady owed $400. Molly’ s terms were 2/10, n/30. Within 10 days, Molly sent a check for $80. The actual credit the buyer gave Molly is as follows:

100% 2% 98% .98 $80 $81.63 Step 2. .98 Step 1.

Step 3.

$80 1 .02

Discount rate

$400.00 81.63 partial payment—although sent in $80 $318.37 outstanding balance

Note: We do not multiply .02 $80 because the seller did not base the original discount on $80. When Molly makes a payment within the 10-day discount period, 98 cents pays each $1 she owes. Before buyers take discounts on partial payments, they must have permission from the seller . Not all states allow partial payments. You have completed another unit. Let’ s check your progress.

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Learning Unit 7–2

LU 7–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

187

Complete the following table: Date of invoice July 6 February 19 May 9 May 12 May 29

1. 2. 3. 4. 5.

✓

3.

4. 5. 6.

Last day* of discount period

End of credit period

Metro Corporation sent Vasko Corporation an invoice for equipment with an $8,000 list price. Metro dated the invoice May 26. Terms were 2/10 EOM. Vasko receives a 20% trade discount and paid the invoice on July 3. What was the cost of equipment for Vasko? (A blueprint aid will be in the solution to help dissect this problem.) Complete amount to be credited and balance outstanding: Amount of invoice: $600 Terms: 2/10, 1/15, n/30 Date of invoice: September 30 Paid October 3: $400

7.

2.

June 9

Terms 2/10, n/30 3/10, n/30 ROG 4/10, 1/30, n/60 2/10 EOM 2/10 EOM

*If more than one discount, assume date of last discount.

6.

1.

Date goods received

Solutions End of discount period: July 6 10 days July 16 End of credit period: By Table 7.1, July 6 187 days 30 days 217 search End of discount period: June 9 10 days June 19 End of credit period: By Table7.1, June 9 160 days 30 days 190 search End of discount period: By Table 7.1, May 9 129 days 30 days 159 search End of credit period: By Table 7.1, May 9 129 days 60 days 189 search End of discount period: June 10 End of credit period: June 10 20 June 30 End of discount period: July 10 End of credit period: July 10 20 July 30 Vasko Corporation’s cost of equipment:

Aug. 5

July 9 June 8 July 8

The facts

Solving for?

Steps to take

Key points

List price: $8,000.

Cost of equipment.

Net price List price Complement of trade discount rate.

Trade discounts are deducted before cash discounts are taken.

Trade discount rate: 20%. Terms: 2/10 EOM. Invoice date: 5/26. Date paid: 7/3.

EOM before 25th: Discount period is 1st 10 days of month that follows sale.

Cash discounts are not taken on freight or returns.

Steps to solving problem 1. Calculate the net price.

$8,000 .80 $6,400

2. Calculate the discount period.

Until July 10

3. Calculate the cost of office equipment.

$6,400 .98 $6,272 a

100% b 2%

100% 20%

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$400 $408.16, amount credited. .98 $600 $408.16 $191.84, balance outstanding.

7.

LU 7–2a:

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 190)

Complete the following table: Date of invoice July 8 February 24 May 12 April 14 April 27

1. 2. 3. 4. 5.

Date goods received June 12

Terms 2/10, n/30 3/10, n/30 ROG 4/10, 1/30, n/60 2/10 EOM 2/10 EOM

Last day of discount period*

End of credit period

*If more than one discount, assume date of last discount.

Metro Corporation sent Vasko Corporation an invoice for equipment with a $9,000 list price. Metro dated the invoice June 29. Terms were 2/10 EOM. Vasko receives a 30% trade discount and paid the discount on August 9. What was the cost of equipment for Vasko? Complete amount to be credited and balance outstanding:

6.

7.

Amount of invoice: $700 Terms: 2/10, 1/15, n/30 Date of invoice: September 28 Paid October 3: $600

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Trade discount amount, p. 171

Trade discount List Trade discount amount price rate

$600 list price 30% trade discount rate Trade discount amount $600 .30 $180

Calculating net price, p. 172

Net price

List Trade discount price amount

or List Complement of trade price discount price Freight, p. 173

Calculating list price when net price and trade discount rate are known, p. 175

FOB shipping point—buyer pays freight. FOB destination—seller pays freight.

List price

Net price Complement of trade discount rate

$600 list price 30% trade discount rate Net price $600 .70 $420 1.00 .30 .70 Moose Company of New York sells equipment to Agee Company of Oregon. Terms of shipping are FOB New York. Agee pays cost of freight since terms are FOB shipping point. 40% trade discount rate Net price, $120 $120 $200 list price .60 (1.00 .40)

(continues)

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189

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (continued) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Chain discounts, p. 176

Successively lower base.

5/10 on a $100 list item $ 100 $ 95 .05 .10 $5.00 $9.50 (running balance) $95.00 9.50 $85.50 net price

Actual cost List Net price to buyer price equivalent rate Take complement of each chain discount and multiply—do not round. Trade discount List Actual cost amount price to buyer

Given: 5/10 on $1,000 list price Take complement: .95 .90 .855 (net price equivalent)

Single equivalent discount rate, p. 177

Trade discount List 1 Net price amount price equivalent rate

See preceding example for facts: 1 .855 .145 .145 $1,000 $145

Cash discounts, p. 179

Cash discounts, due to prompt payment, are not taken on freight, returns, etc.

Gross $1,000 (includes freight) Freight $25 Terms, 2/10, n/30 Returns $25 Purchased: Sept. 9; paid Sept. 15

Net price equivalent rate, p. 176

$1,000 .855 $855 (actual cost or net price) $1,000 855 $ 145 trade discount amount

Cash discount $950 .02 $19 Calculating due dates, p. 180

Option 1: Thirty days has September, April, June, and November; all the rest have 31 except February has 28, and 29 in leap years. Option 2: Knuckles—31-day month; in between knuckles are short months. Option 3: Days-in-a-year table.

Invoice $500 on March 5; terms 2/10, n/30 End of discount period:

End of credit period by Table 7.1:

March 5 64 days 30 94 days

Search in Table 7.1 Common terms of sale a. Ordinary dating, p. 182

Discount period begins from date of invoice. Credit period ends 20 days from the end of the discount period unless otherwise stipulated; example, 2/10, n/60—the credit period ends 50 days from end of discount period.

March 5 10 March 15

April 4

Invoice $600 (freight of $100 included in price) dated March 8; payment on March 16; 3/10, n/30. March 8 10 End of discount period: March 18

End of credit period by Table 7.1:

March 8 67 days 30 97 days

Search in Table 7.1

April 7

If paid on March 16: .97 $500 $485 100 freight $585 (continues)

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Chapter 7 Discounts: Trade and Cash

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

b. Receipt of goods (ROG), p. 183

Discount period begins when goods are received. Credit period ends 20 days from end of discount period.

4/10, n/30, ROG. $600 invoice; no freight; dated August 5; goods received October 2, payment made October 20. October 2 10 End of discount period: October 12 October 2 275 End of 30 credit period 305 by Table 7.1: Search in Table 7.1 Payment on October 20: No discount, pay $600

c. End of month (EOM), p. 184

On or before 25th of the month, discount period is 10 days after month following sale. After 25th of the month, an additional month is gained. Partial payment 1 Discount rate

Partial payments, p. 186

Amount credited

KEY TERMS

Cash discount, p. 179 Chain discounts, p. 176 Complement, p. 174 Credit period, p. 179 Discount period, p. 179 Due dates, p. 180 End of credit period, p. 180 End of month (EOM), p. 184 FOB destination, p. 173

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 7–1a (p. 179) 1. $4,800 TD; $11,200 NP 2. $500 3. $9,891.84 NP; TD $4,108.16

November 1

$1,000 invoice dated May 12; no freight or returns; terms 2/10 EOM. End of discount period June 10

End of credit period

June 30

$200 invoice, terms 2/10, n/30, dated March 2, paid $100 on March 5. $100 $100 $102.04 1 .02 .98

FOB shipping point, p. 173 Freight terms, p. 173 Invoice, p. 171 List price, p. 171 Net price, p. 172 Net price equivalent rate, p. 176 Ordinary dating, p. 182 Receipt of goods (ROG), p. 183

1. 2. 3. 4. 5. 6. 7.

Series discounts, p. 176 Single equivalent discount rate, p. 177 Single trade discount, p. 174 Terms of the sale, p. 179 Trade discount, p. 171 Trade discount amount, p. 171 Trade discount rate, p. 171

LU 7–2a (p. 188) July 18; Aug. 7 June 22; July 12 June 11; July 11 May 10; May 30 June 10; June 30 $6,174 a) $612.24 b) $87.76

Critical Thinking Discussion Questions 1. What is the net price? June Long bought a jacket from a catalog company. She took her trade discount of f the original price plus freight. What is wrong with June’ s approach? Who would benefit from June’ s approach—the buyer or the seller? 2. How do you calculate the list price when the net price and trade discount rate are known? A publisher tells the bookstore its net price of a book along with a suggested trade

discount of 20%. The bookstore uses a 25% discount rate. Is this ethical when textbook prices are rising? 3. Explain FOB shipping point and FOB destination. Think back to your last major purchase. Was it FOB shipping point or FOB destination? Did you get a trade or a cash discount? 4. What are the steps to calculate the net price equivalent rate? Why is the net price equivalent rate not rounded?

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

191

5. What are the steps to calculate the single equivalent discount 7. Explain the following credit terms of sale: rate? Is this rate off the list or net price? Explain why this cala. 2/10, n/30. culation of a single equivalent discount rate may not always b. 3/10, n/30 ROG. be needed. c. 1/10 EOM (on or before 25th of month). 6. What is the difference between a discount and credit period? d. 1/10 EOM (after 25th of month). Are all cash discounts taken before trade discounts? Agree or 8. Explain how to calculate a partial payment. Whom does a disagree? Why? partial payment favor—the buyer or the seller?

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Classroom Notes

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS For all problems, round your final answer to the nearest cent. Do not round net price equivalent rates or single equivalent discount rates. Complete the following:

Item

List price

Chain discount

7–1. Apple iPod

$300

5/2

7–2. Panasonic DVD player

$199

8/4/3

7–3. IBM scanner

$269

7/3/1

Net price equivalent rate (in decimals)

Single equivalent discount rate (in decimals)

Trade discount

Net price

Complete the following: Item

List price

Chain discount

7–4. Trotter treadmill

$3,000

9/4

7–5. Maytag dishwasher

$450

8/5/6

7–6. Hewlett-Packard scanner

$320

3/5/9

7–7. Land Rover roofrack

$1,850

12/9/6

Net price

Trade discount

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7–8. Which of the following companies, A or B, gives a higher discount? Use the single equivalent discount rate to make your choice (convert your equivalent rate to the nearest hundredth percent). Company A 8/10/15/3

Company B 10/6/16/5

Complete the following: Invoice 7–9. June 18

Dates when goods received

Terms 1/10, n/30

7–10. Nov. 27

7–11. May 15

Final day bill is due (end of credit period)

2/10 EOM

June 5

3/10, n/30, ROG

7–12. April 10

2/10, 1/30, n/60

7–13. June 12

3/10 EOM

7–14. Jan. 10

Last day* of discount period

Feb. 3 (no leap year)

4/10, n/30, ROG

*If more than one discount, assume date of last discount.

Complete the following by calculating the cash discount and net amount paid: Gross amount of invoice (freight charge already included) 7–15. $7,000

Freight charge $100

Date of invoice 4/8

Terms of invoice 2/10, n/60

Date of payment 4/15

7–16. $600

None

8/1

3/10, 2/15, n/30

8/13

7–17. $200

None

11/13

1/10 EOM

12/3

7–18. $500

$100

11/29

1/10 EOM

1/4

Cash discount

Net amount paid

Complete the following: Amount of invoice 7–19. $700

194

Terms 2/10, n/60

Invoice date 5/6

Actual partial payment made $400

Date of partial payment 5/15

Amount of payment to be credited

Balance outstanding

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4/10, n/60

7/5

$400

7/14

WORD PROBLEMS (Round to Nearest Cent as Needed) 7–21. The list price of a Luminox watch is $475. Barry Katz receives a trade discount of 40%. Find the trade discount amount and the net price.

7–22. A model NASCAR race car lists for $79.99 with a trade discount of 40%. What is the net price of the car? 7–23. An article in The (Biloxi, MS) Sun Herald on August 4, 2006, discussed quantity discounts for schools and businesses. Publisher, Andrews McMeel’s books are available at quantity discounts with bulk purchases for educational or business use. School district 510 purchased 50 books at $26.95 each with a quantity discount of 5%. (a) What was total list price for the books? (b) What was the total discount amount? (c) What was the total net price for the books? Round to the nearest cent.

7–24. Levin Furniture buys a living room set with a $4,000 list price and a 55% trade discount. Freight (FOB shipping point) of $50 is not part of the list price. What is the delivered price (including freight) of the living room set, assuming a cash discount of 2/10, n/30, ROG? The invoice had an April 8 date. Levin received the goods on April 19 and paid the invoice on April 25. 7–25. A manufacturer of skateboards offered a 5/2/1 chain discount to many customers. Bob’ s Sporting Goods ordered 20 skateboards for a total $625 list price. What was the net price of the skateboards? What was the trade discount amount?

7–26. Home Depot wants to buy a new line of shortwave radios. Manufacturer A offers a 21/13 chain discount. Manufacturer B offers a 26/8 chain discount. Both manufacturers have the same list price. What manufacturer should Home Depot buy from?

7–27. Maplewood Supply received a $5,250 invoice dated 4/15/06. The $5,250 included $250 freight. Terms were 4/10, 3/30, n/60. (a) If Maplewood pays the invoice on April 27, what will it pay? (b) If Maplewood pays the invoice on May 21, what will it pay?

7–28. Sport Authority ordered 50 pairs of tennis shoes from Nike Corporation. The shoes were priced at $85 for each pair with the following terms: 4/10, 2/30, n/60. The invoice was dated October 15. Sports Authority sent in a payment on October 28. What should have been the amount of the check? 7–29. Macy of New York sold Marriott of Chicago of fice equipment with a $6,000 list price. Sale terms were 3/10, n/30 FOB New York. Macy agreed to prepay the $30 freight. Marriott pays the invoice within the discount period. What does Marriott pay Macy?

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7–30. Royal Furniture bought a sofa for $800. The sofa had a $1,400 list price. What was the trade discount rate Royal received? Round to the nearest hundredth percent.

7–31. Amazon.com paid a $6,000 net price for textbooks. The publisher offered a 30% trade discount. What was the publisher ’s list price? Round to the nearest cent.

7–32. Bally Manufacturing sent Intel Corporation an invoice for machinery with a $14,000 list price. Bally dated the invoice July 23 with 2/10 EOM terms. Intel receives a 40% trade discount. Intel pays the invoice on August 5. What does Intel pay Bally? 7–33. On August 1, Intel Corporation (Problem 7–32) returns $100 of the machinery due to defects. What does Intel pay Bally on August 5? Round to nearest cent.

7–34. Stacy’s Dress Shop received a $1,050 invoice dated July 8 with 2/10, 1/15, n/60 terms. On July 22, Stacy’ s sent a $242 partial payment. What credit should Stacy’s receive? What is Stacy’s outstanding balance?

7–35. On March 11, Jangles Corporation received a $20,000 invoice dated March 8. Cash discount terms were 4/10, n/30. On March 15, Jangles sent an $8,000 partial payment. What credit should Jangles receive? What is Jangles’ outstanding balance?

ADDITIONAL SET OF WORD PROBLEMS 7–36. In the February 2007 issue of The Tax Adviser, it was reported that trade discounts are not income. Westpac Pacific Food agreed to buy a minimum quantity of merchandise and receive a volume discount. Westpac Pacific Food received a 4 percent quantity discount. Total amount of an order placed by Westpac amounted to $20,500. What was the net price paid by Westpac?

7–37. Borders.com paid a $79.99 net price for each calculus textbook. The publisher offered a 20% trade discount. What was the publisher’s list price?

7–38. Home Office.com buys a computer from Compaq Corporation. The computers have a $1,200 list price with a 30% trade discount. What is the trade discount amount? What is the net price of the computer? Freight char ges are FOB destination. 7–39. Vail Ski Shop received a $1,201 invoice dated July 8 with 2/10, 1/15, n/60 terms. On July 22, Vail sent a $485 partial payment. What credit should Vail receive? What is Vail’s outstanding balance?

7–40. True Value received an invoice dated 4/15/02. The invoice had a $5,500 balance that included $300 freight. Terms were 4/10, 3/30, n/60. True Value pays the invoice on April 29. What amount does True Value pay?

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7–41. Staples purchased seven new computers for $850 each. It received a 15% discount because it purchased more than five and an additional 6% discount because it took immediate delivery . Terms of payment were 2/10, n/30. Staples pays the bill within the cash discount period. How much should the check be? Round to the nearest cent.

7–42. On May 14, Talbots of Boston sold Forrest of Los Angeles $7,000 of fine clothes. Terms were 2/10 EOM FOB Boston. Talbots agreed to prepay the $80 freight. If Forrest pays the invoice on June 8, what will Forrest pay? If Forrest pays on June 20, what will Forrest pay?

7–43. Sam’s Ski Boards.com offers 5/4/1 chain discounts to many of its customers. The Ski Hut ordered 20 ski boards with a total list price of $1,200. What is the net price of the ski boards? What was the trade discount amount? Round to the nearest cent.

7–44. Majestic Manufacturing sold Jordans Furniture a living room set for an $8,500 list price with 35% trade discount. The $100 freight (FOB shipping point) was not part of the list price. Terms were 3/10, n/30 ROG. The invoice date was May 30. Jordans received the goods on July 18 and paid the invoice on July 20. What was the final price (include cost of freight) of the living room set?

7–45. Boeing Truck Company received an invoice showing 8 tires at $1 10 each, 12 tires at $160 each, and 15 tires at $180 each. Shipping terms are FOB shipping point. Freight is $400; trade discount is 10/5; and a cash discount of 2/10, n/30 is offered. Assuming Boeing paid within the discount period, what did Boeing pay?

7–46. The Greeley Tribune (Greeley, CO) on February 24, 2007 reported on discounts. Republican Representative Kevin Lundberg, of Berthoud, Colorado, said the law defined “cost” as the wholesale price of goods plus any overhead costs, but stores sell things below cost all the time. Jim Riesber g purchased slacks for $25.00, with an original price of $125. What was the percent discount Jim received?

7–47. Verizon offers to sell cellular phones listing for $99.99 with a chain discount of 15/10/5. Cellular Company of fers to sell its cellular phones that list at $102.99 with a chain discount of 25/5. If Irene is to buy 6 phones, how much could she save if she buys from the lower -priced company?

197

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7–48. Bryant Manufacture sells its furniture to wholesalers and retailers. It of fers to wholesalers a chain discount of 15/10/5 and to retailers a chain discount of 15/10. If a sofa lists for $500, how much would the wholesaler and retailer pay?

CHALLENGE PROBLEMS 7–49. The original price of a 2003 Honda Insight to the dealer is $17,995, but the dealer will pay only $16,495. If the dealer pays Honda within 15 days, there is a 1% cash discount. (a) How much is the rebate? (b) What percent is the rebate? Round to nearest hundredth percent. (c) What is the amount of the cash discount if the dealer pays within 15 days? (d) What is the dealer ’s final price? (e) What is the dealer ’s total savings? Round answer to the nearest hundredth.

7–50. On March 30, Century Television received an invoice dated March 28 from ACME Manufacturing for 50 televisions at a cost of $125 each. Century received a 10/4/2 chain discount. Shipping terms were FOB shipping point. ACME prepaid the $70 freight. Terms were 2/10 EOM. When Century received the goods, 3 sets were defective. Century returned these sets to ACME. On April 8, Century sent a $150 partial payment. Century will pay the balance on May 6. What is Century’s final payment on May 6? Assume no taxes.

DVD SUMMARY PRACTICE TEST (Round to the Nearest Cent as Needed) Complete the following: (p. 172) Item 1. Apple iPod 2. Palm Pilot

198

List price

Single trade discount

$350

5% 10%

Net price

$190

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Calculate the net price and trade discount (use net price equivalent rate and single equivalent discount rate) for the following: (p. 176) Item 3. Sony HD flat-screen TV

List price

Chain discount

$899

5/4

Net price

Trade discount

4. From the following, what is the last date for each discount period and credit period? (p. 187)

a.

Date of invoice

Terms

Nov. 4

2/10, n/30

b. Oct. 3, 2009

c.

May 2

d. Nov. 28

End of discount period

End of credit period

3/10, n/30 ROG (Goods received March 10, 2010) 2/10 EOM 2/10 EOM

5. Best Buy buys an iPod from a wholesaler with a $300 list price and a 5% trade discount. What is the trade discount amount? What is the net price of the iPod? (p. 182)

6. Jordan’s of Boston sold Lee Company of New York computer equipment with a $7,000 list price. Sale terms were 4/10, n/30 FOB Boston. Jordan’s agreed to prepay the $400 freight. Lee pays the invoice within the discount period. What does Lee pay Jordan’s? (p. 174) 7. Julie Ring wants to buy a new line of Tonka trucks for her shop. Manufacturer A offers a 14/8 chain discount. Manufacturer B offers a 15/7 chain discount. Both manufacturers have the same list price. Which manufacturer should Julie buy from? (p. 177)

8. Office.com received a $8,000 invoice dated April 10. Terms were 2/10, 1/15, n/60. On April 14, Office.com sent an $1,900 partial payment. What credit should Office.com receive? What is Office.com’s outstanding balance? Round to the nearest cent. (p. 186)

9. Logan Company received from Furniture.com an invoice dated September 29. Terms were 1/10 EOM. List price on the invoice was $8,000 (freight not included). Logan receives a 8/7 chain discount. Freight char ges are Logan’s responsibility, but Furniture.com agreed to prepay the $300 freight. Logan pays the invoice on November 7. What does Logan Company pay Furniture.com? (p. 177)

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Personal Finance A KIPLINGER APPROACH

T EC H

| Cable, phone and Internet packages

month. In southern California, bundles from Time Warner Cable with Internet, phone and cable start at about dangle attractive prices. By Jeff Bertolucci $100. And in areas of Massachusetts and other states where Verizon has wired homes with its FiOS high-speed fiber-optic service, subscribers can get Internet, telephone and nearly 200 digital cable TV and music channels for $105 per month. Verizon offers bundles with yi ng up your telecom satellite TV in services in a single packother markets. age is the lure many local Although price telephone and cable comis a big draw, a panies are casting in bundle isn’t worth selected areas around the U.S. For it if it excludes about $100 a month, you can get cable services you want. or satellite TV, local and long-distance The AT&T Quad telephone service, plus high-speed Pack, for instance, Internet service. In addition to paying allows only 100 just one bill, you have just one comminutes per month pany to call if you have a technical or of direct-dial calls billing issue. Then again, this onefrom your home. stop-shop approach can backfire if More long-disyour vendor’s customer service stinks. tance minutes cost Many bundled deals (often marketed 9 cents each. as “triple plays” or “triple packs”) are And there may limited-time offers ranging from three be other drawto 12 months. The Comcast Triple backs. If a single Play, for instance, includes Internet, high-speed line phone and cable service for $99 brings all commuper month for one year. After the year nications to your is up, will the hammer fall—and the home, you could price skyrocket? Not necessarily. lose your phone, You can expect Comcast’s package to G About $100 a month will buy you TV, phone and Internet service. cable and Internet cost “about $130 per month,” says service at the same scriber, check your vendor’s Web site company spokeswoman Jenni Moyer. time if the line goes down. Some digiPatrick Matters, who lives in Indital phone services that use the Internet for bundled discounts. anapolis, signed up for Comcast’s Some vendors are offering quadruple for voice calls don’t support faxing— Triple Play about a year ago. He pays plays that add wireless phone service. a significant shortcoming for home$100 to $110 per month (“a little more AT&T’s Quad Pack, for instance, based businesses. if my daughters buy a movie”), a savbundles Internet, telephone, Dish NetAnd bundles make it more difficult ings of more than $50 over his previous work satellite TV and Cingular Wireto change providers for a specific serva la carte plans. At $130 per month, less service for $123 per month. Its ice—for instance, switching from cable he’d still be ahead. Triple Pack—Internet, telephone and to satellite TV. Of course, from a teleThe Triple Play is for new customers wireless—costs $95 per month. com company’s perspective, that’s the only. But current Comcast subscribers whole idea. can also get discounts if they add new Regional offers. Bundles vary dependStill, the convenience and relatively services. For example, a Comcast cable- ing on where you live. For example, in low prices make bundled services apTV customer can sign up for the comQwest’s 14-state region, the starting pealing. And there should be plenty of pany’s phone service for $33 per month price for a package including Internet, competition as telephone and cable for one year. If you’re already a subphone and DirecTV is about $90 per companies duke it out.

Save a BUNDLE on telecom services

BUSINESS MATH ISSUE Kiplinger’s © 2007

If you get a bundled package you will always save. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

200

W H I T E PA C K E R T/G E T T Y I M A G E S

T

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A Pick a product and how this article will change your buying habits.

2006 Wall Street Journal ©

b site text We he e e S : s t T t Projec /slater9e) and e. Interne ce Guid m r o u .c o e s h e h R t .m e (www Intern ss Math Busine

201

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Video Case

HILLERICH & BRADSBY COMPANY “LOUISVILLE SLUGGER” According to Bob Hill, author of Crack of the Bat: The Louisville Slugger Story, in 1884 the star outfielder Pete “The Gladiator” Browning of the Louisville Eclipses was in a batting slump. Bud Hillerich, son of J.F. Hillerich, made Browning a bat. After Browning got three hits with his new bat, his teammates began clamoring for the Hillerich bat. In 1910, during the rebuilding process following a factory fire, Hillerich hired Frank Bradsby to oversee the company’s sales policy. In 1916, Bradsby’s salesman skills won him a partnership and the company’s name was changed to Hillerich & Bradsby (H & B) Company. After 118 years, H & B remains the leading manufacturer of baseball bats. H & B makes customized bats for players according to their specifications for bat weight, length, and wood preference (white ash or maple). Players rarely use bats over 34.5 inches long and 33 ounces in weight. Maple bats are denser and heavier than ash. About 20% of the 200,000 big league bats produced by H & B are made of maple. Big league teams pay $41 for a white ash bat and $51 for a maple bat. In the early 1970s, aluminum bats became very popular with amateur players. Aluminum bats are considered safer and more durable than wood bats. Aluminum bats are also

PROBLEM

1

The video stated that in 1974, when the NCAA legalized aluminum bats in college, production of wood bats dropped from 7,000,000 to 800,000. The average retail price of a wood bat is $46. (a) What was the percent decrease in production? Round to the nearest hundredth percent. (b) How much did revenue decrease? PROBLEM

2

The video states that the Ontario, California, plant produces over 300 different models of aluminum baseball bats. The plant produces 5,500 bats each day. Assume the average price is $175 per bat. (a) What would be the revenue generated for a week’s production (5-day week)? (b) How many bats would be produced annually? (c) What would be the total annual sales? PROBLEM

3

The April 11, 2003, issue of the Dayton Daily News reported that the Massachusetts Interscholastic Athletic Association created a stir by switching from aluminum bats to wood, citing safety concerns. Of the state’s 40 leagues, 62.5% have decided to use wood during the regular season. How many have decided to stay with aluminum bats? PROBLEM

4

On April 14, 2003, Forbes reported that H & B receives nearly 34 of its $110 million in annual revenue from baseball and softball bats. (a) What is the total amount received for baseball and softball bats? (b) How much revenue is received from other items?

202

more economical. Since batters can swing the aluminum bats faster, the ball travels farther. The decline in white ash availability led to the increased cost of wood bats and increased use of aluminum bats. While aluminum bats rarely break, a college team would go through more than 350 wood bats per season. Basic models of aluminum bats sell for around $100; high-tech metal bats sell for as much as $500. Interest in bats was renewed after Sammy Sosa’s bat broke, spraying cork in the infield. Players and fans wanted to know how bats differ and how batters benefited from different types of bats. In 1971, aluminum bats were approved for Little League play; in 1975 they were approved for college play. As early as 1970, H & B contracted an outside aluminum company to manufacture H & B aluminum bats; however, H & B remained focused on Louisville Slugger wood bats. H & B felt aluminum bats would detract from the game of baseball. After aluminum bats were introduced, the Dayton Daily News reported that Louisville Slugger wood bat sales shrank from seven million to 800,000. Now new sales are back up to one million. After being in a slump, it looks like H & B hit a “home run.”

PROBLEM

5

The average number of bats used by a Major Leaguer in a season is 90. There are 30 teams with a roster of 25 players. The cost of a bat is $41 to $51. (a) What is the percent increase? Round to the nearest tenth. (b) Using an average price, what is the total amount of dollars spent on baseball bats by Major Leaguers? PROBLEM

6

On April 13, 2003, the Chicago Sun-Times reported aluminum bats typically cost $100 to $250 and have one-year warranties. Wood bats typically cost $35 to $90. Aluminum bats are cheaper in the long run because they don’t break. A high school hitter can go through four wood bats in one season. (a) What is the most a high school hitter would pay for wood bats during the season? (b) What is the least amount paid for wood bats during the season? (c) What is the average percent savings using aluminum as compared to wood? PROBLEM

7

On September 30, 2003, the Chicago Sun-Times reported that a 34-inch, 38-ounce Louisville Slugger commemorative bat honoring the Chicago Cubs 2003 National League Central Division Champions hit the market. The bats retail at $129.95 plus shipping. Personalized (recipient’s name engraved) bats are sold by H & B for $54.00. (a) What is the percent change for the commemorative bat? Round to the nearest tenth. (b) If shipping costs for the commemorative bats are an additional 11.54%, what is the amount charged for shipping? Round to the nearest dollar. (c) What would be the total cost for the commemorative bat?

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CHAPTER

8

Markups and Markdowns; Perishables and Breakeven Analysis

LEARNING UNIT OBJECTIVES LU 8–1: Markups1 Based on Cost (100%)

urnal © 2006 Wall Street Jo

• Calculate dollar markup and percent markup on cost (p. 205). • Calculate selling price when you know the cost and percent markup on cost (p. 206). • Calculate cost when dollar markup and percent markup on cost are known (p. 207). • Calculate cost when you know the selling price and percent markup on cost (p. 208).

LU 8–2: Markups Based on Selling Price (100%) • • • • •

Calculate dollar markup and percent markup on selling price (p. 210). Calculate selling price when dollar markup and percent markup on selling price are known (p. 211). Calculate selling price when cost and percent markup on selling price are known (p. 211). Calculate cost when selling price and percent markup on selling price are known (p. 212). Convert from percent markup on cost to percent markup on selling price and vice versa (p. 213).

LU 8–3: Markdowns and Perishables • Calculate markdowns; compare markdowns and markups (p. 216). • Price perishable items to cover spoilage loss (p. 217).

LU 8–4: Breakeven Analysis • Calculate contribution margin (p. 219). • Calculate breakeven point (p. 219). Some texts use the term markon (selling price minus cost).

1

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Chapter 8 Markups and Markdowns; Perishables and Breakeven Analysis

Reprinted by permission of The Wall Street Journal, © 2003 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

The McGraw-Hill Companies, Ken Cavanagh photographer

Are you one of the many shoppers who shop at Target? If you wear jeans, you may be interested in the Wall Street Journal clipping “Levi Strauss Sells Low-Cost Jeans to Target in Bid to Increase Sales.” The clipping states that Levi Strauss & Co. has begun selling its Levi Strauss Signature™ brand jeans to Target. If you are familiar with products from Levi Strauss & Co. and shop at the mass-channel retail stores that carry Levi Strauss Signature™ brand jeans, such as Target and Wal-Mart, you will probably look at these lower -cost jeans. Levi Strauss & Co. wants to boost sales with their Levi Strauss Signature™ brand products by appealing to a new group of value-conscious consumers who don’ t buy their other branded products. Before we study the two pricing methods available to Target (percent markup on cost and percent markup on selling price), we must know the following terms: • • •

• •

Selling price. The price retailers char ge consumers. The total selling price of all the goods sold by a retailer (like Target) represents the retailer ’s total sales. Cost. The price retailers pay to a manufacturer or supplier to bring the goods into the store. Markup, margin, or gross profit. These three terms refer to the dif ference between the cost of bringing the goods into the store and the selling price of the goods. As an example of high-mar gin sales, Sharper Image customers are buying more high-mar gin gadgets. This helps the company fuel a 15% rise in same-store sales. Operating expenses or overhead. The regular expenses of doing business such as wages, rent, utilities, insurance, and advertising. Net profit or net incomes. The profit remaining after subtracting the cost of bringing the goods into the store and the operating expenses from the sale of the goods (including any returns or adjustments). In Learning Unit 8–4 we will take a closer look at the point at which costs and expenses are covered. This is called the breakeven point.

From these definitions, we can conclude that markup represents the amount that retailers must add to the cost of the goods to cover their operating expenses and make a profit.2 Let’s assume Target pays Levi Strauss & Co. $18 for a pair for jeans and sells them for $23. 3

In this chapter , we concentrate on the markup of retailers. Manufacturers and suppliers also use markup to determine selling price. 2

Amounts used are hypothetical; prices and markups may vary .

3

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Learning Unit 8–1

Basic selling price formula

205

Selling price (S)

Cost (C)

Markup (M)

$23

$18 (price paid to bring jeans into store)

$5 (amount in dollars to cover operating expenses and make a profit)

In the Levi Strauss example, the markup is a dollar amount, or a dollar markup. Markup is also expressed in percent. When expressing markup in percent, retailers can choose a percent based on cost (Learning Unit 8–1) or a percent based on selling price (Learning Unit 8–2). When you study the Wall Street Journal clipping “Shirt Tale,” you will see how the clothing sourcing for some retailers is divided by supply regions. These retailers include H&M, Gap, French Connection, and Wal-Mart. Do you shop at any of these retailers?

Wall Street Journal © 2005

Learning Unit 8–1: Markups Based on Cost (100%) In Chapter 6 you were introduced to the portion formula, which we used to solve percent problems. We also used the portion formula in Chapter 7 to solve problems involving trade and cash discounts. In this unit you will see how we use the basic selling price formula and the portion formula to solve percent markup situations based on cost. We will be using blueprint aids to show how to dissect and solve all word problems in this chapter . Many manufacturers mark up goods on cost because manufacturers can get cost information more easily than sales information. Since retailers have the choice of using percent markup on cost or selling price, in this unit we assume Target has chosen percent markup on cost. In Learning Unit 8–2 we show how Target would determine markup if it decided to use percent markup on selling price. Businesses that use percent markup on cost recognize that cost is 100%. This 100% represents the base of the portion formula. All situations in this unit use cost as 100%. To calculate percent markup on cost, we will use the Levi Strauss Signature™ brand jeans sold at Target and begin with the basic selling price formula given in the chapter introduction. When we know the dollar markup, we can use the portion formula to find the percent markup on cost. Markup expressed in dollars: Selling price ($23) Cost ($18) Markup ($5) Markup expressed as a percent markup on cost: Cost 100.00% Cost is 100%—the base. Dollar markup Markup 27.78 is the portion, and percent markup on Selling price 127.78% cost is the rate.

In Situation 1 (p. 206) we show why Target has a 27.78% markup based on cost by presenting the Levi Strauss Signature™ brand jeans as a word problem. We solve the problem with the blueprint aid used in earlier chapters. In the second column, however , you will see footnotes after two numbers. These refer to the steps we use below the blueprint aid to solve the problem. Throughout the chapter , the numbers that we are solving for are in red. Remember that cost is the base for this unit.

Situation 1: Calculating Dollar Markup and Percent Markup on Cost Dollar markup is calculated with the basic selling price formula know the cost and selling price of goods, reverse the formula to cost from the selling price, and you have the dollar markup.

S C M. When you M S C. Subtract the

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Dollar markup

Portion Base Rate Cost

4:07 PM

Percent markup on cost

The percent markup on cost is calculated with the portion formula. For Situation 1 the portion (P) is the dollar markup, which you know from the selling price formula. In this unit the rate (R) is always the percent markup on cost and the base (B) is always the cost (100%). To find the percent markup on cost ( R), use the portion formula R PB and divide the dollar markup ( P) by the cost ( B). Convert your answer to a percent and round if necessary. Now we will look at the Target example to see how to calculate the 27.78% markup on cost. The Word Problem Target buys Levi Strauss Signature™ brand jeans for $18 and plans to

sell them for $23. What is Target’s dollar markup? What is the percent markup on cost (round to the nearest hundredth percent)? The facts

Solving for?

Steps to take

Signature™ jeans cost: $18.

% $ C 100.00% $18 M 27.782 51 S 127.78% $23

Dollar Selling Cost. markup price

Signature™ jeans selling price: $23.

1

Dollar markup. 2 Percent markup on cost.

Key points Dollar markup

Percent Dollar markup markup Cost on cost

Portion ($5) Base Rate ($18) (?) Cost

Steps to solving problem Dollar markup Selling price Cost

1. Calculate the dollar markup.

$5 2. Calculate the percent markup on cost.

$23

$18

Dollar markup Cost $5 27.78% $18

Percent markup on cost

To check the percent markup on cost, you can use the basic selling price formula S C M. Convert the percent markup on cost found with the portion formula to a decimal and multiply it by the cost. This gives the dollar markup. Then add the cost and the dollar markup to get the selling price of the goods. You could also check the cost ( B) by dividing the dollar markup ( P) by the percent markup on cost ( R). Check Selling price Cost Markup

$23 $23

$18 .2778($18) $18 $5

$23

$23

or

Cost (B)

Dollar markup (P ) Percent markup on cost (R)

$5 $18 .2778 Parentheses mean that you multiply the percent markup on cost in decimal by the cost.

Situation 2: Calculating Selling Price When You Know Cost and Percent Markup on Cost When you know the cost and the percent markup on cost, you calculate the selling price with the basic selling formula S C M. Remember that when goods are marked up on cost, the cost is the base (100%). So you can say that the selling price is the cost plus the markup in dollars (percent markup on cost times cost). Now let’s look at Mel’ s Furniture where we calculate Mel’ s dollar markup and selling price. The Word Problem Mel’s Furniture bought a lamp that cost $100. To make Mel’ s desired profit, he needs a 65% markup on cost. What is Mel’s dollar markup? What is his selling price?

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Learning Unit 8–1

The facts

Solving for?

Steps to take

Lamp cost: $100.

% $ C 100% $100 65 651 M S 165% $1652

Dollar markup:

Markup on cost: 65%.

207

Key points Selling price

S C M. Portion (?)

or Percent S Cost ° 1 markup ¢ on cost

1

Dollar markup. Selling price.

2

Base Rate ($100) (1.65) 100% +65%

Cost

Steps to solving problem 1. Calculate the dollar markup.

SCM

S $100 .65($100) S $100 $65 2. Calculate the selling price.

Parentheses mean you multiply the percent markup in decimal by the cost.

Dollar markup

S $165

You can check the selling price with the formula P B R. You are solving for the portion (P)—the selling price. Rate ( R) represents the 100% cost plus the 65% markup on cost. Since in this unit the markup is on cost, the base is the cost. Convert 165% to a decimal and multiply the cost by 1.65 to get the selling price of $165. Check Selling price Cost (1 Percent markup on cost) (P ) (B) (R)

$100 1.65 $165

Situation 3: Calculating Cost When You Know Selling Price and Percent Markup on Cost When you know the selling price and the percent markup on cost, you calculate the cost with the basic selling formula S C M. Since goods are marked up on cost, the percent markup on cost is added to the cost. Let’s see how this is done in the following Jill Sport example. The Word Problem Jill Sport, owner of Sports, Inc., sells tennis rackets for $50. To make her desired profit, Jill needs a 40% markup on cost. What do the tennis rackets cost Jill? What is the dollar markup? The facts

Solving for?

Steps to take

Selling price: $50.

$ % C 100% $35.711 M 40 14.292 S 140% $50.00

S C M.

Markup on cost: 40%.

1

Cost. Dollar markup.

2

Key points Selling price

or Cost

Selling price Percent 1 markup on cost

Portion ($50) Base Rate (?) (1.40)

M S C. Cost

100% +40%

Steps to solving problem 1. Calculate the cost.

SCM $50.00 C .40C $50.00 1.40C 1.40 1.40 C $35.71

This means 40% times cost. C is the same as 1C. Adding .40C to 1C gives the percent markup on cost of 1.40C in decimal.

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MSC M $50.00 $35.71 M $14.29

2. Calculate the dollar markup.

You can check your cost answer with the portion formula B PR. Portion (P) is the selling price. Rate ( R) represents the 100% cost plus the 40% markup on cost. Convert the percents to decimals and divide the portion by the rate to find the base, or cost. Check Cost (B)

Selling price (P ) 1 Percent markup on cost (R)

$50.00 $35.71 1.40

Now try the following Practice Quiz to check your understanding of this unit.

LU 8–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Solve the following situations (markups based on cost): 1. Irene Westing bought a desk for $400 from an of fice supply house. She plans to sell the desk for $600. What is Irene’s dollar markup? What is her percent markup on cost? Check your answer . 2. Suki Komar bought dolls for her toy store that cost $12 each. To make her desired profit, Suki must mark up each doll 35% on cost. What is the dollar markup? What is the selling price of each doll? Check your answer . 3. Jay Lyman sells calculators. His competitor sells a new calculator line for $14 each. Jay needs a 40% markup on cost to make his desired profit, and he must meet price competition. At what cost can Jay af ford to bring these calculators into the store? What is the dollar markup? Check your answer .

✓ 1.

Solutions Irene’s dollar markup and percent markup on cost:

The facts

Solving for?

Desk cost: $400.

% C 100% M 502 S 150%

Desk selling price: $600.

Steps to take $ $400 2001 $600

1

Dollar markup. Percent markup on cost.

Key points

Dollar Selling Cost. markup price

Dollar markup

Percent Dollar markup markup Cost on cost

Portion ($200)

2

Base Rate ($400) (?) Cost

Steps to solving problem Dollar markup Selling price Cost

1. Calculate the dollar markup.

$200

$600

$400

Dollar markup Cost $200 50% $400

Percent markup on cost

2. Calculate the percent markup on cost.

Check Selling price Cost Markup $600 $400 .50($400) $600 $400 $200 $600 $600

or

Cost (B)

Dollar markup (P) Percent markup on cost (R) $200 $400 .50

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Learning Unit 8–1

2.

Dollar markup and selling price of doll: The facts

Solving for?

Steps to take

Doll cost: $12 each.

% $ C 100% $12.00 35 4.201 M S 135% $16.202

Dollar markup:

Markup on cost: 35%.

1

Dollar markup. Selling price.

2

Key points Selling price

S C M. Portion (?)

or Percent S Cost ° 1 markup ¢ on cost

Base Rate ($12) (1.35) 100% +35%

Cost

Steps to solving problem 1. Calculate the dollar markup.

SCM S $12.00 .35($12.00) S $12.00 $4.20

2. Calculate the selling price.

Dollar markup

S $16.20

Check Selling price Cost (1 Percent markup on cost) $12.00 1.35 $16.20 (P) (B) (R) 3.

Cost and dollar markup: The facts

Solving for?

Steps to take

Selling price: $14.

% $ C 100% $101 40 M 42 S 140% $14

S C M.

Markup on cost: 40%.

1

Cost. Dollar markup.

2

Key points Selling price

or Cost

Selling price Percent 1 markup on cost

Portion ($14) Base Rate (?) (1.40)

M S C. Cost

Steps to solving problem 1. Calculate the cost.

SCM $14 C .40C 1.40C $14 1.40 1.40 $10 C

2. Calculate the dollar markup.

MSC M $14 $10 M $4

Check Cost (B)

$14 Selling price (P) $10 1 Percent markup on cost (R) 1.40

100% +40%

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LU 8–1a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 222)

Solve the following situations (markups based on cost): 1. Irene Westing bought a desk for $800 from an of fice supply house. She plans to sell the desk for $1,200. What is Irene’ s dollar markup? What is her percent markup on cost? Check your answer . 2. Suki Komar bought dolls for her toy store that cost $14 each. To make her desired profit, Suki must mark up each doll 38% on cost. What is the dollar markup? What is the selling price of each doll? Check your answer . 3. Jay Lyman sells calculators. His competitor sells a new calculator line for $16 each. Jay needs a 42% markup on cost to make his desired profit, and he must meet price competition. At what cost can Jay af ford to bring these calculators into the store? What is the dollar markup? Check your answer .

Learning Unit 8–2: Markups Based on Selling Price (100%) Many retailers mark up their goods on the selling price since sales information is easier to get than cost information. These retailers use retail prices in their inventory and report their expenses as a percent of sales. Businesses that mark up their goods on selling price recognize that selling price is 100%. We begin this unit by assuming Target has decided to use percent markup based on selling price. We repeat Target’s selling price formula expressed in dollars. Markup expressed in dollars: Selling price ($23) Cost ($18) Markup ($5) Markup expressed as percent markup on selling price: Cost Markup Selling price

78.26% 21.74 100.00%

Selling price is 100%—the base. Dollar markup is the portion, and percent markup on selling price is the rate.

In Situation 1 (below) we show why Target has a 21.74% markup based on selling price. In the last unit, markups were on cost. In this unit, markups are on selling price.

Situation 1: Calculating Dollar Markup and Percent Markup on Selling Price The dollar markup is calculated with the selling price formula used in Situation 1, Learning Unit 8–1: M S C. To find the percent markup on selling price, use the portion formula R PB, where rate (the percent markup on selling price) is found by dividing the portion (dollar markup) by the base (selling price). Note that when solving for percent markup on cost in Situation 1, Learning Unit 8–1, you divided the dollar markup by the cost. The Word Problem Target buys Levi Strauss Signature™ brand jeans for $18 and plans

to sell them for $23. What is Target’s dollar markup? What is its percent markup on selling price? (Round to nearest hundredth percent.) The facts

Solving for?

Signature™ jeans cost: $18.

% C 78.26% M 21.74%2 S 100.00%

Signature™ jeans selling price: $23.

1

Steps to take $ $18 51 $ 23

Dollar markup. Percent markup on selling price.

2

Key points

Dollar Selling Cost. markup price Dollar Percent markup markup on Selling selling price price

Dollar markup Portion ($5) Base Rate ($23) (?) Selling price

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211

Learning Unit 8–2

Steps to solving problem Dollar markup Selling price Cost

1. Calculate the dollar markup.

$5 $23 $18 Percent markup Dollar markup on selling price Selling price $5 21.74% $23

2. Calculate the percent markup on selling price.

You can check the percent markup on selling price with the basic selling price formula S C M. You can also use the portion formula by dividing the dollar markup ( P) by the percent markup on selling price ( R). Check Selling price Cost Markup

Selling price (B)

or

$23

$18 .2174($23)

$23

$18 $5

$23

$23

Dollar markup (P ) Percent markup on selling price (R)

$5 $23 .2174

Parentheses mean you multiply the percent markup on selling price in decimal by the selling price.

Situation 2: Calculating Selling Price When You Know Cost and Percent Markup on Selling Price When you know the cost and percent markup on selling price, you calculate the selling price with the basic selling formula S C M. Remember that when goods are marked up on selling price, the selling price is the base (100%). Since you do not know the selling price, the percent markup is based on the unknown selling price. To find the dollar markup after you find the selling price, use the selling price formula M S C. The Word Problem Mel’s Furniture bought a lamp that cost $100. To make Mel’s desired

profit, he needs a 65% markup on selling price. dollar markup? The facts

Solving for?

Lamp cost: $100.

% 35% C 65 M S 100%

Markup on selling price: 65%.

What are Mel’ s selling price and his

Steps to take $ $100.00 185.712 $285.711

S C M.

Selling price. Dollar markup.

2

Cost

or

S

1

Key points

Cost Percent markup 1 on selling price

Portion ($100) Base Rate (?) (.35) Selling price

100% –65%

Steps to solving problem 1. Calculate the selling price. 1.00S .65S .35S 2. Calculate the dollar markup.

S C M S $100.00 .65S .65S .65S .35S $100.00 .35 .35 S $285.71 M

S

Do not multiply the .65 times $100.00. The 65% is based on selling price not cost.

C

$185.71 $285.71 $100.00

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You can check your selling price with the portion formula B ⫽ PR. To find the selling price ( B), divide the cost ( P) by the rate (100% ⫺ percent markup on selling price). Check Selling price (B) ⫽

⫽

Cost (P) 1 ⫺ Percent markup on selling price (R)

$100.00 $100.00 ⫽ ⫽ $285.71 1 ⫺ .65 .35

Situation 3: Calculating Cost When You Know Selling Price and Percent Markup on Selling Price When you know the selling price and the percent markup on selling price, you calculate the cost with the basic formula S ⫽ C ⫹ M. To find the dollar markup, multiply the markup percent by the selling price. When you have the dollar markup, subtract it from the selling price to get the cost. The Word Problem Jill Sport, owner of Sports, Inc., sells tennis rackets for $50. To make her desired profit, Jill needs a 40% markup on the selling price. What is the dollar markup? What do the tennis rackets cost Jill?

The facts

Solving for?

Selling price: $50.

% C 60% ⫹M 40 ⫽ S 100%

Markup on selling price: 40%.

Steps to take

Key points

S ⫽ C ⫹ M.

$ $302 201 $50

Cost

or Portion (?)

Cost ⫽ Selling price ⫻ a1 ⫺

1

Dollar markup. Cost.

2

Percent markup b on selling price

Base ⫻ Rate ($50) (.60) Selling price

100% –40%

Steps to solving problem 1. Calculate the dollar markup.

S⫽C⫹

M

$50 ⫽ C ⫹ .40($50) 2. Calculate the cost.

$50 ⫽ C ⫹ $20 ⫺ 20

Dollar markup

⫺ 20

$30 ⫽ C

To check your cost, use the portion formula Cost ( selling price ⫺ Percent markup on selling price) ( R).

P) ⫽ Selling price ( B) ⫻ (100%

Check Selling Percent markup Cost ⫽ price ⫻ a1 ⫺ b on selling price (P ) (B) (R)

⫽ $50 ⫻ .60 ⫽ $30

(1.00 ⫺ .40)

In Table 8.1, we compare percent markup on cost with percent markup on retail (selling price). This table is a summary of the answers we calculated from the word problems in Learning Units 8–1 and 8–2. The word problems in the units were the same except in Learning Unit 8–1, we assumed markups were on cost, while in Learning Unit 8–2, markups were on selling price. Note that in Situation 1, the dollar markup is the same $5, but the percent markup is dif ferent. Let’s now look at how to convert from percent markup on cost to percent markup on selling price and vice versa. We will use Situation 1 from Table 8.1.

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Learning Unit 8–2

TABLE

8.1

Comparison of markup on cost versus markup on selling price

Markup based on cost— Learning Unit 8–1

213

Markup based on selling price— Learning Unit 8–2

Situation 1: Calculating dollar amount of markup and percent markup on cost.

Situation 1: Calculating dollar amount of markup and percent markup on selling price.

Signature™ jeans cost, $18.

Signature™ jeans cost, $18.

Signature™ jeans selling price, $23.

Signature™ jeans selling price, $23.

MSC

MSC

M $23 $18 $5 markup (p. 206)

M $23 $18 $5 markup (p. 211)

M C $5 $18 27.78%

M S $5 $23 21.74%

Situation 2: Calculating selling price on cost.

Situation 2: Calculating selling price on selling price.

Lamp cost, $100. 65% markup on cost

Lamp cost, $100. 65% markup on selling price

S C (1 Percent markup on cost)

S C (1 Percent markup on selling price)

S $100 1.65 $165 (p. 207)

S $100.00 .35

(100% 65% 165% 1.65)

S $285.71 (p. 211)

Situation 3: Calculating cost on cost.

Situation 3: Calculating cost on selling price.

Tennis racket selling price, $50. 40% markup on cost

Tennis racket selling price, $50. 40% markup on selling price

C S (1 Percent markup on cost)

C S (1 Percent markup on selling price)

C $50.00 1.40

C $50 .60 $30 (p. 212)

(100% 65% 35% .35)

(100% 40% 140% 1.40)

C $35.71 (p. 207)

(100% 40% 60% .60)

Formula for Converting Percent Markup on Cost to Percent Markup on Selling Price To convert percent markup on cost to percent markup on selling price: .2778 21.74% 1 .2778

Percent markup on cost 1 Percent markup on cost

Formula for Converting Percent Markup on Selling Price to Percent Markup on Cost To convert percent markup on selling price to percent markup on cost:

Percent markup on selling price 1 Percent markup on selling price

.2174 27.78% 1 .2174 Key point: A 21.74% markup on selling price or a 27.78% markup on cost results in same dollar markup of $5. Now let’s test your knowledge of Learning Unit 8–2.

LU 8–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Solve the following situations (markups based on selling price). Note numbers 1, 2, and 3 are parallel problems to those in Practice Quiz 8–1. 1. Irene Westing bought a desk for $400 from an of fice supply house. She plans to sell the desk for $600. What is Irene’s dollar markup? What is her percent markup on selling price (round to the nearest tenth percent)? Check your answer . Selling price will be slightly of f due to rounding.

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Suki Komar bought dolls for her toy store that cost $12 each. To make her desired profit, Suki must mark up each doll 35% on the selling price. What is the selling price of each doll? What is the dollar markup? Check your answer . Jay Lyman sells calculators. His competitor sells a new calculator line for $14 each. Jay needs a 40% markup on the selling price to make his desired profit, and he must meet price competition. What is Jay’ s dollar markup? At what cost can Jay af ford to bring these calculators into the store? Check your answer . Dan Flow sells wrenches for $10 that cost $6. What is Dan’ s percent markup at cost? Round to the nearest tenth percent. What is Dan’ s percent markup on selling price? Check your answer .

2.

3.

4.

✓

Solutions 1. Irene’s dollar markup and percent markup on selling price: The facts

Solving for?

Desk cost: $400.

% C 66.7% ⫹M 33.32 ⫽ S 100%

Desk selling price: $600.

Steps to take

Key points

Dollar Selling ⫽ ⫺ Cost markup price

$ $400 2001 $600

Markup Portion ($200)

Dollar Percent markup markup on ⫽ Selling selling price price

1

Dollar markup. Percent markup on selling price.

2

Base ⫻ Rate ($600) (?) Selling price

Steps to solving problem Dollar markup ⫽ Selling price ⫺ Cost

1. Calculate the dollar markup.

$200 ⫽

⫺ $400

$600

Percent markup Dollar markup ⫽ on selling price Selling price

2. Calculate the percent markup on selling price.

⫽

$200 ⫽ 33.3% $600

Check Selling ⫽ Cost ⫹ Markup price

or

Dollar markup (P) Selling ⫽ Percent markup on selling price (R) price (B)

$600 ⫽ $400 ⫹ .333($600)

⫽ $200 ⫽ $600.60* .333 (not exactly $600 due to rounding)

$600 ⫽ $400 ⫹ $199.80 $600 ⫽ $599.80* *Off due to rounding.

2.

Selling price of doll and dollar markup: The facts

Solving for?

Steps to take

Doll cost: $12 each.

% $ C 65% $12.00 ⫹M 35 6.462 ⫽ S 100% $18.461

S ⫽ C ⫹ M.

Markup on selling price: 35%.

1

Selling price. Dollar markup.

2

Key points Cost

or

S⫽

Cost Percent markup 1⫺ on selling price

Portion ($12) Base ⫻ Rate (?) (.65) Selling price

100% –35%

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Learning Unit 8–2

Steps to solving problem 1. Calculate the selling price.

S C M S $12.00 .35S .35S .35S .65S $12.00 .65 .65 S $18.46 M $6.46

2. Calculate the dollar markup.

S $18.46

C $12.00

Check Selling price (B)

Cost (P) $12.00 $18.46 1 Percent markup on selling price (R) .65

Dollar markup and cost:

3.

The facts

Solving for?

Selling price: $14.

% C 60% M 40 S 100%

Markup on selling price: 40%.

Steps to take

Key points

S C M.

$ $ 8.402 5.601 $14.00

Cost

or Portion (?)

Cost Selling price a1

1

Dollar markup. Cost.

2

Percent markup b on selling price

Base Rate ($14) (.60) Selling price

100% –40%

Steps to solving problem 1. Calculate the dollar markup.

SCM $14.00 C .40($14.00)

2. Calculate the cost.

$14.00 C $5.60 5.60

Dollar markup

5.60

$8.40 C

Check Cost Selling price (1 Percent markup on selling price) $14.00 .60 $8.40 (P) (B) (R)

4.

LU 8–2a

$4 Cost 66.7% $6 $4 Selling price 40% $10

(1.00 .40) .40 2 .40 66.7% 1 .40 .60 3 .667 .667 40% (due to rounding) 1 .667 1.667

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 222)

Solve the following situations (markups based on selling price). 1. Irene Westing bought a desk for $800 from an of fice supply house. She plans to sell the desk for $1,200. What is Irene’ s dollar markup? What is her percent markup on selling price (round to the nearest tenth percent)? Check your answer . Selling price will be slightly of f due to rounding. 2. Suki Komar bought dolls for her toy store that cost $14 each. To make her desired profit, Suki must mark up each doll 38% on selling price. What is the selling price of each doll? What is the dollar markup? Check your answer .

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3.

4.

Jay Lyman sells calculators. His competitor sells a new calculator line for $16 each. Jay needs a 42% markup on the selling price to make his desired profit, and he must meet price competition. What is Jay’ s dollar markup? At what cost can Jay af ford to bring these calculators into the store? Check your answer . Dan Flow sells wrenches for $12 that cost $7. What is Dan’ s percent markup at cost? Round to the nearest tenth percent. What is Dan’ s percent markup on selling price? Check your answer .

Learning Unit 8–3: Markdowns and Perishables Have you ever wondered how your local retail store determines a typical markdown on clothing? The following Wall Street Journal clipping “Sale Rack Shuffle” explains the typical markdown money arrangement between a clothing vendor and a retailer. Evidently, the retailer does not always take the entire financial loss when a piece of clothing is marked down until it sells.

Wall Street Journal © 2005

Barron’s © 2005

This learning unit focuses your attention on how to calculate markdowns. Then you will learn how a business prices perishable items that may spoil before customers buy them.

Markdowns Markdowns are reductions from the original selling price caused by seasonal changes, special promotions, style changes, and so on. We calculate the markdown percent as follows: Markdown percent

Dollar markdown Selling price (original)

Let’s look at the following Kmart example: Dollar markdown Portion ($7.20) Base Rate ($18) (?) Original selling price

Kmart marked down an $18 video to $10.80. Calculate the dollar markdown and the markdown percent.

EXAMPLE

$18.00 Original selling price 10.80 Sale price $ 7.20 Markdown

Dollar markdown, $7.20 40% Selling price (original), $18.00

Calculating a Series of Markdowns and Markups Often the final selling price is the result of a series of markdowns (and possibly a markup in between markdowns). We calculate additional markdowns on the previous selling price. Note in the following example how we calculate markdown on selling price after we add a markup. Jones Department Store paid its supplier $400 for a TV. On January 10, Jones marked the TV up 60% on selling price.As a special promotion, Jones marked theTV down 30% on February 8 and another 20% on February 28. No one purchased the TV, so Jones marked it up 10% on March 11. What was the selling price of the TV on March 11?

EXAMPLE

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Learning Unit 8–3

January 10: Selling price ⫽ Cost S ⫽ $400 ⫺ .60S .40S .40 S Check

217

⫹ Markup ⫹ .60S ⫺ .60S

$400 .40 ⫽ $1,000 ⫽ S⫽

Cost 1 ⫺ Percent markup on selling price

S⫽

$400 $400 ⫽ ⫽ $1,000 1 ⫺ .60 .40

February 8 markdown:

100% ⫺ 30 70%

.70 ⫻ $1,000 ⫽ $700 selling price

February 28 additional markdown:

100% ⫺ 20 80%

.80 ⫻ $700 ⫽ $560

March 11 additional markup:

100% ⫹ 10 110%

1.10 ⫻ $560 ⫽ $616

Pricing Perishable Items The following formula can be used to determine the price of goods that have a short shelf life such as fruit, flowers, and pastry . (W e limit this discussion to obviously perishable items.) Selling price of perishables ⫽

Total dollar sales Number of units produced ⫺ Spoilage

The Word Problem Audrey’s Bake Shop baked 20 dozen bagels. Audrey expects 10% of the bagels to become stale and not salable. The bagels cost Audrey $1.20 per dozen. Audrey wants a 60% markup on cost. What should Audrey char ge for each dozen bagels so she will make her profit? Round to the nearest cent. The facts

Solving for?

Steps to take

Key points

Bagels cost: $1.20 per dozen.

Price of a dozen bagels.

Total cost.

Markup is based on cost.

Total dollar markup.

Not salable: 10%.

Total selling price.

Baked: 20 dozen.

Bagel loss.

Markup on cost: 60%.

TS ⫽ TC ⫹ TM.

Steps to solving problem 1. Calculate the total cost. 2. Calculate the total dollar markup.

TC ⫽ 20 dozen ⫻ $1.20 ⫽ $24.00

TS ⫽ TC ⫹ TM TS ⫽ $24.00 ⫹ .60($24.00) TS ⫽ $24.00 ⫹ $14.40

3. Calculate the total selling price.

TS ⫽ $38.40

4. Calculate the bagel loss.

20 dozen ⫻ .10 ⫽ 2 dozen

5. Calculate the selling price for a dozen bagels.

$38.40 ⫽ $2.13 per dozen 18

It’s time to try the Practice Quiz.

Total dollar markup

Total selling price 20 ⫺ 2

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Chapter 8 Markups and Markdowns; Perishables and Breakeven Analysis

LU 8–3

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

Sunshine Music Shop bought a stereo for $600 and marked it up 40% on selling price. To promote customer interest, Sunshine marked the stereo down 10% for one week. Since business was slow , Sunshine marked the stereo down an additional 5%. After a week, Sunshine marked the stereo up 2%. What is the new selling price of the stereo to the nearest cent? What is the markdown percent based on the original selling price to the nearest hundredth percent?

1.

DVD

Alvin Rose owns a fruit and vegetable stand. He knows that he cannot sell all his produce at full price. Some of his produce will be markdowns, and he will throw out some produce. Alvin must put a high enough price on the produce to cover markdowns and rotted produce and still make his desired profit. Alvin bought 300 pounds of tomatoes at 14 cents per pound. He expects a 5% spoilage and marks up tomatoes 60% on cost. What price per pound should Alvin charge for the tomatoes?

2.

✓

Solutions S C

1.

M

Check

S $600 .40S .40S .40S .60S $600 .60 .60 S $1,000 First markdown: Second markdown: Markup:

S

Cost 1 Percent markup on selling price

S

$600 $600 $1,000 1 .40 .60

.90 $1,000 $900 selling price .95 $900 $855 selling price 1.02 $855 $872.10 final selling price

$1,000 $872.10

$127.90 12.79% $1,000

Price of tomatoes per pound.

2.

The facts

Solving for?

Steps to take

Key points

300 lb. tomatoes at $.14 per pound.

Price of tomatoes per pound.

Total cost.

Markup is based on cost.

Total dollar markup.

Spoilage: 5%.

Total selling price.

Markup on cost: 60%.

Spoilage amount.

TS TC TM.

Steps to solving problem 1. Calculate the total cost.

TC 300 lb. $.14 $42.00

2. Calculate the total dollar markup.

TS TC TM TS $42.00 .60($42.00) TS $42.00 $25.20

Total dollar markup

3. Calculate the total selling price.

TS $67.20

4. Calculate the tomato loss.

300 pounds .05 15 pounds spoilage $67.20 $.24 per pound (rounded to 285 nearest hundredth)

5. Calculate the selling price per pound of tomatoes.

Total selling price

(300 15)

LU 8–3a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 222)

1.

Sunshine Music Shop bought a stereo for $800 and marked it up 30% on selling price. To promote customer interest, Sunshine marked the stereo down 10% for one week. Since business was slow , Sunshine marked the stereo down an additional 5%. After a week, Sunshine marked the stereo up 2%. What is the new selling price of the stereo to the nearest cent? What is the markdown percent based on the original selling price to the nearest hundredth percent?

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Learning Unit 8–4

2.

219

Alvin Rose owns a fruit and vegetable stand. He knows that he cannot sell all his produce at full price. Some of his produce will be markdowns, and he will throw out some produce. Alvin must put a high enough price on the produce to cover markdowns and rotted produce and still make his desired profit. Alvin bought 500 pounds of tomatoes at 16 cents per pound. He expects a 10% spoilage and marks up tomatoes 55% on cost. What price per pound should Alvin charge for the tomatoes?

Learning Unit 8–4: Breakeven Analysis So far in this chapter , cost is the price retailers pay to a manufacturer or supplier to bring the goods into the store. In this unit, we view costs from the perspective of manufacturers or suppliers who produce goods to sell in units, such as pens, calculators, lamps, and so on. These manufacturers or suppliers deal with two costs—fixed costs ( FC ) and variable costs (FC ). To understand how the owners of manufacturers or suppliers that produce goods per unit operate their businesses, we must understand fixed costs ( FC ), variable costs ( VC ), contribution margin (CM), and breakeven point ( BE ). Carefully study the following definitions of these terms: • • • •

•

Fixed costs (FC). Costs that do not change with increases or decreases in sales; they include payments for insurance, a business license, rent, a lease, utilities, labor , and so on. Variable costs (VC). Costs that do change in response to changes in the volume of sales; they include payments for material, some labor , and so on. Selling price (S). In this unit we focus on manufacturers and suppliers who produce goods to sell in units. Contribution margin (CM ). The dif ference between selling price ( S ) and variable costs (VC ). This difference goes first to pay of f total fixed costs ( FC ); when they are covered, profits (or losses) start to accumulate. Breakeven point (BE). The point at which the seller has covered all expenses and costs of a unit and has not made any profit or suf fered any loss. Every unit sold after the breakeven point ( BE ) will bring some profit or cause a loss.

Learning Unit 8–4 is divided into two sections: calculating a contribution mar gin (CM) and calculating a breakeven point ( BE ). You will learn the importance of these two concepts and the formulas that you can use to calculate them. Study the example given for each concept to help you understand why the success of business owners depends on knowing how to use these two concepts.

Calculating a Contribution Margin (CM ) Before we calculate the breakeven point, we must first calculate the contribution mar The formula is as follows:

gin.

Contribution margin (CM) ⫽ Selling price (S ) ⫺ Variable cost (VC ) Assume Jones Company produces pens that have a selling price S( ) of $2.00 and a variable cost (VC ) of $.80. We calculate the contribution margin (CM) as follows:

EXAMPLE

Contribution margin (CM ) ⫽ $2.00 (S) ⫺ $.80 (VC ) CM ⫽ $1.20 This means that for each pen sold, $1.20 goes to cover fixed costs ( FC ) and results in a profit. It makes sense to cover fixed costs ( FC ) first because the nature of a FC is that it does not change with increases or decreases in sales. Now we are ready to see how Jones Company will reach a breakeven point ( BE).

Calculating a Breakeven Point (BE ) Sellers like Jones Company can calculate their profit or loss by using a concept called the breakeven point (BE). This important point results after sellers have paid all their expenses and costs. Study the following formula and the example:

Breakeven point (BE) ⫽

Fixed costs (FC) Contribution margin (CM)

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EXAMPLE Jones Company produces pens. The company has a fixed cost (FC ) of $60,000. Each pen sells for $2.00 with a variable cost ( VC) of $.80 per pen.

Fixed cost ( FC ) Selling price ( S) per pen Variable cost ( VC ) per pen Breakeven point (BE) ⫽

$60,000 $2.00 $.80

$60,000 (FC ) $60,000 (FC ) ⫽ ⫽ 50,000 units (pens) $2.00 (S) ⫺ $.80 (VC ) $1.20 (CM)

At 50,000 units (pens), Jones Company is just covering its costs. Each unit after 50,000 brings in a profit of $1.20 ( CM). It is time to try the Practice Quiz.

LU 8–4

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

Blue Company produces holiday gift boxes. Given the following, calculate (1) the contribution margin ( CM) and (2) the breakeven point ( BE ) for Blue Company . Fixed cost ( FC ) Selling price ( S) per gift box Variable cost ( VC ) per gift box

DVD ✓

Solutions Contribution margin (CM ) ⫽ $20 (S) ⫺ $8 (VC ) ⫽ $12 $45,000(FC ) $45,000 (FC ) Breakeven point (BE) ⫽ ⫽ ⫽ 3,750 units (gift boxes) $20 (S) ⫺ $8 (VC ) $12 (CM)

1. 2.

LU 8–4a

$45,000 $20 $8

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 222)

Angel Company produces car radios. Given the following, calculate (1) the contribution margin ( CM) and (2) the breakeven point ( BE) for Angel Company. Fixed cost ( FC ) Selling price ( S) per radio Variable cost ( VC ) per radio

$96,000 $240 $80

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Markups based on cost: Cost is 100% (base), p. 205

Selling price (S) ⫽ Cost (C) ⫹ Markup (M)

Percent markup on cost, p. 206

Dollar markup (portion) Percent markup ⫽ Cost (base) on cost (rate)

Cost, p. 206

C⫽

Calculating selling price, p. 207

S⫽C⫹M Check S ⫽ Cost ⫻ (1 ⫹ Percent markup on cost)

Dollar markup Percent markup on cost

Example(s) to illustrate situation $400 ⫽ $300 ⫹ $100 S ⫽ C ⫹ M $100 1 1 ⫽ ⫽ 33 % $300 3 3 $100 ⫽ $303 Off slightly .33 due to rounding Cost, $6; percent markup on cost, 20% S ⫽ $6 ⫹ .20($6) Check S ⫽ $6 ⫹ $1.20 S ⫽ $7.20 $6 ⫻ 1.20 ⫽ $7.20

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

221

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (continued) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Calculating cost, p. 207

S⫽C⫹M Check

S ⫽ $100; M ⫽ 70% of cost S⫽C⫹M $100 ⫽ C ⫹ .70C Remember,≤ ¢ $100 ⫽ 1.7C C ⫽ 1.00C Check $100 ⫽C 1.7 $100 $58.82 ⫽ C ⫽ $58.82 1 ⫹ .70

Cost ⫽

Selling price 1 ⫹ Percent markup on cost

Markups based on selling price: selling price is 100% (Base), p. 210

Dollar markup ⫽ Selling price ⫺ Cost

Percent markup on selling price, p. 210

Dollar markup (portion) Percent markup ⫽ Selling price (base) selling price (rate)

Selling price, p. 211

S⫽

Calculating selling price, p. 211

S⫽C⫹M Check

M⫽S⫺C $600 ⫽ $1,000 ⫺ $400

Dollar markup Percent markup on selling price

Selling price ⫽

Cost Percent markup 1⫺ on selling price

$600 ⫽ 60% $1,000 $600 ⫽ $1,000 .60 Cost, $400; percent markup on S, 60% S⫽C⫹M S ⫽ $400 ⫹ .60S S ⫺ .60S ⫽ $400 ⫹ .60S ⫺ .60S .40S $400 S ⫽ $1,000 ⫽ .40 .40 Check

Calculating cost, p. 212

S⫽C⫹M Check Cost ⫽

Conversion of markup percent, p. 213

$1,000 ⫽ C ⫹ 60%($1,000) $1,000 ⫽ C ⫹ $600

Selling Percent markup b ⫻ a1 ⫺ price on selling price

Percent markup on cost

to

Percent markup on selling price

Percent markup on cost 1 ⫹ Percent markup on cost Percent markup on selling price

to

Percent markup on cost

Percent markup on selling price 1 ⫺ Percent markup on selling price

Markdowns, p. 216

$400 $400 ⫽ ⫽ $1,000 1 ⫺ .60 .40

Markdown percent ⫽

Dollar markdown Selling price (original)

$400 ⫽ C Check

$1,000 ⫻ (1 ⫺ .60) $1,000 ⫻ .40 ⫽ $400

Round to nearest percent: 54% markup on cost 35% markup on selling price .54 .54 ⫽ ⫽ 35% 1 ⫹ .54 1.54 35% markup on selling price

54% markup on cost

.35 .35 ⫽ ⫽ 54% 1 ⫺ .35 .65 $40 selling price 10% markdown $40 ⫻ .10 ⫽ $4 markdown $4 ⫽ 10% $40

(continues)

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Chapter 8 Markups and Markdowns; Perishables and Breakeven Analysis

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Pricing perishables, p. 217

1. Calculate total cost and total selling price. 2. Calculate selling price per unit by dividing total sales in Step 1 by units expected to be sold after taking perishables into account.

50 pastries cost 20 cents each; 10 will spoil before being sold. Markup is 60% on cost.

Breakeven point (BE ), p. 219

BE

Fixed cost (FC) Contribution margin (CM) (Selling price, S Variable cost, VC)

50 $.20 $10 TC TM $10 .60($10) $10 $6 $16 $16 2. $.40 per pastry 40 pastries

1. TC TS TS TS TS

Fixed cost (FC ) Selling price (S ) Variable cost (VC ) BE

KEY TERMS

Breakeven point, p. 219 Contribution margin, p. 219 Cost, p. 204 Dollar markdown, p. 216 Dollar markup, p. 205 Fixed cost, p. 219 Gross profit, p. 204

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 8–1a (p. 210) 1. $400; 50% 2. $5.32; $19.32 3. $11.27; $4.73

1. 2. 3. 4.

$60,000 $60,000 1,000 $90 $30 $60 units

Margin, p. 204 Markdowns, p. 216 Markup, p. 204 Net profit (net income), p. 204 Operating expenses (overhead), p. 204

LU 8–2a (p. 215) $400; 33.3% $22.58; $8.58 $6.72; $9.28 71.4%; 41.7%

$60,000 $90 $30

Percent markup on cost, p. 205 Percent markup on selling price, p. 210 Perishables, p. 217 Selling price, p. 204 Variable cost, p. 219

LU 8–3a (p. 218) 1. $996.68; 12.79% 2. .28

LU 8–4a (p. 220) 1. $160; $600

Critical Thinking Discussion Questions 1. Assuming markups are based on cost, explain how the portion formula could be used to calculate cost, selling price, dollar markup, and percent markup on cost. Pick a company and explain why it would mark goods up on cost rather than on selling price. 2. Assuming markups are based on selling price, explain how the portion formula could be used to calculate cost, selling price, dollar markup, and percent markup on selling price. Pick a company and explain why it would mark up goods on selling price rather than on cost. 3. What is the formula to convert percent markup on selling price to percent markup on cost? How could you explain that a 40% markup on selling price, which is a 66.7% markup on cost, would result in the same dollar markup?

4. Explain how to calculate markdowns. Do you think stores should run one-day-only markdown sales? Would it be better to offer the best price “all the time”? 5. Explain the five steps in calculating a selling price for perishable items. Recall a situation where you saw a store that did not follow the five steps. How did it sell its items? 6. Explain how Wal-Mart uses breakeven analysis. Give an example.

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Assume markups in Problems 8–1 to 8–6 are based on cost. Find the dollar markup and selling price for the following problems. Round answers to the nearest cent. Item 8–1. Apple iPod

Cost $300

Markup percent 40%

8–2. Luminox Navy Seal watch

$300

30%

Dollar markup

Selling price

Solve for cost (round to the nearest cent): 8–3. Selling price of office furniture at Staples, $6,000 Percent markup on cost, 40% Actual cost? 8–4. Selling price of lumber at Home Depot, $4,000 Percent markup on cost, 30% Actual cost? Complete the following: Cost

Selling price

Dollar markup

Percent markup on cost*

8–5. $15.10

$22.00

?

?

8–6. ?

?

$4.70

102.17%

*Round to the nearest hundredth percent.

Assume markups in Problems 8–7 to 8–12 are based on selling price. Find the dollar markup and cost (round answers to the nearest cent): Item

Selling price

Markup percent

8–7. Panasonic plasma TV

$450

40%

8–8. IBM scanner

$80

30%

Dollar markup

Cost

223

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Solve for the selling price (round to the nearest cent): 8–9. Selling price of a complete set of pots and pans at Wal-Mart? 40% markup on selling price Cost, actual, $66.50

8–10. Selling price of a dining room set at Macy’s? 55% markup on selling price Cost, actual, $800

Complete the following: Selling price

Dollar markup

Percent markup on selling price (round to nearest tenth percent)

8–11. $14.80

$49.00

?

?

8–12. ?

?

$4

20%

Cost

By conversion of the markup formula, solve the following (round to the nearest whole percent as needed): Percent markup on cost

Percent markup on selling price

8–13. 12.4%

?

8–14. ?

13%

Complete the following: 8–15. Calculate the final selling price to the nearest cent and markdown percent to the nearest hundredth percent: Original selling price

First markdown

Second markdown

Markup

Final markdown

$5,000

20%

10%

12%

5%

Item 8–16. Brownies

224

Total quantity bought

Unit cost

20

$.79

Total cost

Percent markup on cost

Total selling price

Percent that will spoil

Selling price per brownie

?

60%

?

10%

?

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Complete the following: Breakeven point

Fixed cost

Contribution margin

Selling price per unit

Variable cost per unit

8–17.

$65,000

$5.00

$1.00

8 – 18.

$90,000

$9.00

$4.00

WORD PROBLEMS 8 – 19. Matthew Kaminsky bought an old Walter Lantz Woody Woodpecker oil painting for $10,000. He plans to resell it on eBay for $15,000. What are the dollar markup and percent markup on cost? Check the cost figure.

8 – 20. Chin Yov, store manager for Best Buy, does not know how to price a GE freezer that cost the store $600. Chin knows his boss wants a 45% markup on cost. Help Chin price the freezer.

8 – 21. Cecil Green sells golf hats. He knows that most people will not pay more than $20 for a golf hat. Cecil needs a 40% markup on cost. What should Cecil pay for his golf hats? Round to the nearest cent.

8 – 22. Macy’s was selling Calvin Klein jean shirts that were originally priced at $58.00 for $8.70. (a) What was the amount of the markdown? (b) Based on the selling price, what is the percent markdown?

8–23. The Miami Herald, on January 31, 2007, ran a story on Super Bowl ticket prices. Ticket reseller Stubhub.com reported the average Super Bowl seat was selling for $4,445 with a face value of $700. (a) What is the percent markup based on cost? (b) What is the percent markup based on selling price? Round to the nearest hundredth percent.

225

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8–24. The February 3, 2007 issue of Billboard reported on hefty markups by leading music merchants. Canadian Indies say they generally sell all products to independent distributors at between $8.00 and $9.50 per unit, which is then supplied to retailers at between $13.50 and $14.50. (a) What is the percent markup on cost for the lower price? (b) What is the percent markup on cost for the higher price? Round to the nearest hundredth percent.

8–25. Misu Sheet, owner of the Bedspread Shop, knows his customers will pay no more than $120 for a comforter. Misu wants a 30% markup on selling price. What is the most that Misu can pay for a comforter?

8–26. Assume Misu Sheet (Problem 8–25) wants a 30% markup on cost instead of on selling price. What is Misu’s cost? Round to the nearest cent.

8–27. Misu Sheet (Problem 8–25) wants to advertise the comforter as “percent markup on cost.” What is the equivalent rate of percent markup on cost compared to the 30% markup on selling price? Check your answer. Is this a wise marketing decision? Round to the nearest hundredth percent.

8 – 28. DeWitt Company sells a kitchen set for $475. To promote July 4, DeWitt ran the following advertisement: Beginning each hour up to 4 hours we will mark down the kitchen set 10%. At the end of each hour, we will mark up the set 1%. Assume Ingrid Swenson buys the set 1 hour 50 minutes into the sale. What will Ingrid pay? Round each calculation to the nearest cent. What is the markdown percent? Round to the nearest hundredth percent.

226

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8–29. Angie’s Bake Shop makes birthday chocolate chip cookies that cost $2 each. Angie expects that 10% of the cookies will crack and be discarded. Angie wants a 60% markup on cost and produces 100 cookies. What should Angie price each cookie? Round to the nearest cent.

8–30. Assume that Angie (Problem 8–29) can sell the cracked cookies for $1.10 each. What should Angie price each cookie?

8–31. Jane Corporation produces model toy cars. Each sells for $29.99. Its variable cost per unit is $14.25. What is the breakeven point for Jane Corporation assuming it has a fixed cost of $314,800?

ADDITIONAL SET OF WORD PROBLEMS 8–32. PFS Fitness bought a treadmill for $700. PFS has a 70% markup on selling price. What is the selling price of the treadmill (to the nearest dollar)?

8–33. Sachi Wong, store manager for Hawk Appliance, does not know how to price a GE dishwasher that cost the store $399. Sachi knows her boss wants a 40% markup on cost. Can you help Sachi price the dishwasher?

8–34. Working off an 18% margin, with markups based on cost, the Food Co-op Club boasts that they have 5,000 members and a 200% increase in sales. The markup is 36% based on cost. What would be their percent markup if selling price were the base? Round to the nearest hundredth percent.

8–35. At a local Bed and Bath Superstore, the manager, Jill Roe, knows her customers will pay no more than $300 for a bedspread. Jill wants a 35% markup on selling price. What is the most that Jill can pay for a bedspread?

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8–36. U.S. News & World Report October 9, 2006, reported RV dealer markups can top 40 percent. Jim Abbott purchased

a $60,000 RV with a 40 percent markup on selling price. (a) What was the amount of the dealer’s markup? (b) What was the dealers original cost?

8–37. Circuit City sells a hand-held personal planner for $199.99. Circuit City marked up the personal planner 35% on the selling price. What is the cost of the hand-held personal planner?

8–38. Arley’s Bakery makes fat-free cookies that cost $1.50 each. Arley expects 15% of the cookies to fall apart and be discarded. Arley wants a 45% markup on cost and produces 200 cookies. What should Arley price each cookie? Round to the nearest cent.

8–39. Assume that Arley (Problem 8–38) can sell the broken cookies for $1.40 each. What should Arley price each cookie?

8–40. An Apple Computer Center sells computers for $1,258.60. Assuming the computers cost $10,788 per dozen, find for each computer the (a) dollar markup, (b) percent markup on cost, and (c) percent markup on selling price (nearest hundredth percent).

Prove (b) and (c) of the above problem using the equivalent formulas.

8–41. Pete Corporation produces bags of peanuts. Its fixed cost is $17,280. Each bag sells for $2.99 with a unit cost of $1.55. What is Pete’s breakeven point?

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CHALLENGE PROBLEMS 8–42. Virtual dealer, Dirt Cheap, says it marks up its jewelry a mere 8%. That is why Peter Bertling could buy a two-carat pair of diamond earrings for $5,000—49% of what he would pay at a conventional retailer. (a) Based on selling price, what is Dirt Cheap’s cost? (b) What is Dirt Cheap’s markup amount? (c) What was the selling price of the conventional retailer? (d) How much did Peter save? Round to the nearest hundredth.

8–43. On July 8, 2009, Leon’s Kitchen Hut bought a set of pots with a $120 list price from Lambert Manufacturing. Leon’s receives a 25% trade discount. Terms of the sale were 2/10, n/30. On July 14, Leon’s sent a check to Lambert for the pots. Leon’s expenses are 20% of the selling price. Leon’s must also make a profit of 15% of the selling price. A competitor marked down the same set of pots 30%. Assume Leon’s reduces its selling price by 30%. a.

What is the sale price at Kitchen Hut?

b. What was the operating profit or loss?

DVD SUMMARY PRACTICE TEST 1. Sunset Co. marks up merchandise 40% on cost. A DVD player costs Sunset $90. What is Sunset’s selling price? Round to the nearest cent. (p. 206)

2. JCPenney sells jeans for $49.50 that cost $38.00. What is the percent markup on cost? Round to the nearest hundredth percent. Check the cost. (p. 206)

3. Best Buy sells a flat-screen high-definition TV for $700. Best Buy marks up the TV 45% on cost. What is the cost and dollar markup of the TV? (p. 207)

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4. Sports Authority marks up New Balance sneakers $30 and sells them for $109. Markup is on cost. What are the cost and percent markup to the nearest hundredth percent? (p. 206)

5. The Shoe Outlet bought boots for $60 and marks up the boots 55% on the selling price. What is the selling price of the boots? Round to the nearest cent. (p. 211)

6. Office Max sells a desk for $450 and marks up the desk 35% on the selling price. What did the desk cost Office Max? Round to the nearest cent. (p. 212)

7. Zales sells diamonds for $1,100 that cost $800. What is Zales’s percent markup on selling price? Round to the nearest hundredth percent. Check the selling price. (p. 211)

8. Earl Miller, a customer of J. Crew, will pay $400 for a new jacket. J. Crew has a 60% markup on selling price. What is the most that J. Crew can pay for this jacket? (p. 212)

9. Home Liquidators mark up its merchandise 35% on cost. What is the company’s equivalent markup on selling price? Round to the nearest tenth percent. (p. 213)

10. The Muffin Shop makes no-fat blueberry muffins that cost $.70 each. The Muffin Shop knows that 15% of the muffins will spoil. If The Muffin Shop wants 40% markup on cost and produces 800 muffins, what should The Muffin Shop price each muffin? Round to the nearest cent. (p. 217)

11. Angel Corporation produces calculators selling for $25.99. Its unit cost is $18.95. Assuming a fixed cost of $80,960, what is the breakeven point in units? (p. 219)

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Personal Finance A KIPLINGER APPROACH

INSURANCE

| Yes, you can afford coverage that

pays nursing-home costs. By Kimberly Lankford

A fresh look at LONG TERM care

C

A shorter benefit period. For

starters, you probably don’t need a policy that pays lifetime benefits. Milliman, an actuarial consulting firm, recently stud-

The best deal of all may be a sharedcare policy, which gives you and your spouse a pool of benefits. If you each buy, say, a fiveyear shared-care policy, you actually get ten years to split between you. Most long-term-care insurers offer such policies, which generally cost about 10% more than separate policies with the same benefit period. Shared care.

MIKI DUISTERHOF

ould you afford to withdraw $250,000 from your retirement savings to pay for one year in a nursing home? Based on current charges, that’s the projected cost in 25 years, when today’s 55-yearold is likely to need care. And with nursing-home stays averaging about 2.5 years, your total bill could top $600,000—which could quickly drain your retirement accounts, leaving you and your spouse with little savings and your heirs without an inheritance. Buying long-term-care insurance is the best way to protect your retirement savings from astronomical bills. And a new law, which makes it more difficult to qualify for medicaid coverage of nursing-home costs, gives long-termcare policies a boost (see “Medicaid Gets Tough,” on page 86). Long-term-care coverage doesn’t come cheap. Prices for new policies have jumped by 20% to 40% over the past few years. It can now cost a 55year-old nearly $5,000 per year for a lifetime policy with a $200 daily benefit (the average nursing-home cost nationwide), 5% compound inflation protection and a 60-day waiting period before benefits begin. That’s nearly $7,000 for a married couple, even with a spousal discount. But with some smart planning, you can buy all the coverage you need for a fraction of that amount.

cut your premiums significantly. A John Hancock policy with a $200 daily benefit and a five-year benefit period would cost a 55-year-old $2,900 per year—about $2,000 less than lifetime coverage, or $3,000 less per couple annually. For example, Glen and Joan Berwick of South Glastonbury, Conn., each bought a six-year policy with a $150 daily benefit from John Hancock four years ago, when Glen was 63 ● Glen and Joan Berwick and Joan was 59, stretched their premium saving them thoudollars by buying six-year sands of dollars. shared-care policies. One caveat: Of the 8% of nursinghome residents likely to need extended care, many will have chronic conditions, such as Alzheimer’s. If you have a family history of a chronic disease, you’re better off with a policy with a ten-year benefit period, which would still cost a 55-year-old $1,000 less a year than a lifetimebenefits policy.

ied more than 1.6 million long-termcare policies and found that only about 8% of 70-year-old claimants are likely to need care for longer than five years—leaving 92% with claims of five years or fewer. Dawn Helwig, the study’s co-author, points out that the average claim period is even shorter, because most people don’t activate their policies until they are in their eighties. Shortening the benefit period can

BUSINESS MATH ISSUE Kiplingers © 2006

There is little markup on long-term care and thus it is a bargain at any age. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A Using an example, explain the following: product cost; other direct costs; net profit margin.

232

b site text We he e e S : s t T t Projec /slater9e) and e. Interne m ce Guid r o u .c o e s h e h R t .m e (www Intern ss Math Busine

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DVD

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Video Case HOTEL MONACO CHICAGO

You may not have heard of the Kimpton Group, but you have probably heard the name of at least one of their 40 stylish boutique hotels that combine affordability with personality. The company believes it is cheaper to renovate old downtown buildings into charming hotels featuring popular restaurants than build new chain hotels. Developing a classy boutique property costs $150,000 per room compared to $350,000 a room for a new chain hotel. Investors also see a more rapid return. The late Bill Kimpton, a former Lehman Bros. investment banker, created the boutique hotel concept and founded the Kimpton Hotel & Restaurant Group, a San Francisco– based chain, in 1981. Today the Kimpton Group runs 40 luxury hotels and 36 restaurants in the United States and Canada. Rates run from $100 to $200 per night, the average being $140, which is usually about 25% to 30% less than comparative nearby hotels. The average occupancy is 62%.

PROBLEM

1

As shown in the video, Hotel Monaco’s clients are 65% business travelers, of which 35% are with groups. Hotel Monaco has 192 rooms with the occupancy rate running about 62%. (a) On a given evening, how many guests would be business travelers? (b) How many would be group business travelers? Round to the nearest whole number. PROBLEM

2

On May 16, 2003, the Chicago Sun Times reported Chicago downtown hotels were averaging occupancy of 50.5% for the first two months of 2003, with rates averaging $121. Regionwide, occupancy was 47.9%, up almost 4 percentage points from the same period a year ago, but average room rates inched down to $92 from $93 last year. Hotel Monaco has 192 rooms, averaging $199 per room. (a) What is the percent change in Hotel Monaco’s average rate compared to the industry average rate? (b) What is the percent change in regionwide rates? Round to the nearest hundredths. (c) What would be the revenue generated by Hotel Monaco for one evening? Round to the nearest dollar. PROBLEM

3

Hotel Monaco’s occupancy rate topped 62% in 2001 and 70% in 2000. The average rate is $199 per evening. What would be the dollar change in total revenue for one week (7 days) based on 192 rooms? Round final answer to the nearest dollar. PROBLEM

4

The average daily hotel room rate totaled $104.32 at yearend 2002, $113.12 at year-end 2001, and $116.42 at yearend 2000. (a) With 2000 as the base year, what were the percent changes each year? (b) Using 2002 as the base year, what were the percent changes each year? Round to the nearest hundredth.

The Kimpton Group is known for its innovative ideas. All seven hotels offer the “Guppy Love” goldfish service. Steve Pinetti, senior vice president–sales and marketing, came up with the goldfish idea. He suggested providing complimentary goldfish to guests. Among the other concepts that Kimpton claims to have originated are pet-friendly hotels, custom-made “tall” beds, and complimentary wine hours for guests. U.S. hotel operators look to Kimpton for inspiration and credit it with inventing boutique hotels. When a new hotel is planned, Kimpton invites people from theaters, galleries, and department stores in the area to offer ideas that might suit the particular location. Kimpton’s philosophy is that travelers want something different and exciting in a hotel, and the element of excitement should not be underestimated no matter what the age group or location. Goldfish seem to be hooking customers for Kimpton.

PROBLEM

5

On April 9, 2002, USA Today reported total nationwide revenue for boutique hotels dropped 13% to $1.6 billion in 2001 from the previous year. Revenue per available room— another measure of the hotel industry’s financial health—fell 16% in 2001 for boutique hotels, compared with 6% for all hotels. (a) What had been the total revenue in 2000? Round to the nearest tenth. (b) Based on an average room rate of $199, what was the dollar change for boutique hotels? (c) With the same average rate, what was the dollar change for hotels? Round to the nearest whole dollar. PROBLEM

6

Thomas LaTour, CEO of the Kimpton Group, stated it is more profitable to develop boutique hotels than large chain hotels because it is cheaper to renovate an old downtown building than build new chain hotels. To develop a classy boutique property costs $150,000 per room compared to $350,000 per room for a new chain hotel. (a) What is the percent increase in the cost of a new chain hotel? Round to the nearest hundredth. (b) What would be the total cost of a 192-room boutique? (c) What would be the total cost of a 192-room new chain hotel? PROBLEM

7

Revenue in 2001 for the private Kimpton Group fell 15% to $350 million. CEO Thomas LaTour predicted sales growth of 5% for 2002. Room occupancy rates, which sank to 45% after September 11, have risen to 66% during the month of April. (a) What had been the total revenue in 2002? Round to the nearest million. (b) What is the amount of sales growth projected for 2002? (c) With 192 rooms, what is the change in room occupancy? Round to the nearest whole number.

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CUMULATIVE REVIEW

A Word Problem Approach—Chapters 6, 7, 8 1.

Assume Kellogg’s produced 715,000 boxes of Corn Flakes this year. This was 110% of the annual production last year. What was last year’s annual production? (p. 150)

2.

A new Sony camcorder has a list price of $420. The trade discount is 10/20 with terms of 2/10, n/30. If a retailer pays the invoice within the discount period, what is the amount the retailer must pay? (p. 175)

3.

JCPenney sells loafers with a markup of $40. If the markup is 30% on cost, what did the loafers cost JCPenney? Round to the nearest dollar. (p. 207)

4.

Aster Computers received from Ring Manufacturers an invoice dated August 28 with terms 2/10 EOM. The list price of the invoice is $3,000 (freight not included). Ring offers Aster a 9/8/2 trade chain discount. Terms of freight are FOB shipping point, but Ring prepays the $150 freight. Assume Aster pays the invoice on October 9. How much will Ring receive? (p. 175)

5.

Runners World marks up its Nike jogging shoes 25% on selling price. The Nike shoe sells for $65. How much did the store pay for them? (p. 212)

6.

Ivan Rone sells antique sleds. He knows that the most he can get for a sled is $350. Ivan needs a 35% markup on cost. Since Ivan is going to an antiques show, he wants to know the maximum he can offer a dealer for an antique sled. (p. 207)

7.

Bonnie’s Bakery bakes 60 loaves of bread for $1.10 each. Bonnie’s estimates that 10% of the bread will spoil. Assume a 60% markup on cost. What is the selling price of each loaf? If Bonnie’s can sell the old bread for one-half the cost, what is the selling price of each loaf? (p. 217)

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CHAPTER

9

Payroll

LEARNING UNIT OBJECTIVES LU 9–1: Calculating Various Types of Employees’ Gross Pay • Define, compare, and contrast weekly, biweekly, semimonthly, and monthly pay periods (p. 236). • Calculate gross pay with overtime on the basis of time (p. 237). • Calculate gross pay for piecework, differential pay schedule, straight commission with draw, variable commission scale, and salary plus commission (pp. 238–240).

LU 9–2: Computing Payroll Deductions for Employees’ Pay; Employers’ Responsibilities • Prepare and explain the parts of a payroll register (p. 241–244). • Explain and calculate federal and state unemployment taxes (p. 244).

urnal © 2005 Wall Street Jo

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Chapter 9 Payroll

Wall Street Journal © 2005

The Wall Street Journal clipping “Delta, Pilots Agree on Interim Pay Cuts” shows how pilots have agreed to a 14% pay cut. Note that the average pilot salary is $170,000. A 14% pay cut means a loss of $23,800. This chapter discusses (1) the type of pay people work for , (2) how employers calculate paychecks and deductions, and (3) what employers must report and pay in taxes.

Tom Uhlman/AP Wide World

Learning Unit 9–1: Calculating Various Types of Employees’ Gross Pay Logan Company manufactures dolls of all shapes and sizes. These dolls are sold worldwide. We study Logan Company in this unit because of the variety of methods Logan uses to pay its employees. Companies usually pay employees weekly, biweekly, semimonthly, or monthly. How often employers pay employees can af fect how employees manage their money . Some employees prefer a weekly paycheck that spreads the inflow of money . Employees who have monthly bills may find the twice-a-month or monthly paycheck more convenient. All employees would like more money to manage. Let’s assume you earn $50,000 per year . The following table shows what you would earn each pay period. Remember that 13 weeks equals one quarter . Four quarters or 52 weeks equals a year . Salary paid

Period (based on a year)

Earnings for period (dollars)

Weekly

52 times (once a week)

$ 961.54 ($50,000 ⫼ 52)

Biweekly

26 times (every two weeks)

$1,923.08 ($50,000 ⫼ 26)

Semimonthly

24 times (twice a month)

$2,083.33 ($50,000 ⫼ 24)

Monthly

12 times (once a month)

$4,166.67 ($50,000 ⫼ 12)

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Learning Unit 9–1

237

Now let’s look at some pay schedule situations and examples of how Logan Company calculates its payroll for employees of dif ferent pay status.

Situation 1: Hourly Rate of Pay; Calculation of Overtime The Fair Labor Standards Act sets minimum wage standards and overtime regulations for employees of companies covered by this federal law . The law provides that employees working for an hourly rate receive time-and-a-half pay for hours worked in excess of their regular 40-hour week. The current hourly minimum wage is $5.85, rising to $6.55 in summer of 2008 and then $7.25 in summer of 2009. Many managerial people, however , are exempt from the time-and-a-half pay for all hours in excess of a 40-hour week.

Ryan McVay/Getty Images

Wall Street Journal © 2005

In addition to many managerial people being exempt from time-and-a-half pay for more than 40 hours, other workers may also be exempt. Note in the Wall Street Journal clipping “As Tech Matures, Workers File a Spate of Salary Complaints” that many employees in the tech sector plan to sue if they do not get overtime pay . Now we return to our Logan Company example. Logan Company is calculating the weekly pay of Ramon Valdez who works in its manufacturing division. For the first 40 hours Ramon works, Logan calculates his gross pay (earnings before deductions) as follows: Gross pay Hours employee worked Rate per hour

Ramon works more than 40 hours in a week. For every hour over his 40 hours, Ramon must be paid an overtime pay of at least 1.5 times his regular pay rate. The following formula is used to determine Ramon’ s overtime: Hourly overtime pay rate Regular hourly pay rate 1.5

Logan Company must include Ramon’ s overtime pay with his regular pay . To determine Ramon’s gross pay , Logan uses the following formula: Gross pay Earnings for 40 hours Earnings at time-and-a-half rate (1.5)

We are now ready to calculate Ramon’ s gross pay from the following data:

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Chapter 9 Payroll

EXAMPLE

Employee Ramon Valdez

M 13

T 821

W 10

6112 total hours 40 regular hours 2112 hours overtime1

Th 8

F 1141

S 1034

Total hours 6112

Rate per hour $9

Time-and-a-half pay: $9 1.5 $13.50

Gross pay (40 hours $9) (2112 hours $13.50) $360

$290.25

$650.25 Note that the $13.50 overtime rate came out even. However , throughout the text, if an overtime rate is greater than two decimal places, do not round it. Round only the final answer. This gives greater accuracy.

Situation 2: Straight Piece Rate Pay Some companies, especially manufacturers, pay workers according to how much they produce. Logan Company pays R yan Foss for the number of dolls he produces in a week. This gives Ryan an incentive to make more money by producing more dolls. R yan receives $.96 per doll, less any defective units. The following formula determines R yan’s gross pay: Gross pay = Number of units produced Rate per unit

Companies may also pay a guaranteed hourly wage and use a piece rate as a bonus. However, Logan uses straight piece rate as wages for some of its employees. During the last week of April, Ryan Foss produced 900 dolls. Using the above formula, Logan Company paid Ryan $864.

EXAMPLE

Gross pay 900 dolls $.96 $864

Situation 3: Differential Pay Schedule Some of Logan’ s employees can earn more than the $.96 straight piece rate for every doll they produce. Logan Company has set up a differential pay schedule for these employees. The company determines the rate these employees make by the amount of units the employees produce at different levels of production. EXAMPLE

Logan Company pays Abby Rogers on the basis of the following schedule: Units produced

Amount per unit

First 50

1–50

Next 100

51–150

$ .50

Next 50

151–200

.75

Over 200

1.25

.62

Last week Abby produced 300 dolls. What is Abby’s gross pay? Logan calculated Abby’s gross pay as follows: (50 $.50) (100 $.62) (50 $.75) (100 $1.25) $25

1

$62

$37.50

$125

$249.50

Some companies pay overtime for time over 8 hours in one day; Logan Company pays overtime for time over 40 hours per week.

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Learning Unit 9–1

239

Now we will study some of the other types of employee commission payment plans.

Situation 4: Straight Commission with Draw Companies frequently use straight commission to determine the pay of salespersons. This commission is usually a certain percentage of the amount the salesperson sells. An example of one group of companies ceasing to pay commissions is the rental-car companies. Companies such as Logan Company allow some of its salespersons to draw against their commission at the beginning of each month. A draw is an advance on the salesperson’s commission. Logan subtracts this advance later from the employee’ s commission earned based on sales. When the commission does not equal the draw , the salesperson owes Logan the difference between the draw and the commission. Commission

Portion Base Rate Net sales

Commission rate

Logan Company pays Jackie Okamoto a straight commission of 15% on her net sales (net sales are total sales less sales returns). In May, Jackie had net sales of $56,000. Logan gave Jackie a $600 draw in May . What is Jackie’ s gross pay? Logan calculated Jackie’ s commission minus her draw as follows:

EXAMPLE

$56,000 .15

$8,400 600 $7,800

Logan Company pays some people in the sales department on a variable commission scale. Let’s look at this, assuming the employee had no draw .

Situation 5: Variable Commission Scale A company with a variable commission scale uses dif ferent commission rates for dif ferent levels of net sales. Last month, Jane Ring’ s net sales were $160,000. based on the following schedule?

EXAMPLE

Up to $35,000 Excess of $35,000 to $45,000 Over $45,000

What is Jane’ s gross pay

4% 6% 8%

Gross pay ($35,000 .04) ($10,000 .06) ($115,000 .08)

$1,400

$600

$9,200

$11,200

Situation 6: Salary Plus Commission Logan Company pays Joe Roy a $3,000 monthly salary plus a 4% commission for sales over $20,000. Last month Joe’ s net sales were $50,000. Logan calculated Joe’ s gross monthly pay as follows: Gross pay Salary (Commission Sales over $20,000) $3,000 $3,000

(.04 $30,000) $1,200

$4,200 Before you take the Practice Quiz, you should know that many managers today receive overrides. These managers receive a commission based on the net sales of the people they supervise.

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Chapter 9 Payroll

LU 9–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

Jill Foster worked 52 hours in one week for Delta Airlines. Jill earns $10 per hour. What is Jill’s gross pay, assuming overtime is at time-and-a-half? Matt Long had $180,000 in sales for the month. Matt’ s commission rate is 9%, and he had a $3,500 draw. What was Matt’s end-of-month commission? Bob Meyers receives a $1,000 monthly salary . He also receives a variable commission on net sales based on the following schedule (commission doesn’ t begin until Bob earns $8,000 in net sales): $8,000–$12,000 1% Excess of $20,000 to $40,000 5% Excess of $12,000 to $20,000 3% More than $40,000 8% Assume Bob earns $40,000 net sales for the month. What is his gross pay?

1. 2.

DVD

3.

✓

Solutions

1.

2.

3.

LU 9–1a

40 hours $10.00 $400.00 12 hours $15.00 180.00 ($10.00 1.5 $15.00) $580.00 $180,000 .09 $16,200 3,500 $12,700 Gross pay $1,000 ($4,000 .01) ($8,000 .03) ($20,000 .05) $1,000 $40 $240 $1,000 $2,280

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 247)

1. 2. 3.

Jill Foster worked 54 hours in one week for Delta Airlines. Jill earns $12 per hour. What is Jill’s gross pay, assuming overtime is at time-and-a-half? Matt Long had $210,000 in sales for the month. Matt’ s commission rate is 8%, and he had a $4,000 draw. What was Matt’s end-of-month commission? Bob Myers receives a $1,200 monthly salary. He also receives a variable commission on net sales based on the following schedule (commission doesn’ t begin until Bob earns $9,000 in net sales). $9,000–$12,000 1% Excess of $20,000 to $40,000 5% Excess of $12,000 to $20,000 3% More than $40,000 8% Assume Bob earns $60,000 net sales for the month. What is his gross pay?

Learning Unit 9–2: Computing Payroll Deductions for Employees’ Pay; Employers’ Responsibilities Did you know that Wal-Mart is the lar gest employer in twenty-one states? Can you imagine the accounting involved to pay all these employees? This unit begins by dissecting a paycheck. Then we give you an insight into the tax responsibilities of employers.

Computing Payroll Deductions for Employees Companies often record employee payroll information in a multicolumn form called a payroll register. The increased use of computers in business has made computerized registers a timesaver for many companies. Glo Company uses a multicolumn payroll register . On page 241 is Glo’ s partial payroll register showing the payroll information for Alice Rey during week 44. Let’ s check each column to see if Alice’s take-home pay of $1,324.36 is correct. Note how the circled letters in the register correspond to the explanations that follow .

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241

Learning Unit 9–2

GLO COMPANY Payroll Register Week #44 Allow. & Employee marital Cum. name status earn.

Sal. per week

Reg.

Ovt.

Rey, Alice

2,250

—

M-2

96,750

2,250

A

B

C

Payroll Register Explanations A —Allowance and marital status B , C , D —Cumulative earnings before payroll, salaries, earnings E —Cumulative earnings after payroll

Earnings

FICA Taxable Earnings

Deductions FICA

Gross

Cum. earn.

S.S.

Med.

S.S.

2,250

99,000

750

2,250

46.50

E

F

G

H

D

FIT

32.63 355.96 I

J

SIT

Health ins.

Net pay

135

100

1,579.91

K

L

M

When Alice was hired, she completed the W-4 (Employee’s Withholding Allowance Certificate) form shown in Figure 9.1 stating that she is married and claims an allowance (exemption) of 2. Glo Company will need this information to calculate the federal income tax ● J. Before this pay period, Alice has earned $96,750 (43 weeks ⫻ $2,250 salary per week). Since Alice receives no overtime, her $2,250 salary per week represents her gross pay (pay before any deductions). After this pay period, Alice has earned $99,000 ($96,750 ⫹ $2,250). The Federal Insurance Contribution Act (FICA) funds the Social Security program. The program includes Old Age and Disability, Medicare, Survivor Benefits, and so on. The FICA tax requires separate reporting for Social Security and Medicare. We will use the following rates for Glo Company: Rate

F , G —Taxable earnings for Social Security and Medicare

Med.

Base

Social Security

6.20%

$97,500

Medicare

1.45

No base

These rates mean that Alice Rey will pay Social Security taxes on the first $97,500 she earns this year . After earning $97,500, Alice’s wages will be exempt from Social Security . Note that Alice will be paying Medicare taxes on all wages since Medicare has no base cutoff. To help keep Glo’s record straight, the taxable earnings column only shows what wages will be taxed. This amount is not the tax. For example, in week 44, only $750 of Alice’s salary will be taxable for Social Security . $97,500 Social Security base B ⫺ 96,750 ● $ 750

FIGURE

9.1

Employee’s W-4 form

Alice

Rey

021 36 9494 X

2 Roundy Road Marblehead, MA 01945

Alice Rey

2

1/1

XX

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Chapter 9 Payroll

TABLE

9.1

Percentage method income tax withholding table

Dept. of the Treasury, Internal Revenue Service Publication 15, Jan. 2007.

H —Social Security

To calculate Alice’s Social Security tax, we multiply $750

F ●

by 6.2%:

$750 .062 $46.50 I —Medicare

Since Medicare has no base, Alice’s entire weekly salary is taxed 1.45%, which is multiplied by $2,250. $2,250 .0145 $32.63

J —FIT

Using the W-4 form Alice completed, Glo deducts federal income tax withholding (FIT). The more allowances an employee claims, the less money Glo deducts from the employee’ s paycheck. Glo uses the percentage method to calculate FIT .2 The Percentage Method3 Today, since many companies do not want to store the tax tables, they use computers for their payroll. These companies use the percentage method. For this method we use Table 9.1 and Table 9.2 from Circular E to calculate Alice’s FIT. Step 1.

In Table 9.1, locate the weekly withholding for one allowance. Multiply this number by 2. $65.38 2 $130.76

Step 2.

Subtract $130.76 in Step 1 from Alice’s total pay.

Step 3.

$2,250.00 130.76 $2,119.24

In Table 9.2, locate the married person’s weekly pay table. The $2,119.24 falls between $1,360 and $2,573. The tax is $166.15 plus 25% of the excess over $1,360.00. $2,119.24 1,360.00 $ 759.24 Tax

$166.15 .25 ($759.24) $166.15 $189.81 $355.96

We assume a 6% state income tax (SIT).

K —SIT

$2,250 .06 $135.00 L —Health insurance M —Net pay

Alice contributes $100 per week for health insurance. Alice’s net pay is her gross pay less all deductions. $2,250.00 46.50 32.63 355.96 135.00 100.00 $1,579.91

gross Social Security Medicare FIT SIT health insurance net pay

2

The Business Math Handbook has a sample of the wage bracket method.

3

An alternative method is called the wage bracket method that is shown in the Business Math Handbook.

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Learning Unit 9–2

TABLE

9.2

Percentage method income tax withholding taxes

Dept. of the Treasury, Internal Revenue Service Publication 15, Jan. 2007.

243

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Chapter 9 Payroll

Employers’ Responsibilities

Wall Street Journal © 2005

In the first section of this unit, we saw that Alice contributed to Social Security and Medicare. Glo Company has the legal responsibility to match her contributions. Besides matching Social Security and Medicare, Glo must pay two important taxes that employees do not have to pay—federal and state unemployment taxes. Note that the Wall Street Journal clipping “W orkers’ Comp Costs Increase Nearly 10%” states that employers now face an increase in their worker ’s compensation costs.

Federal Unemployment Tax Act (FUTA) The federal government participates in a joint federal-state unemployment program to help unemployed workers. At this writing, employers pay the government a 6.2% FUTA tax on the first $7,000 paid to employees as wages during the calendar year . Any wages in excess of $7,000 per worker are exempt wages and are not taxed for FUT A. If the total cumulative amount the employer owes the government is less than $100, the employer can pay the liability yearly (end of January in the following calendar year). If the tax is greater than $100, the employer must pay it within a month after the quarter ends. Companies involved in a state unemployment tax fund can usually take a 5.4% credit against their FUT A tax. In reality, then, companies are paying .8% (.008) to the federal unemployment program. In all our calculations, FUT A is .008.

RF/Corbis

EXAMPLE Assume a company had total wages of $19,000 in a calendar year . No employee earned more than $7,000 during the calendar year . The FUTA tax is .8% (6.2% minus the company’s 5.4% credit for state unemployment tax). How much does the company pay in FUT A tax? The company calculates its FUT A tax as follows:

6.2% FUTA tax 5.4% credit for SUTA tax .8% tax for FUTA .008 $19,000 $152 FUTA tax due to federal government State Unemployment Tax Act (SUTA) The current SUTA tax in many states is 5.4% on the first $7,000 the employer pays an employee. Some states of fer a merit rating system that results in a lower SUT A rate for companies with a stable employment period. The federal government still allows 5.4% credit on FUT A tax to companies entitled to the lower SUT A rate. Usually states also charge companies with a poor employment record a higher SUT A rate. However, these companies cannot take any more than the 5.4% credit against the 6.2% federal unemployment rate. Assume a company has total wages of $20,000 and $4,000 of the wages are exempt from SUT A. What are the company’ s SUT A and FUT A taxes if the company’ s SUTA rate is 5.8% due to a poor employment record? The exempt wages (over $7,000 earnings per worker) are not taxed for SUT A or FUTA. So the company owes the following SUT A and FUTA taxes:

EXAMPLE

$20,000 4,000 (exempt wages) $16,000 .058 $928 SUTA

Federal FUTA tax would then be: $16,000 .008 $128

You can check your progress with the following Practice Quiz.

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Learning Unit 9–2

LU 9–2

245

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1.

DVD

2.

✓

Calculate Social Security taxes, Medicare taxes, and FIT for Joy Royce. Joy’s company pays her a monthly salary of $9,500. She is single and claims 1 deduction. Before this payroll, Joy’s cumulative earnings were $94,000. (Social Security maximum is 6.2% on $97,500, and Medicare is 1.45%.) Calculate FIT by the percentage method. Jim Brewer, owner of Arrow Company, has three employees who earn $300, $700, and $900 a week. Assume a state SUTA rate of 5.1%. What will Jim pay for state and federal unemployment taxes for the first quarter?

Solutions

1.

Social Security $97,500 94,000 $ 3,500 .062 $217.00

Medicare $9,500 .0145 $137.75

FIT Percentage method: $9,500.00 283.33 (Table 9.1) $283.33 1 $9,216.67 $6,423 to $13,567 $9,216.67 6,423.00 $2,793.67 .28

$1,262.20 plus 28% of excess over $6,423 (Table 9.2)

$ 782.23* 1,262.20 $2,044.43

*Due to rounding.

2.

13 weeks $300 $ 3,900 13 weeks $700 9,100 ($9,100 $7,000) 13 weeks $900 11,700 ($11,700 $7,000) $24,700 $24,700 $6,800 $17,900 taxable wages SUTA .051 $17,900 $912.90 FUTA .008 $17,900 $143.20

LU 9–2a

$2,100 Exempt wages 4,700 ¶ (not taxed for FUTA or SUTA) $6,800

Note: FUTA remains at .008 whether SUTA rate is higher or lower than standard.

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 247)

1.

2.

Calculate Social Security taxes, Medicare taxes, and FIT for Joy Royce. Joy’s company pays her a monthly salary of $10,000. She is single and claims 1 deduction. Before this payroll, Joy’s cumulative earnings were $97,000. (Social Security maximum is 6.2% on $97,500, and Medicare is 1.45%.) Calculate FIT by the percentage method. Jim Brewer, owner of Arrow Company, has three employees who earn $200, $800, and $950 a week. Assume a state SUTA rate of 5.1%. What will Jim pay for state and federal unemployment taxes for the first quarter?

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Chapter 9 Payroll

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Gross pay, p. 237

Hours employee Rate per worked hour

$6.50 per hour at 36 hours

Overtime, p. 237

Gross Earnings at Regular earnings pay overtime rate (pay) (112)

$6 per hour; 42 hours Gross pay (40 $6) (2 $9) $240 $18 $258

Straight piece rate, p. 238

Gross Number of units Rate per pay produced unit

1,185 units; rate per unit, $.89 Gross pay 1,185 $.89 $1,054.65

Differential pay schedule, p. 238

Rate on each item is related to the number of items produced.

1–500 at $.84; 501–1,000 at $.96; 900 units produced. Gross (500 $.84) (400 $.96) pay $420 $384 $804

Straight commission, p. 239

Total sales Commission rate Any draw would be subtracted from earnings.

$155,000 sales; 6% commission

Variable commission scale, p. 239

Sales at different levels pay different rates of commission.

Up to $5,000, 5%; $5,001 to $10,000, 8%; over $10,000, 10% Sold: $6,500 Solution: ($5,000 .05) ($1,500 .08) $250 $120 $370

Salary plus commission, p. 239

Regular wages Commissions (fixed) earned

Base $400 per week 2% on sales over $14,000 Actual sales: $16,000 $400 (base) (.02 $2,000) $440

Payroll register, p. 241

Multicolumn form to record payroll. Married and paid weekly. (Table 9.2) Claims 1 allowance. FICA rates from chapter.

FICA, p. 241 Social Security Medicare

6.2% on $97,500 (S.S.) 1.45% (Med.)

Gross pay 36 $6.50 $234

$155,000 .06 $9,300

Earnings Gross 1,100

Deductions Net FICA pay S.S. Med. FIT 68.20 15.95 117.34 898.51

If John earns $99,000, what did he contribute for the year to Social Security and Medicare? S.S.: $97,500 .062 $6,045.00 Med.: $99,000 .0145 $1,435.50

FIT calculation (percentage method), p. 242

Facts: Al Doe: Married Claims: 2 Paid weekly: $1,600

$1,600.00 130.76 ($65.38 2) Table 9.1 $1,469.24 By Table 9.2 $1,469.24 1,360.00 $ 109.24 $166.15 .25($109.24) $166.15 $27.31 $193.46

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

247

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

State and federal unemployment, p. 244

Employer pays these taxes. Rates are 6.2% on $7,000 for federal and 5.4% for state on $7,000. 6.2% 5.4% .8% federal rate after credit. If state unemployment rate is higher than 5.4%, no additional credit is taken. If state unemployment rate is less than 5.4%, the full 5.4% credit can be taken for federal unemployment.

Cumulative pay before payroll, $6,400; this week’s pay, $800. What are state and federal unemployment taxes for employer, assuming a 5.2% state unemployment rate?

KEY TERMS

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

Biweekly, p. 236 Deductions, p. 237 Differential pay schedule, p. 238 Draw, p. 239 Employee’s Withholding Allowance Certificate (W-4), p. 241 Fair Labor Standards Act, p. 237 Federal income tax withholding (FIT), p. 242

.052 $600 $31.20

Federal .008 $600 $4.80 ($6,400 $600 $7,000 maximum)

Federal Insurance Contribution Act (FICA), p. 241 Federal Unemployment Tax Act (FUTA), p. 244 Gross pay, p. 237 Medicare, p. 241 Monthly, p. 236 Net pay, p. 242 Overrides, p. 239 Overtime, p. 237 Payroll register, p. 240

LU 9–1a (p. 240) 1. $732 2. $12,800 3. $4,070

State

Percentage method, p. 242 Semimonthly, p. 236 Social Security, p. 241 State income tax (SIT), p. 242 State Unemployment Tax Act (SUTA), p. 244 Straight commission, p. 239 Variable commission scale, p. 239 W-4, p. 241 Weekly, p. 236

LU 9–2a (p. 245) 1. $31; 145; $2,184.43 2. $846.60; $132.80

Critical Thinking Discussion Questions 1. Explain the dif ference between biweekly and semimonthly . Explain what problems may develop if a retail store hires someone on straight commission to sell cosmetics. 2. Explain what each column of a payroll register records (p. 241) and how each number is calculated. Social Security

tax is based on a specific rate and base; Medicare tax is based on a rate but no base. Do you think this is fair to all taxpayers? 3. What taxes are the responsibility of the employer? How can an employer benefit from a merit-rating system for state unemployment?

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Classroom Notes

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Complete the following table: Employee

M

T

W

Th

F

Hours

Rate per hour

9–1. Tom Bradey

11

7

8

7

6

$7.50

9–2. Kristina Shaw

5

9

10

8

8

$8.10

Gross pay

Complete the following table (assume the overtime for each employee is a time-and-a-half rate after 40 hours):

Employee M

T

W

Th

F

Sa

Total regular hours

Total overtime hours

Regular rate

9–3. Blue

12

9

9

9

9

3

$8.00

9–4. Tagney

14

8

9

9

5

1

$7.60

Overtime rate

Gross earnings

Calculate gross earnings: Number of units produced

Rate per unit

9–5. Lang

510

$2.10

9–6. Swan

846

$ .58

Worker

Gross earnings

Calculate the gross earnings for each apple picker based on the following dif ferential pay scale: 1–1,000: $.03 each Apple picker

1,001–1,600: $.05 each Number of apples picked

9–7. Ryan

1,600

9–8. Rice

1,925

Employee 9–9. Reese

Over 1,600: $.07 each

Gross earnings

Total sales

Commission rate

Draw

$300,000

7%

$8,000

End-of-month commission received

249

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Ron Company has the following commission schedule: Commission rate 2% 3.5% 4%

Sales Up to $80,000 Excess of $80,000 to $100,000 More than $100,000

Calculate the gross earnings of Ron Company’s two employees: Employee

Total sales

9–10. Bill Moore

$ 70,000

9–11. Ron Ear

$155,000

Gross earnings

Complete the following table, given that A Publishing Company pays its salespeople a weekly salary plus a 2% commission on all net sales over $5,000 (no commission on returned goods): Gross Employee sales

Net Return sales

Given quota

Commission Commission Total sales rates commission

Regular wage

9–12. Ring

$ 8,000

$ 25

$5,000

2%

$250

9–13. Porter

$12,000

$100

$5,000

2%

$250

Total wage

Calculate the Social Security and Medicare deductions for the following employees (assume a tax rate of 6.2% on $97,500 for Social Security and 1.45% for Medicare): Cumulative earnings before this pay period

Pay amount this period

9–14. Lee

$96,500

$2,000

9–15. Chin

$90,000

$8,000

9–16. Davis

$500,000

$4,000

Employee

Social Security

Medicare

Complete the following payroll register . Calculate FIT by the percentage method for this weekly period; Social Security and Medicare are the same rates as in the previous problems. No one will reach the maximum for FICA. Marital status

Allowances claimed

Gross pay

9–17. Jim Day

M

2

$1,400

9–18. Ursula Lang

M

4

$1,900

Employee

250

FICA FIT

S.S.

Med.

Net pay

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9–19. Given the following, calculate the state (assume 5.3%) and federal unemployment taxes that the employer must pay for each of the first two quarters. The federal unemployment tax is .8% on the first $7,000. PAYROLL SUMMARY Quarter 1

Quarter 2

$4,000

$ 8,000

Rich Haines

8,000

14,000

Alice Smooth

3,200

3,800

Bill Adams

†

WORD PROBLEMS 9–20. On February 7, 2007 the San Jose Mercury News reported on Bay Area workers average pay. Bay Area workers pocketed 23 percent more pay last year than workers in Los Angeles with $26.10 compared to $21.21 an hour . Jim Moody, a Bay Area worker, worked 1014, 812, 934, 834 and 914 hours last week. Jim is paid an overtime pay of 1.5 times his regular pay . What is Jim’s total gross pay for the week? Round to the nearest cent.

9–21. The Telegraph (Nashau, NH) on December 6, 2006, described the living wage needed in New Hampshire. Jessica Bullard is a single parent with one child and claims 2. She needs to make $17.71 an hour to get by in Hillsborough County. However Jessica Bullard only earns $13.00 an hour . Jessica works 40 hours a week. Social Security tax is 6.2 percent and Medicare is 1.45 percent (a) What is her gross pay per week? (b) How much is deducted for Social Security Tax? (c) How much is deducted for Medicare? (d) How much is withheld for FIT , assuming she claims 2? (e) What is her net pay? Round to the nearest cent.

251

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9–22. The Social Security Administration increased the taxable wage base from $94,200 to $97,500. The 6.2% tax rate is unchanged. Joe Burns earned over $100,000 each of the past two years. (a) What is the percent increase in the base? Round to the nearest hundredth percent. (b) What is Joe’s increase in Social Security tax for the new year?

9–23. Dennis Toby is a salesclerk at Northwest Department Store. Dennis receives $8 per hour plus a commission of 3% on all sales. Assume Dennis works 30 hours and has sales of $1,900. What is his gross pay? 9–24. Owing to a bill signed by Governor Arnold Schwarzenegger that increased the minimum wage from $6.75 to $8.00 an hour, by 2008, The Business Press (San Bernardino, CA) on September 18, 2006 reports firms are weighing leaving the state. Donna Carter, single, works 37 hours per week with one withholding exemption. Using the same percentage withholding for wages paid in 2007, (a) what is the amount of FIT withheld for 2007? (b) What would be the amount of FIT withheld for 2008? Round to the nearest cent.

9–25. Robin Hartman earns $600 per week plus 3% of sales over $6,500. Robin’ s sales are $14,000. How much does Robin earn? 9–26. Pat Maninen earns a gross salary of $2,100 each week. What are Pat’s first week’s deductions for Social Security and Medicare? Will any of Pat’s wages be exempt from Social Security and Medicare for the calendar year? Assume a rate of 6.2% on $97,500 for Social Security and 1.45% for Medicare.

9–27. Richard Gaziano is a manager for Health Care, Inc. Health Care deducts Social Security , Medicare, and FIT (by percentage method) from his earnings. Assume the same Social Security and Medicare rates as in Problem 9–26. Before this payroll, Richard is $1,000 below the maximum level for Social Security earnings. Richard is married, is paid weekly , and claims 2 exemptions. What is Richard’s net pay for the week if he earns $1,300?

9–28. Len Mast earned $2,200 for the last two weeks. He is married, is paid biweekly , and claims 3 exemptions. What is Len’s income tax? Use the percentage method.

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9–29. Westway Company pays Suzie Chan $2,200 per week. By the end of week 50, how much did Westway deduct for Suzie’s Social Security and Medicare for the year? Assume Social Security is 6.2% on $97,500 and 1.45% for Medicare. What state and federal unemployment taxes does Westway pay on Suzie’s yearly salary? The state unemployment rate is 5.1%. FUTA is .8%.

9–30. Morris Leste, owner of Carlson Company, has three employees who earn $400, $500, and $700 per week. What are the total state and federal unemployment taxes that Morris owes for the first 1 1 weeks of the year and for week 30? Assume a state rate of 5.6% and a federal rate of .8%.

CHALLENGE PROBLEMS 9–31. The Victorville, California, Daily Press stated that the San Bernardino County Fair hires about 150 people during fair time. Their wages range from $6.75 to $8.00. California has a state income tax of 9%. Sandy Denny earns $8.00 per hour; George Barney earns $6.75 per hour. They both worked 35 hours this week. Both are married; however , Sandy claims 2 exemptions and George claims 1 exemption. Assume a rate of 6.2% on $97,500 for Social Security and 1.45% for Medicare. (a) What is Sandy’s net pay after FIT, Social Security tax, state income tax, and Medicare have been taken out? (b) What is George’s net pay after the same deductions? (c) How much more is Sandy’s net pay versus George’s net pay? Round to the nearest cent.

9–32. Bill Rose is a salesperson for Boxes, Inc. He believes his $1,460.47 monthly paycheck is in error . Bill earns a $1,400 salary per month plus a 9.5% commission on sales over $1,500. Last month, Bill had $8,250 in sales. Bill believes his traveling expenses are 16% of his weekly gross earnings before commissions. Monthly deductions include Social Security, $126.56; Medicare, $29.60; FIT, $239.29; union dues, $25.00; and health insurance, $16.99. Calculate the following: (a) Bill’s monthly take-home pay, and indicate the amount his check was under - or overstated, and (b) Bill’s weekly traveling expenses. Round your final answer to the nearest dollar .

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DVD SUMMARY PRACTICE TEST 1. Calculate Sam’s gross pay (he is entitled to time-and-a-half). (p. 237) M

T

W

Th

F

914

941

1021

821

1112

Total hours

Rate per hour

Gross pay

$8.00

2. Mia Kaminsky sells shoes for Macy’s. Macy’s pays Mia $12 per hour plus a 5% commission on all sales. Assume Mia works 37 hours for the week and has $7,000 in sales. What is Mia’s gross pay? (p. 237)

3. Lee Company pays its employees on a graduated commission scale: 6% on the first $40,000 sales, 7% on sales from $40,001 to $80,000, and 13% on sales of more than $80,000. May West, an employee of Lee, has $230,000 in sales. What commission did May earn? (p. 239)

4. Matty Kim, an accountant for Vernitron, earned $90,000 from January to June. In July , Matty earned $20,000. Assume a tax rate of 6.2% for Social Security on $97,500 and 1.45% on Medicare. Ho w much are the July tax es for Social Security and Medicare? (p. 241)

5. Grace Kelley earns $2,000 per week. She is married and claims 2 e xemptions. What is Grace’s income tax? Use the percentage method. (p. 242)

6. Jean Michaud pays his two employees $900 and $1,200 per week. Assume a state unemployment tax rate of 5.7% and a federal unemployment tax rate of .8%. What state and federal unemployment taxes will Jean pay at the end of quarter 1 and quarter 2? (p. 244)

254

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Personal Finance A KIPLINGER APPROACH G Short-term policies

CORBIS

can pay the bills if a young adult needs medical care.

| Earning a diploma often means losing medical coverage.

INSURANCE

HEALTHY choices

A

lthough finding a job may be the top priority for most new college graduates, parents are often more concerned about continuing their children’s health coverage. Insurers typically drop kids from their parents’ health plan once they grab that diploma (or by the time they turn 25). Sandy D’Annunzio, a nurse in Sterling Heights, Mich., bought short-term health-insurance policies from Golden Rule (www .goldenrule.com) for her two daughters, Jennifer and Kelly. D’Annunzio pays about $57 a month for each policy, both of which have a $1,000 deductible

and 20% co-insurance (meaning the insurer picks up 80% of a claim after the deductible is met). “If one of them broke a leg, it could cost 70 times as much,” says D’Annunzio. Most short-term policies last six months to a year, after which you may reapply, as long as you remain healthy. But they don’t typically cover preventive care or preexisting conditions, so they’re really just a temporary fix. For longer coverage, consider an individual policy with a high deductible. For a policy with a $1,500 deductible and 20% coinsurance, a young female nonsmoker would pay $124

per month in Chicago. That’s more expensive than short-term insurance, but it covers many of the medical expenses that short-term policies exclude. If graduation is still a few months away, buying student health insurance may be a cheaper way to go. But don’t delay. Assurant Health, a major provider of such plans, requires that coverage begin at least 31 days before a student graduates. Like short-term health insurance, student health coverage has a long list of exclusions. But in most cases, it is less expensive than a short-term policy and is renewable. For example, a 22-year-old female nonsmoker in Chicago would pay $66 a month for a student health policy with a $1,000 deductible and 20% co-insurance through eHealthInsurance.com. A similar short-term policy would cost $104 a month. If your child has a medical condition, such as asthma or depression, buying individual health insurance can be tough. In that case, take advantage of COBRA; the law allows your adult child to remain on your policy for up to 36 months. COBRA coverage isn’t cheap because you have to pay both the employer share and the employee share of your group premium, but it can serve as a safety net while you look into other options. A number of states are taking steps to extend coverage for young adults.

— THOMAS M. ANDERSON

BUSINESS MATH ISSUE Kiplingers © 2006

If you’re young you really don’t need health insurance. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

255

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A Do you think FedEx employees should be independent contractors?

2006 Wall Street Journal ©

PROJECT B Why do you think Indian law forbids overtime?

b site text We he e e S : s t T t Projec /slater9e) and e. Interne ce Guid m r o u .c o e s h e h R t .m e (www Intern ss Math Busine

2006 Wall Street Journal ©

256

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Video Case WASHBURN GUITARS

Washburn International, founded in 1883, makes 80 models of instruments, both custom and for the mass market. Washburn is a privately held company with over 100 employees and annual sales of $48 million. This compares to its annual sales of $300,000 when Rudy Schlacher took over in 1976. When he acquired the company, about 250 guitars were produced per month; now 15,000 are produced each month. The Washburn tradition of craftsmanship and innovation has withstood the tests of economics, brand competition, and fashion. Since its birth in Chicago, the name Washburn has been branded into the world’s finest stringed instruments. To maintain quality, Washburn must have an excellent pool of qualified employees who are passionate about craftsmanship.

PROBLEM

1

$120,000 was paid to 16 of Washburn’s salespeople in override commissions. (a) What was the average amount paid to each salesperson? (b) What amount of the average sales commission will go toward the salesperson’s Social Security tax? (c) What amount will go toward Medicare? PROBLEM

2

Washburn is seeking a Sales and Marketing Coordinator with a bachelor’s degree or equivalent experience, knowledgeable in Microsoft Office. This position pays $25,000 to $35,000, depending on experience. Assume a person is paid weekly and earns $32,500. Using the percentage method, what would be the taxes withheld for a married person who claims 3 exemptions? PROBLEM

3

Guitarists hoping for a little country music magic in their playing can now buy an instrument carved out of oak pews from the former home of the Grand Ole Opry. Only 243 of the Ryman Limited Edition Acoustic Guitars are being made, each costing $6,250. Among the first customers were singers Vince Gill, Amy Grant, and Loretta Lynn, Ms. Lynn purchased two guitars. What would be the total revenue received by Washburn if all the guitars are sold? PROBLEM

4

Under Washburn’s old pay system, phone reps received a commission of 1.5% only on instruments they sold. Now the phone reps are paid an extra .75% commission on field sales made in their territory; the outside salespeople still get a commission up to 8%, freeing them to focus on introducing new products and holding in-store clinics. Assume sales were $65,500: (a) How much would phone reps receive? (b) How much would the outside salespeople receive?

Washburn consolidated its four divisions in an expansive new 130,000 square foot plant in Mundelein, Illinois. The catalyst for consolidating operations in Mundelein was a chronic labor shortage in Elkhart and Chicago. The Mundelein plant was the ideal home for all Washburn operations because it had the necessary space, was cost effective, and gave Washburn access to a labor pool. To grow profitably, Washburn must also sell its other products. To keep Washburn’s 16 domestic salespeople tuned in to the full line, the company offers an override incentive. It is essential that to produce quality guitars, Washburn must keep recruiting dedicated, well-qualified, and team-oriented employees and provide them with profitable incentives.

PROBLEM

5

Washburn introduced the Limited Edition EA27 Gregg Allman Signature Series Festival guitar—only 500 guitars were produced with a selling price of $1,449.90. If Washburn’s markup is 35% on selling price, what was Washburn’s total cost for the 500 guitars? PROBLEM

6

Retailers purchased $511 million worth of guitars from manufacturers—some 861,300 guitars—according to a study done by the National Association of Music Merchants. (a) What would be the average selling price of a guitar? (b) Based on the average selling price, if manufacturer’s markup on cost is 40%, what would be the average cost? PROBLEM

7

A Model NV 300 acoustic-electric guitar is being sold for a list price of $1,899.90, with a cash discount of 3/10, n/30. Sales tax is 7% and shipping is $30.40. How much is the final price if the cash discount period was met? PROBLEM

8

A Model M3SWE mandolin has a list price of $1,299.90, with a chain discount of 5/3/2. (a) What would be the trade discount amount? (b) What would be the net price? PROBLEM

9

A purchase was made of 2 Model J282DL six-string acoustic guitars at $799.90 each, with cases priced at $159.90, and 3 Model EA10 festival series acoustic-electric guitars at $729.90, with cases listed at $149.90. If sales tax is 6%, what is the total cost? PROBLEM

10

Production of guitars has increased by what percent since Rudy Schlacher took over Washburn?

257

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CHAPTER

10

Simple Interest

LEARNING UNIT OBJECTIVES LU 10–1: Calculation of Simple Interest and Maturity Value • Calculate simple interest and maturity value for months and years (p. 259). • Calculate simple interest and maturity value by (a) exact interest and (b) ordinary interest (pp. 260–261).

LU 10–2: Finding Unknown in Simple Interest Formula • Using the interest formula, calculate the unknown when the other two (principal, rate, or time) are given (pp. 262–263).

LU 10–3: U.S. Rule—Making Partial Note Payments before Due Date • List the steps to complete the U.S. Rule (pp. 264–265). • Complete the proper interest credits under the U.S. Rule (pp. 264–265).

Wall Street Jo urnal © 2005

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Learning Unit 10–1

259

Digital Vision/Getty Images Wall Street Journal © 2005

Are you careless about making your credit payments on time? Do you realize that some penalty rates can increase when the Federal Reserve increases its short-term interest rate? The Wall Street Journal clipping “Major Issuers Boost Costs for Late Payment Past 30% Amid Rising Interest Rates” shows how expensive it can be if you do not pay your credit bills on time, if you bounce checks, or if you exceed your credit limit. In this chapter , you will study simple interest. The principles discussed apply whether you are paying interest or receiving interest. Let’ s begin by learning how to calculate simple interest.

Learning Unit 10–1: Calculation of Simple Interest and Maturity Value Jan Carley, a young attorney, rented an office in a professional building. Since Jan recently graduated from law school, she was short of cash. To purchase of fice furniture for her new of fice, Jan went to her bank and borrowed $30,000 for 6 months at an 8% annual interest rate. The original amount Jan borrowed ($30,000) is the principal (face value) of the loan. Jan’s price for using the $30,000 is the interest rate (8%) the bank char ges on a yearly basis. Since Jan is borrowing the $30,000 for 6 months, Jan’ s loan will have a maturity value of $31,200—the principal plus the interest on the loan. Thus, Jan’s price for using the furniture before she can pay for it is $1,200 interest, which is a percent of the principal for a specific time period. To make this calculation, we use the following formula: Maturity value (MV ) Principal (P ) Interest (I ) $31,200

$30,000

$1,200

Jan’s furniture purchase introduces simple interest—the cost of a loan, usually for 1 year or less. Simple interest is only on the original principal or amount borrowed. Let’ s examine how the bank calculated Jan’ s $1,200 interest.

Simple Interest Formula To calculate simple interest, we use the following

simple interest formula:

Simple interest (I ) Principal (P ) Rate (R ) Time (T )

In this formula, rate is expressed as a decimal, fraction, or percent; and time is expressed in years or a fraction of a year .

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EXAMPLE Jan Carley borrowed $30,000 for office furniture. The loan was for 6 months at an annual interest rate of 8%. What are Jan’s interest and maturity value? Using the simple interest formula, the bank determined Jan’ s interest as follows: In your calculator, multiply $30,000 times .08 times 6. Divide your answer by 12. You could also use the % key—multiply $30,000 times 8% times 6 and then divide your answer by 12.

Step 1.

Step 2.

Calculate the interest.

Calculate the maturity value.

6 12 (R) ( T )

I $30,000 .08

(P) $1,200 MV $30,000 $1,200 (P) (I )

$31,200 Now let’ s use the same example and assume Jan borrowed $30,000 for 1 year bank would calculate Jan’ s interest and maturity value as follows: Step 1.

Calculate the interest.

Step 2.

Calculate the maturity value.

. The

I $30,000 .08 1 year (P) (R) (T ) $2,400 MV $30,000 $2,400 (P) (I) $32,400

Let’s use the same example again and assume Jan borrowed $30,000 for 18 months. Then Jan’s interest and maturity value would be calculated as follows: Step 1.

Step 2.

Calculate the interest.

Calculate the maturity value.

I $30,000 .08

181 12 (T)

(P) (R) $3,600 MV $30,000 $3,600 (P) (I) $33,600

Next we’ll turn our attention to two common methods we can use to calculate simple interest when a loan specifies its beginning and ending dates.

Two Methods for Calculating Simple Interest and Maturity Value Method 1: Exact Interest (365 Days) The Federal Reserve banks and the federal gov-

ernment use the exact interest method. The exact interest is calculated by using a 365-day year. For time, we count the exact number of days in the month that the borrower has the loan. The day the loan is made is not counted, but the day the money is returned is counted as a full day . This method calculates interest by using the following fraction to represent time in the formula:

From the Business Math Handbook July 6 187th day March 4 63rd day

March

April May June July

124 days (exact time of loan) 31 4 27 30 31 30 6

Time

Exact number of days 365

Exact interest

For this calculation, we use the exact days-in-a-year calendar from the Handbook. You learned how to use this calendar in Chapter 7, p. 181.

Business Math

On March 4, Peg Carry borrowed $40,000 at 8% interest. Interest and principal are due on July 6. What is the interest cost and the maturity value?

EXAMPLE

Step 1.

Calculate the interest.

IPRT $40,000 .08

124 365

$1,087.12 (rounded to nearest cent)

124 days 1

This is the same as 1.5 years.

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Learning Unit 10–1

Step 2.

Calculate the maturity value.

261

MV P I $40,000 $1,087.12 $41,087.12

Method 2: Ordinary Interest (360 Days) In the ordinary interest method, time in the formula I P R T is equal to the following: Time

Exact number of days 360

Ordinary interest

Since banks commonly use the ordinary interest method, it is known as the Banker’s Rule. Banks charge a slightly higher rate of interest because they use 360 days instead of 365 in the denominator . By using 360 instead of 365, the calculation is supposedly simplified. Consumer groups, however , are questioning why banks can use 360 days, since this benefits the bank and not the customer . The use of computers and calculators no longer makes the simplified calculation necessary. For example, after a court case in Oregon, banks began calculating interest on 365 days except in mortgages. Now let’s replay the Peg Carry example we used to illustrate Method 1 to see the difference in bank interest when we use Method 2. On March 4, Peg Carry borrowed $40,000 at 8% interest. Interest and principal are due on July 6. What are the interest cost and the maturity value?

EXAMPLE

Step 1.

Step 2.

Calculate the interest. Calculate the maturity value.

I $40,000 .08

124 360

$1,102.22 MV P I $40,000 $1,102.22 $41,102.22

Note: By using Method 2, the bank increases its interest by $15.10. $1,102.22 1,087.12 $ 15.10

Method 2 Method 1

Now you should be ready for your first Practice Quiz in this chapter

LU 10–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

.

Calculate simple interest (round to the nearest cent): 1. $14,000 at 4% for 9 months 2. $25,000 at 7% for 5 years 1 3. $40,000 at 10 2% for 19 months 4. On May 4, Dawn Kristal borrowed $15,000 at 8%. Dawn must pay the principal and interest on August 10. What are Dawn’ s simple interest and maturity value if you use the exact interest method? 5. What are Dawn Kristal’ s (Problem 4) simple interest and maturity value if you use the ordinary interest method?

✓ 1. 2. 3.

Solutions 9 $420 12 $25,000 .07 5 $8,750 19 $40,000 .105 $6,650 12 $14,000 .04

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Chapter 10 Simple Interest

4.

August 10 May 4

5.

LU 10–1a

222

$15,000 .08

98 $322.19 365

124 MV $15,000 $322.19 $15,322.19

98 98 $15,000 .08 $326.67 360

MV $15,000 $326.67 $15,326.67

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 267)

Calculate simple interest (round to the nearest cent):

$16,000 at 3% for 8 months $15,000 at 6% for 6 years $50,000 at 7% for 18 months On May 6, Dawn Kristal borrowed $20,000 at 7%. Dawn must pay the principal and interest on August 14. What are Dawn’s simple interest and maturity value if you use the exact interest method? 5. What are Dawn Kristal’ s (Problem 4) simple interest and maturity value if you use the ordinary interest method? 1. 2. 3. 4.

Learning Unit 10–2: Finding Unknown in Simple Interest Formula This unit begins with the formula used to calculate the principal of a loan. Then it explains how to find the principal, rate, and time of a simple interest loan. In all the calculations, we use 360 days and round only final answers.

Finding the Principal Tim Jarvis paid the bank $19.48 interest at 9.5% for 90 days. How much didTim borrow using ordinary interest method? The following formula is used to calculate the principal of a loan:

EXAMPLE

Interest ($19.48)

Principal

Principal Rate Time ? (.095) 90 360

( (

Note how we illustrated this in the mar gin. The shaded area is what we are solving for When solving for principal, rate, or time, you are dividing. Interest will be in the numerator, and the denominator will be the other two elements multiplied by each other .

M .

Step 3. When using a calculator, press

.

$19.48 90 .095 360

Step 1.

Set up the formula.

P

Step 2.

Multiply the denominator.

.095 times 90 divided by 360 (do not round)

Step 2. When using a calculator, press .095 90 360

Interest Rate Time

$19.48 .02375 P $820.21 P

Step 3.

Divide the numerator by the result of Step 2.

Step 4.

Check your answer.

19.48 MR .

$19.48 $820.21 .095 (I )

(P)

(R)

90 360 (T )

Finding the Rate Tim Jarvis borrowed $820.21 from a bank. Tim’s interest is $19.48 for 90 days. What rate of interest did Tim pay using ordinary interest method? The following formula is used to calculate the rate of interest:

EXAMPLE Interest ($19.48) Principal Rate Time 90 ($820.21) ? 360

( (

Rate

Interest Principal Time

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Learning Unit 10–2

Step 1.

Set up the formula.

Multiply the denominator. Do not round the answer. Step 3. Divide the numerator by the result of Step 2. Check your answer.

$820.21

90 360

$19.48 $205.0525

R

Step 2.

Step 4.

$19.48

R

263

R 9.5%

$19.48 $820.21 .095 (I)

(P)

(R)

90 360 (T )

Finding the Time Tim Jarvis borrowed $820.21 from a bank. Tim’s interest is $19.48 at 9.5%. How much time does Tim have to repay the loan using ordinary interest method? The following formula is used to calculate time:

EXAMPLE

Interest ($19.48) Principal Rate Time ($820.21) (.095) ?

Time (in years)

Step 2. When using a calculator, press 820.21 .095 M . Step 3. When using a calculator, press 19.48 MR .

Step 1.

Interest Principal Rate

$19.48 $820.21 .095 $19.48 T $77.91995

Set up the formula.

T

Multiply the denominator. Do not round the answer. Step 3. Divide the numerator by the result of Step 2. Step 2.

Step 4.

Convert years to days (assume 360 days).

Step 5.

Check your answer.

T .25 years

.25 360 90 days $19.48 $820.21 .095 (I)

(P)

(R)

90 360 (T)

Before we go on to Learning Unit 10 –3, let’s check your understanding of this unit.

LU 10–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Complete the following (assume 360 days): Interest Time Principal rate (days) 1. ? 5% 90 days 2. $7,000 ? 220 days 3. $1,000 8% ?

✓ 1.

2.

Simple interest $8,000 350 300

Solutions $8,000 $8,000 I $640,000 P 90 .0125 RT .05 360 $350 $350 I R 8.18% 220 $4,277.7777 PT $7,000 360

(do not round) $300 $300 3.75 360 1,350 days 3. $1,000 .08 $80

T

I PR

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Chapter 10 Simple Interest

LU 10–2a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 267)

Complete the following (assume 360 days): Interest Time Principal rate (days) 1. ? 4% 90 days 2. $6,000 ? 180 days 3. $900 6% ?

Simple interest $9,000 280 190

Learning Unit 10–3: U.S. Rule—Making Partial Note Payments before Due Date Often a person may want to pay of f a debt in more than one payment before the maturity date. The U.S. Rule allows the borrower to receive proper interest credits. This rule states that any partial loan payment first covers any interest that has built up. The remainder of the partial payment reduces the loan principal. Courts or legal proceedings generally use the U.S. Rule. The Supreme Court originated the U.S. Rule in the case of Story v. Livingston. Joe Mill owes $5,000 on an 11%, 90-day note. On day 50, Joe pays $600 on the note. On day 80, Joe makes an $800 additional payment. Assume a 360-day year . What is Joe’s adjusted balance after day 50 and after day 80? What is the ending balance due? To calculate $600 payment on day 50:

EXAMPLE

Calculate interest on principal from date of loan to date of first principal payment. Round to nearest cent. Step 2. Apply partial payment to interest due. Subtract remainder of payment from principal. This is the adjusted balance (principal).

IPRT

Step 1.

To calculate $800 payment on day 80: Step 3. Calculate interest on adjusted balance that starts from previous payment date and goes to new payment date. Then apply Step 2.

I $5,000 .11 I $76.39

50 360

$600.00 payment 76.39 interest $523.61

$5,000.00 principal 523.61 $4,476.39 adjusted balance— principal

Compute interest on $4,476.39 for 30 days (80 50) 30 I $4,476.39 .11 360 I $41.03 $800.00 payment 41.03 interest $758.97

$4,476.39 758.97 $3,717.42 adjusted balance Step 4.

At maturity, calculate interest from last partial payment. Add this interest to adjusted balance.

Ten days are left on note since last payment. I $3,717.42 .11 I = $11.36

10 360

Balance owed $3,728.78

a

$3,717.42 b 11.36

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265

Note that when Joe makes two partial payments, Joe’ s total interest is $128.78 ($76.39 $41.03 $11.36). If Joe had repaid the entire loan after 90 days, his interest payment would have been $137.50—a total savings of $8.72. Let’s check your understanding of the last unit in this chapter .

LU 10–3

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

Polly Flin borrowed $5,000 for 60 days at 8%. On day 10, Polly made a $600 partial payment. On day 40, Polly made a $1,900 partial payment. What is Polly’s ending balance due under the U.S. Rule (assume a 360-day year)?

✓

DVD

Solutions $5,000 .08

$600.00 11.11 $588.89

10 $11.11 360 $5,000.00 588.89 $4,411.11

$4,411.11 .08

LU 10–3a

30 $29.41 360

$1,900.00 29.41 $1,870.59

$4,411.11 1,870.59 $2,540.52

$2,540.52 .08

20 $11.29 360

$ 11.29 2,540.52 $2,551.81

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 267)

Polly Flin borrowed $4,000 for 60 days at 4%. On day 15, Polly made a $700 partial payment. On day 40, Polly made a $2,000 partial payment. What is Polly’s ending balance due under the U.S. Rule (assume a 360-day year)?

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Simple interest for months, p. 259

Interest Principal Rate Time (I ) (P ) (R ) (T )

$2,000 at 9% for 17 months 17 I $2,000 .09 12 I $255

Exact interest, p. 260

T

Exact number of days 365

IPRT

Ordinary interest (Bankers Rule), p. 261

T

Exact number of days 360

IPRT Finding unknown in simple interest formula (use 360 days), p. 262

IPRT

$1,000 at 10% from January 5 to February 20 46 I $1,000 .10 365 Feb. 20: 51 days Jan. 5: 5 46 days I $12.60

I $1,000 .10

46 360

(51 5)

I $12.78 Higher interest costs Use this example for illustrations of simple interest formula parts: $1,000 loan at 9%, 60 days 60 I $1,000 .09 $15 360

(continues)

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CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (continued) Topic

Key point, procedure, formula

Finding the principal, p. 262

P

I RT

Example(s) to illustrate situation I

P

$15 $15 $1,000 60 .015 .09 360

P R T

Finding the rate, p. 262

R

I PT

I

R

$15 60 $1,000 360

$15 .09 166.66666 9%

P R T

Note: We did not round the denominator. Finding the time, p. 263

T

I PR

(in years)

T I

$15 $15 .1666666 $1,000 .09 $90

.1666666 360 59.99 60 days

P R T

Multiply answer by 360 days to convert answer to days for ordinary interest. U.S. Rule (use 360 days), p. 264

Calculate interest on principal from date of loan to date of first partial payment.

Calculate adjusted balance by subtracting from principal the partial payment less interest cost. The process continues for future partial payments with the adjusted balance used to calculate cost of interest from last payment to present payment.

12%, 120 days, $2,000 Partial payments: On day 40; $250 On day 60; $200 First payment: 40 I $2,000 .12 360 I $26.67 $250.00 payment 26.67 interest $223.33 $2,000.00 principal 223.33 $1,776.67 adjusted balance Second payment: 20 I $1,776.67 .12 360 I $11.84 $200.00 payment 11.84 interest $188.16 $1,776.67 188.16 $1,588.51 adjusted balance

Balance owed equals last adjusted balance plus interest cost from last partial payment to final due date.

60 days left: 60 $31.77 360 $1,588.51 $31.77 $1,620.28 balance due $1,588.51 .12

Total interest

$26.67 11.84 31.77 $70.28

(continues)

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CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

KEY TERMS

Adjusted balance, p. 264 Banker’s Rule, p. 261 Exact interest, p. 260 Interest, p. 259

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 10–1a (p. 262) $320 $5,400 $5,250 $20,383.56; Interest $383.56 5. $20,388.89; Interest $388.89 1. 2. 3. 4.

Example(s) to illustrate situation

Maturity value, p. 259 Ordinary interest, p. 261 Principal, p. 259 Simple interest, p. 259 LU 10–2a (p. 264) 1. $900,000 2. 9.33% 3. 1,267 days

Simple interest formula, p. 259 Time, p. 263 U.S. Rule, p. 267 LU 10–3a (p. 265) $1,318.78

Critical Thinking Discussion Questions 1. What is the dif ference between exact interest and ordinary interest? With the increase of computers in banking, do you think that the ordinary interest method is a dinosaur in business today? 2. Explain how to use the portion formula to solve the unknowns in the simple interest formula. Why would round-

ing the answer of the denominator result in an inaccurate final answer? 3. Explain the U.S. Rule. Why in the last step of the U.S. Rule is the interest added, not subtracted?

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Classroom Notes

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Calculate the simple interest and maturity value for the following problems. Round to the nearest cent as needed. Principal

Interest rate

Time

10–1. $16,000

4%

18 mo.

10–2. $19,000

6%

134 yr.

10–3. $18,000

714%

9 mo.

Simple interest

Maturity value

Complete the following, using ordinary interest: Interest rate

Date borrowed

Date repaid

10–4. $1,000

8%

Mar. 8

June 9

10–5. $585

9%

June 5

Dec. 15

10–6. $1,200

12%

July 7

Jan. 10

Principal

Exact time

Interest

Maturity value

Exact time

Interest

Maturity value

Complete the following, using exact interest: Interest rate

Date borrowed

Date repaid

10–7. $1,000

8%

Mar. 8

June 9

10–8. $585

9%

June 5

Dec. 15

10–9. $1,200

12%

July 7

Jan. 10

Principal

269

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Solve for the missing item in the following (round to the nearest hundredth as needed): Interest rate

Time (months or years)

Simple interest

10–10. $400

5%

?

$100

10–11. ?

7%

112 years

$200

10–12. $5,000

?

6 months

$300

Principal

10–13. Use the U.S. Rule to solve for total interest costs, balances, and final payments (use ordinary interest). Given Principal: $10,000, 8%, 240 days Partial payments: On 100th day, $4,000 On 180th day, $2,000

WORD PROBLEMS 10–14. The Kansas City Star on March 11, 2007 featured a story on emer gency savings in the U.S. Money in a checking account will not generate much interest. So Peggy Cooper decides to place her $1,300 in a savings account with a 5 18 percent return. After 7 months, Peggy needs to withdraw her savings. (a) What is the amount of interest she earned? (b) How much will Peggy receive from the bank? Round to the nearest cent.

10–15. Kim Lee borrowed $10,000 to pay for her child’ s education at River Community College. Kim must repay the loan at the end of 11 months in one payment with 6 12% interest. How much interest must Kim pay? What is the maturity value?

270

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10–16. On September 12, Jody Jansen went to Sunshine Bank to borrow $2,300 at 9% interest. Jody plans to repay the loan on January 27. Assume the loan is on ordinary interest. What interest will Jody owe on January 27? What is the total amount Jody must repay at maturity?

10–17. Kelly O’Brien met Jody Jansen (Problem 10–16) at Sunshine Bank and suggested she consider the loan on exact interest. Recalculate the loan for Jody under this assumption.

10–18. May 3, 2007, Leven Corp. negotiated a short-term loan of $685,000. The loan is due October 1, 2007, and carries a 6.86% interest rate. Use ordinary interest to calculate the interest. What is the total amount Leven would pay on the maturity date?

10–19. Gordon Rosel went to his bank to find out how long it will take for $1,200 to amount to $1,650 at 8% simple interest. Please solve Gordon’s problem. Round time in years to the nearest tenth.

10–20. Bill Moore is buying a van. His April monthly interest at 12% was $125. What was Bill’s principal balance at the beginning of April? Use 360 days.

10–21. On April 5, 2008, Janeen Camoct took out an 8 12% loan for $20,000. The loan is due March 9, 2009. Use ordinary interest to calculate the interest. What total amount will Janeen pay on March 9, 2009?

10–22. Sabrina Bowers took out the same loan as Janeen (Problem 10–21). Sabrina’ s terms, however, are exact interest. What is Sabrina’s difference in interest? What will she pay on March 9, 2009?

271

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10–23. Max Wholesaler borrowed $2,000 on a 10%, 120-day note. After 45 days, Max paid $700 on the note. Thirty days later, Max paid an additional $630. What is the final balance due? Use the U.S. Rule to determine the total interest and ending balance due. Use ordinary interest.

ADDITIONAL SET OF WORD PROBLEMS 10–24. Limits are needed on payday-lending businesses, according to an article in the February 14, 2007 issue of The Columbian (Vancouver, WA). Interest rates on payday loans are so outrageous that the payday-lending industry only has itself to blame for states moving to rein them in. A typical $100 loan is payable in two weeks at $115. What is the percent of interest paid on this loan? Do not round denominator before dividing.

10–25. Availability of state and federal disaster loans was the featured article in The Enterprise Ledger (AL) on March 14, 2007. Alabama Deputy Treasurer Anthony Leigh said the state program allows the state treasurer to place state funds in Alabama banks at 2 percent below the market interest rate. The bank then agrees to lend the funds to individuals or businesses for 2 percent below the normal char ge, to help Alabama victims of disaster to secure emer gency short term loans. Laura Harden qualifies for an emer gency loan. She will need $3,500 for 5 months and the local bank has an interest rate of 4 34 percent. (a) What would have been the maturity value of a non-emer gency loan? (b) What will be the maturity value of the emergency loan? Round to the nearest cent.

272

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10–26. On September 14, Jennifer Rick went to Park Bank to borrow $2,500 at 1 134% interest. Jennifer plans to repay the loan on January 27. Assume the loan is on ordinary interest. What interest will Jennifer owe on January 27? What is the total amount Jennifer must repay at maturity?

10–27. Steven Linden met Jennifer Rick (Problem 10–26) at Park Bank and suggested she consider the loan on exact interest. Recalculate the loan for Jennifer under this assumption.

10–28. Lance Lopes went to his bank to find out how long it will take for $1,000 to amount to $1,700 at 12% simple interest. Can you solve Lance’s problem? Round time in years to the nearest tenth.

10–29. Margie Pagano is buying a car. Her June monthly interest at 12 12% was $195. What was Margie’s principal balance at the beginning of June? Use 360 days. Do not round the denominator before dividing.

10–30. Shawn Bixby borrowed $17,000 on a 120-day, 12% note. After 65 days, Shawn paid $2,000 on the note. On day 89, Shawn paid an additional $4,000. What is the final balance due? Determine total interest and ending balance due by the U.S. Rule. Use ordinary interest.

10–31. Carol Miller went to Europe and for got to pay her $740 mortgage payment on her New Hampshire ski house. For her 59 days overdue on her payment, the bank char ged her a penalty of $15. What was the rate of interest char ged by the bank? Round to the nearest hundredth percent (assume 360 days).

10–32. Abe Wolf bought a new kitchen set at Sears. Abe paid off the loan after 60 days with an interest char ge of $9. If Sears charges 10% interest, what did Abe pay for the kitchen set (assume 360 days)?

10–33. Joy Kirby made a $300 loan to Robinson Landscaping at 1 1%. Robinson paid back the loan with interest of $6.60. How long in days was the loan outstanding (assume 360 days)? Check your answer .

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10–34. Molly Ellen, bookkeeper for Keystone Company, forgot to send in the payroll taxes due on April 15. She sent the payment November 8. The IRS sent her a penalty char ge of 8% simple interest on the unpaid taxes of $4,100. Calculate the penalty. (Remember that the government uses exact interest.)

10–35. Oakwood Plowing Company purchased two new plows for the upcoming winter . In 200 days, Oakwood must make a single payment of $23,200 to pay for the plows. As of today, Oakwood has $22,500. If Oakwood puts the money in a bank today, what rate of interest will it need to pay of f the plows in 200 days (assume 360 days)?

CHALLENGE PROBLEMS 10–36. The Downers Grove Reporter ran an ad for a used 1998 Harley-Davidson Sportster 883 for $6,750. Patrick Schmidt is interested in the motorcycle but does not have the money right now . Patrick contacted the owner on October 19, and he agreed to give Patrick a loan plus 5.5% exact interest. The loan must be paid back by December 22 of the same year . The First National Bank will lend the $6,750 at 5%. Patrick would have 3 months to pay of f the loan. (a) What is the total amount Patrick will have to pay the owner of the motorcycle assuming exact interest? (b) What is the total amount Patrick will have to pay the bank? (c) Which option offers the most savings to Patrick? (d) How much will Patrick save?

10–37. Janet Foster bought a computer and printer at Computerland. The printer had a $600 list price with a $100 trade discount and 2/10, n/30 terms. The computer had a $1,600 list price with a 25% trade discount but no cash discount. On the computer, Computerland offered Janet the choice of (1) paying $50 per month for 17 months with the 18th payment paying the remainder of the balance or (2) paying 8% interest for 18 months in equal payments. a.

Assume Janet could borrow the money for the printer at 8% to take advantage of the cash discount. How much would Janet save (assume 360 days)?

b. On the computer, what is the difference in the final payment between choices 1 and 2?

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DVD SUMMARY PRACTICE TEST 1.

Lorna Hall’s real estate tax of $2,010.88 was due on December 14, 2009. Lorna lost her job and could not pay her tax bill until February 27, 2010. The penalty for late payment is 6 12% ordinary interest. (p. 261) a.

What is the penalty Lorna must pay?

b. What is the total amount Lorna must pay on February 27?

2.

Ann Hopkins borrowed $60,000 for her child’s education. She must repay the loan at the end of 8 years in one payment with 521% interest. What is the maturity value Ann must repay? (p. 260)

3.

On May 6, Jim Ryan borrowed $14,000 from Lane Bank at 7 12% interest. Jim plans to repay the loan on March 1 1. Assume the loan is on ordinary interest. How much will Jim repay on March 1 1? (p. 261)

4.

Gail Ross met Jim Ryan (Problem 3) at Lane Bank. After talking with Jim, Gail decided she would like to consider the same loan on exact interest. Can you recalculate the loan for Gail under this assumption? (p. 260)

5.

Claire Russell is buying a car. Her November monthly interest was $210 at 7 34% interest. What is Claire’s principal balance (to the nearest dollar) at the beginning of November? Use 360 days. Do not round the denominator in your calculation. (p. 262)

6.

Comet Lee borrowed $16,000 on a 6%, 90-day note. After 20 days, Comet paid $2,000 on the note. On day 50, Comet paid $4,000 on the note. What are the total interest and ending balance due by the U.S. Rule? Use ordinary interest. (p. 264)

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Personal Finance A KIPLINGER APPROACH

benefits than they are contributing to pension plans. You’re going to spend that benefit check, but it doesn’t count as income. Once again, it looks as if we’re living beyond our means.

It doesn’t sound representative of how we’re really doing, then. It’s not. Corpo-

So as a nation we’re richer than we appear on paper?

If you really want to get rich, take the money you’d have saved and use it to build a business or invest in real estate, stocks or bonds. That involves more risk, but many investors and entrepreneurs wind up rich. The bubble bursts sometimes, too. Capitalism and wealth creation can be two steps forward and one step back.

rate benefits, such as pensions, are another example. Statisticians treat company contributions to a pension plan as income for the employee. But when the company pays out pension checks to its retirees, that money is not treated as income. Companies have been paying more in retirement T

H

E

K

I

P

L

I

N

G

E

R

G Christina Pridgen,

with Sara, 11, and Alex, 9.

| Sometimes it doesn’t make much sense to pay the money you owe. S O LV E D

My unpaid DEBT still haunts me

MONITOR C BLOGS | Americans bare their souls 19 MILLION Estimated number of active blogs in the English language.

12 MILLION Number of American adults who keep a blog.

57 MILLION Number of American adults who read blogs.

26% Percentage of U.S. bloggers who say they write at least three blogs.

76% Percentage of U.S. bloggers who say they write to document their personal experiences and share them with others.

55% Percentage of U.S. bloggers who use a pseudonym.

82% Percentage of U.S. bloggers who think they will still be at it a year from now. — MAGALI RHEAULT S OURCES : Pew Internet & American Life Project, Technorati

hristina Pridgen struggled with debt as she went through a divorce and began a new life for herself and her two kids. She admits she never paid $14,000 in joint creditcard debt incurred while she was married. Collectors have pretty much stopped bugging her, but she wonders: If she could find the resources to pay off the debt, would her credit rating be resurrected? “I had an excellent credit history,” says Pridgen, 34, a nursing student who lives near Charlotte, N.C. “I’d like to start over.” Believe it or not, paying back the $14,000 would do little to repair Pridgen’s credit history. And the black marks on her credit report will disappear in 18 months, anyway. Under federal law, a report of a bad debt must be removed seven and a half years after the first missed

payment. If the creditors had sued and won a judgment against Pridgen, they’d have had at least ten years to collect. But that didn’t happen—probably because of the relatively small amounts involved with each card issuer. The statute of limitations for such suits (three years in North Carolina, but as many as six elsewhere) has passed. There’s no limit on how long debt collectors can try to collect (see “Debt Police Who Go Too Far,” Nov.). For now, Pridgen is best off using her limited resources to finish her education and care for her kids. She can always try to cut a deal to clear the debt—and her conscience—when she’s out of school, working full-time. Do you have a money problem we can solve? E-mail us at [email protected]

BUSINESS MATH ISSUE Kiplinger’s © 2006

Christina should never pay back the debt. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

276

MIKE CARROLL

estate. Rightly or wrongly, they don’t see much point in saving when they’re sitting on gains of several hundred thousand dollars. That alone explains the drop in the savings rate.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work

PROJECT A Visit a local credit union and check their rates versus a regular bank.

urnal © 2006 Wall Street Jo

b site text We he e e S : s t T t Projec /slater9e) and e. Interne ce Guid m r o u .c o e s h e h R t .m e (www Intern ss Math Busine

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CHAPTER

11

Promissory Notes, Simple Discount Notes, and the Discount Process

LEARNING UNIT OBJECTIVES LU 11–1: Structure of Promissory Notes; the Simple Discount Note • Differentiate between interest-bearing and noninterest-bearing notes (pp. 279–280). • Calculate bank discount and proceeds for simple discount notes (p. 280). • Calculate and compare the interest, maturity value, proceeds, and effective rate of a simple interest note with a simple discount note (p. 281). • Explain and calculate the effective rate for a Treasury bill (p. 281).

LU 11–2: Discounting an Interest-Bearing Note before Maturity • Calculate the maturity value, bank discount, and proceeds of discounting an interest-bearing note before maturity (pp. 282–283). • Identify and complete the four steps of the discounting process (p. 283).

Wall Street Jo urnal © 2006

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Learning Unit 11–1

279

Wall Street Journal © 2005

This Wall Street Journal heading states that Saks is having financial problems. Unlike credit cardholders who fail to meet their financial obligations, Saks has the option of tapping a $650 million credit line to help its financial situation. This chapter begins with a discussion of the structure of promissory notes and simple discount notes. We also look at the application of discounting with Treasury bills. The chapter concludes with an explanation of how to calculate the discounting of promissory notes.

Learning Unit 11–1: Structure of Promissory Notes; the Simple Discount Note Although businesses frequently sign promissory notes, customers also sign promissory notes. For example, some student loans may require the signing of promissory notes. Appliance stores often ask customers to sign a promissory note when they buy lar ge appliances on credit. In this unit, promissory notes usually involve interest payments.

Structure of Promissory Notes To borrow money , you must find a lender (a bank or a company selling goods on credit). You must also be willing to pay for the use of the money . In Chapter 10 you learned that interest is the cost of borrowing money for periods of time. Money lenders usually require that borrowers sign a promissory note. This note states that the borrower will repay a certain sum at a fixed time in the future. The note often includes the charge for the use of the money, or the rate of interest. Figure 11.1 shows a sample promissory note with its terms identified and defined. Take a moment to look at each term. In this section you will learn the dif ference between interest-bearing notes and noninterest-bearing notes. Interest-Bearing versus Noninterest-Bearing Notes A promissory note can be interest bearing or noninterest bearing. To be interest bearing, the note must state the rate of interest. Since the promissory note in Figure 1 1.1 states that its interest is 9%, it is an interest-bearing note. When the note matures, Regal Corporation “will pay back the original amount ( face value) borrowed plus interest. The simple interest formula (also known as the interest formula) and the maturity value formula from Chapter 10 are used for this transaction.” Interest Face value (principal) Rate Time Maturity value Face value (principal) Interest

FIGURE

$10,000

11.1

LAWTON, OKLAHOMA

a.

Sixty days

b.

AFTER DATE we

G.J. Equipment Company Ten thousand and 00/100---------------------DOLLARS. Able National Bank PAYABLE AT VALUE RECEIVED WITH INTEREST AT 9% e. REGAL CORPORATION DUE December 1, 2007 NO. 114

Interest-bearing promissory note

THE ORDER OF

g.

a. b. c. d. e. f. g.

October 2, 2007 c.

PROMISE TO PAY TO d.

f.

J.M. Moore TREASURER

Face value: Amount of money borrowed—$10,000. The face value is also the principal of the note. Term: Length of time that the money is borrowed—60 days. Date: The date that the note is issued—October 2, 2007. Payee: The company extending the credit—G.J. Equipment Company. Rate: The annual rate for the cost of borrowing the money—9%. Maker: The company issuing the note and borrowing the money—Regal Corporation. Maturity date: The date the principal and interest rate are due—December 1, 2007.

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Chapter 11 Promissory Notes, Simple Discount Notes, and the Discount Process

TABLE

11.1

Comparison of simple interest note and simple discount note (Calculations from the Pete Runnels example)

Simple interest note (Chapter 10)

Simple discount note (Chapter 11)

1. A promissory note for a loan with a term of usually less than 1 year. Example: 60 days.

1. A promissory note for a loan with a term of usually less than 1 year. Example: 60 days.

2. Paid back by one payment at maturity. Face value equals actual amount (or principal) of loan (this is not maturity value).

2. Paid back by one payment at maturity. Face value equals maturity value (what will be repaid).

3. Interest computed on face value or what is actually borrowed. Example: $186.67.

3. Interest computed on maturity value or what will be repaid and not on actual amount borrowed. Example: $186.67.

4. Maturity value Face value Interest. Example: $14,186.67.

4. Maturity value Face value. Example: $14,000.

5. Borrower receives the face value. Example: $14,000.

5. Borrower receives proceeds Face value Bank discount. Example: $13,813.33.

6. Effective rate (true rate is same as rate stated on note). Example: 8%.

6. Effective rate is higher since interest was deducted in advance. Example: 8.11%.

7. Used frequently instead of the simple discount note. Example: 8%.

7. Not used as much now because in 1969 congressional legislation required that the true rate of interest be revealed. Still used where legislation does not apply, such as personal loans.

If you sign a noninterest-bearing promissory note for $10,000, you pay back $10,000 at maturity. The maturity value of a noninterest-bearing note is the same as its face value. Usually, noninterest-bearing notes occur for short time periods under special conditions. For example, money borrowed from a relative could be secured by a noninterest-bearing promissory note.

Simple Discount Note The total amount due at the end of the loan, or the maturity value (MV), is the sum of the face value (principal) and interest. Some banks deduct the loan interest in advance. When banks do this, the note is a simple discount note. In the simple discount note, the bank discount is the interest that banks deduct in advance and the bank discount rate is the percent of interest. The amount that the borrower receives after the bank deducts its discount from the loan’ s maturity value is the note’s proceeds. Sometimes we refer to simple discount notes as noninterest-bearing notes. Remember, however, that borrowers do pay interest on these notes. In the example that follows, Pete Runnels has the choice of a note with a simple interest rate (Chapter 10) or a note with a simple discount rate (Chapter 1 1). Table 1 1.1 provides a summary of the calculations made in the example and gives the key points that you should remember. Now let’ s study the example, and then you can review Table 11.1. Pete Runnels has a choice of two dif ferent notes that both have a face value (principal) of $14,000 for 60 days. One note has a simple interest rate of 8%, while the other note has a simple discount rate of 8%. For each type of note, calculate (a) interest owed, (b) maturity value, (c) proceeds, and (d) effective rate. EXAMPLE

Simple interest note—Chapter 10

Simple discount note—Chapter 11

Interest

Interest

a. I Face value (principal) R T

a. I Face value (principal) R T

60 I $14,000 .08 360 I $186.67

I $14,000 .08 I $186.67

Maturity value

Maturity value

b. MV Face value Interest

b. MV Face value

MV $14,000 $186.67

60 360

MV $14,000

MV $14,186.67 Proceeds

Proceeds

c. Proceeds Face value

c. Proceeds MV Bank discount

$14,000

$14,000 $186.67 $13,813.33

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Learning Unit 11–1

Simple interest note—Chapter 10

Simple discount note—Chapter 11

Effective rate

Effective rate

Interest d. Rate Proceeds Time

d. Rate

$186.67

60 $14,000 360 8%

281

Interest Proceeds Time $186.67 $13,813.33

60 360

8.11%

Note that the interest of $186.67 is the same for the simple interest note and the simple discount note. The maturity value of the simple discount note is the same as the face value. In the simple discount note, interest is deducted in advance, so the proceeds are less than the face value. Note that the ef fective rate for a simple discount note is higher than the stated rate, since the bank calculated the rate on the face of the note and not on what Pete received. Application of Discounting—Treasury Bills When the government needs money , it sells Treasury bills. A Treasury bill is a loan to the federal government for 28 days (4 weeks), 91 days (13 weeks), or 1 year. Note that the Wall Street Journal clipping Treasury—bill sales will raise $6 billion. Treasury bills can be bought over the phone or on the government website. (See Business Math Scrapbook, page 293, for details.) The purchase price (or proceeds) of a Treasury bill is the value of the Treasury bill less the discount. For example, if you buy a $10,000, 13-week Treasury bill at 8%, you pay $9,800 since you have not yet earned your interest ($10,000 .08 13 52 $200). At maturity— 13 weeks—the government pays you $10,000. You calculate your ef fective yield (8.16% rounded to the nearest hundredth percent) as follows: Wall Street Journal © 2006

$200 ($10,000 $200)

$9,800

13 52

8.16% effective rate

Now it’s time to try the Practice Quiz and check your progress.

LU 11–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1.

DVD

2.

912 %, Warren Ford borrowed $12,000 on a noninterest-bearing, simple discount, 60-day note. Assume ordinary interest. What are (a) the maturity value, (b) the bank’s discount, (c) Warren’s proceeds, and (d) the effective rate to the nearest hundredth percent? Jane Long buys a $10,000, 13-week Treasury bill at 6%. What is her ef fective rate? Round to the nearest hundredth percent.

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Chapter 11 Promissory Notes, Simple Discount Notes, and the Discount Process

✓

Solutions a. Maturity value Face value $12,000 b. Bank discount MV Bank discount rate Time 60 $12,000 .095 360 $190

1.

c. Proceeds MV Bank discount $12,000 $190 $11,810

$10,000 .06

2.

LU 11–1a

13 $150 interest 52

d. Effective rate

$150 $9,850

Interest Proceeds Time $190 $11,810

60 360

$9.65% 13 52

6.09%

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 286)

1.

2.

1/2%, Warren Ford borrowed $14,000 on a noninterest-bearing, simple discount, 4 60-day note. Assume ordinary interest. What are (a) the maturity value, (b) the bank’s discount, (c) Warren’s proceeds, and (d) the effective rate to the nearest hundredth percent? Jane Long buys a $10,000 13-week Treasury bill at 4%. What is her ef fective rate? Round to the nearest hundredth percent.

Learning Unit 11–2: Discounting an Interest-Bearing Note before Maturity Manufacturers frequently deliver merchandise to retail companies and do not request payment for several months. For example, Roger Company manufactures outdoor furniture that it delivers to Sears in March. Payment for the furniture is not due until September . Roger will have its money tied up in this furniture until September . So Roger requests that Sears sign promissory notes. If Roger Company needs cash sooner than September , what can it do? Roger Company can take one of its promissory notes to the bank, assuming the company that signed the note is reliable. The bank will buy the note from Roger . Now Roger has discounted the note and has cash instead of waiting until September when Sears would have paid Roger . Remember that when Roger Company discounts the promissory note to the bank, the company agrees to pay the note at maturity if the maker of the promissory note fails to pay the bank. The potential liability that may or may not result from discounting a note is called a contingent liability. Think of discounting a note as a three-party arrangement. Roger Company realizes that the bank will char ge for this service. The bank’s charge is a bank discount. The actual amount Roger receives is the proceeds of the note. The four steps below and the formulas in the example that follows will help you understand this discounting process. DISCOUNTING A NOTE Step 1.

Calculate the interest and maturity value.

Step 2.

Calculate the discount period (time the bank holds note).

Step 3.

Calculate the bank discount.

Step 4.

Calculate the proceeds.

EXAMPLE

Roger Company sold the following promissory note to the bank:

Date of note March 8

Face value of note $2,000

Length of note 185 days

Interest rate 10%

Bank discount rate 9%

Date of discount August 9

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283

Learning Unit 11–1

What are Roger’s (1) interest and maturity value (MV)? What are the (2) discount period and (3) bank discount? (4) What are the proceeds? 1.

Calculate Roger’s interest and maturity value (MV): Interest $2,000 .10

MV Face value (principal) Interest

185 360

Exact number of days over 360

$102.78 MV $2,000 $102.78 $2,102.78 Calculating days without table: March 31 8 23 April 30 May 31 June 30 July 31 August 9 154 185 days—length of note 154 days Roger held note 31 days bank waits

2.

August 9 March 8

Calculate discount period: Determine the number of days that the bank will have to wait for the note to come due (discount period).

Date of note

221 days 67 154 days passed before note is discounted 185 days 154 31 days bank waits for note to come due

Date of discount

Date note due 31 days

154 days before note is discounted

March 8

bank waits

Aug. 9

Sept. 9

185 days total length of note

By table: March 8

3.

Calculate bank discount (bank charge): $2,102.78 .09

4.

31 $16.30 360

Calculate proceeds: $2,102.78 16.30 $2,086.48 If Roger had waited until September 9, it would have received $2,102.78. Now, on August 9, Roger received $2,000 plus $86.48 interest.

67 days 185 252 search in table

Number of days bank waits for note Bank Bank to come due MV discount discount 360 rate

Step 1 Proceeds MV Bank discount (charge)

Step 3

Now let’s assume Roger Company received a noninterest-bearing note. Then we follow the four steps for discounting a note except the maturity value is the amount of the loan. No interest accumulates on a noninterest-bearing note. Today, many banks use simple interest instead of discounting. Also, instead of discounting notes, many companies set up lines of credit so that additional financing is immediately available.

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Chapter 11 Promissory Notes, Simple Discount Notes, and the Discount Process

Wall Street Journal © 2004

The Wall Street Journal clipping “Finding Funding” shows that 28% of small businesses surveyed use a line of credit to finance their operations. The Practice Quiz that follows will test your understanding of this unit.

LU 11–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Date of note April 8

Face value (principal) of note $35,000

Length of note 160 days

Interest rate 11%

Bank discount rate 9%

Date of discount June 8

From the above, calculate (a) interest and maturity value, (b) discount period, (c) bank discount, and (d) proceeds. Assume ordinary interest.

✓ a.

Solutions 160 $1,711.11 360 MV $35,000 $1,711.11 $36,711.11 I $35,000 .11

b. Discount period 160 61 99 days. April

30 8 22 May 31 53 June 8 61

Or by table: June 8 159 April 8 98 61

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

Bank discount $36,711.11 .09

c.

285

99 $908.60 360

d. Proceeds $36,711.11 $908.60 $35,802.51

LU 11–2a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 286)

From the information below , calculate (a) interest and maturity value, (b) discount period, (c) bank discount, and (d) proceeds. Assume ordinary interest. Date of note April 10

Face value (principal) of note $40,000

Length of note 170 days

Interest rate 5%

Bank discount rate 2%

Date of discount June 10

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Simple discount note, p. 280

Bank Bank discount MV discount Time (interest) rate

$6,000 .09

Interest based on amount paid back and not what received.

Borrower receives $5,910 (the proceeds) and pays back $6,000 at maturity after 60 days.

60 $90 360

A Treasury bill is a good example of a simple discount note. Effective rate, p. 281

Interest Proceeds Time

Example: $10,000 note, discount rate 12% for 60 days.

What borrower receives

I $10,000 .12

(Face value Discount)

Effective rate: $200 60 $9,800 360

60 $200 360

$200 12.24% $1,633.3333

Amount borrower received Discounting an interest-bearing note, p. 282

KEY TERMS

1. Calculate interest and maturity value. I Face value Rate Time MV Face value Interest 2. Calculate number of days bank will wait for note to come due (discount period). 3. Calculate bank discount (bank charge).

Example: $1,000 note, 6%, 60-day, dated November 1 and discounted on December 1 at 8%. 60 1. I $1,000 .06 $10 360 MV $1,000 $10 $1,010

Bank Number of days bank waits MV discount 360 rate 4. Calculate proceeds. MV Bank discount (charge)

2. 30 days

Bank discount, pp. 280, 282 Bank discount rate, p. 280 Contingent liability, p. 282 Discounting a note, p. 282 Discount period, p. 283 Effective rate, p. 281 Face value, p. 279

30 $6.73 360 4. $1,010 $6.73 $1,003.27 3. $1,010 .08

Interest-bearing note, p. 279 Maker, p. 279 Maturity date, p. 279 Maturity value (MV), p. 279 Noninterest-bearing note, p. 280 Payee, p. 279

Proceeds, pp. 280, 283 Promissory note, p. 279 Simple discount note, p. 280 Treasury bill, p. 281

(continues)

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Chapter 11 Promissory Notes, Simple Discount Notes, and the Discount Process

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 11–1a (p. 282) 1. A. $14,000 B. $105 C. $13,895 D. 4.53% 2. 4.04%

LU 11–2a (p. 285) 1. A. Int. = $944.44; $40,944.44 B. 109 days C. $247.94 D. $40,696.50

Critical Thinking Discussion Questions 1. What are the dif ferences between a simple interest note and 3. What is a line of credit? What could be a disadvantage of a simple discount note? Which type of note would have a having a large credit line? higher effective rate of interest? Why? 2. What are the four steps of the discounting process? Could the proceeds of a discounted note be less than the face value of the note?

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Complete the following table for these simple discount notes. Use the ordinary interest method. Amount due at maturity

Discount rate

Time

Bank discount

11–1.

$18,000

414%

300 days

11–2.

$20,000

614%

180 days

Proceeds

Calculate the discount period for the bank to wait to receive its money:

11–3.

Date of note April 12

Length of note 45 days

Date note discounted May 2

11–4.

March 7

120 days

June 8

Discount period

Solve for maturity value, discount period, bank discount, and proceeds (assume for Problems 1 1–5 and 11–6 a bank discount rate of 9%).

11–5.

Face value (principal) $50,000

Rate of interest 11%

Length of note 95 days

Maturity value

Date of note June 10

Date note discounted July 18

June 8

July 10

Discount period

Bank discount

11–6.

$25,000

9%

60 days

11–7.

Calculate the effective rate of interest (to the nearest hundredth percent) of the following Treasury bill. Given: $10,000 Treasury bill, 4% for 13 weeks.

Proceeds

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WORD PROBLEMS Use ordinary interest as needed. 11–8. On March 19, 2006, The Saint Paul Pioneer Press reported on interest loans which include an additional, one time $20 fee. Wilbert McKee’s bank deducts interest in advance and also deducts $20.00 fee in advance. Wilbert needs a loan for $500. The bank charges 5% interest. Wilbert will need the loan for 90 days. What is the effective rate for this loan? Round to the nearest hundredth percent. Do not round denominator in calculation.

11–9.

Jack Tripper signed a $9,000 note at Fleet Bank. Fleet charges a 914% discount rate. If the loan is for 200 days, find (a) the proceeds and (b) the effective rate charged by the bank (to the nearest tenth percent).

11–10. On January 18, 2007, BusinessWeek reported yields on Treasury bills. Bruce Martin purchased a $10,000 13 week Treasury bill at $9,881.25. (a) What was the amount of interest? (b) What was the effective rate of interest? Round to the nearest hundredth percent.

11–11. On September 5, Sheffield Company discounted at Sunshine Bank a $9,000 (maturity value), 120-day note dated June 5. Sunshine’s discount rate was 9%. What proceeds did Sheffield Company receive?

11–12. The Treasury Department auctioned $21 billion in three month bills in denominations of ten thousand dollars at a discount rate of 4.965%, according to the March 13, 2007 issue of the Chicago Sun-Times. What would be the effective rate of interest? Round your answer to the nearest hundredth percent.

11–13. Annika Scholten bought a $10,000, 13-week Treasury bill at 5%. What is her effective rate? Round to the nearest hundredth percent.

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11–14. Ron Prentice bought goods from Shelly Katz. On May 8, Shelly gave Ron a time extension on his bill by accepting a $3,000, 8%, 180-day note. On August 16, Shelly discounted the note at Roseville Bank at 9%. What proceeds does Shelly Katz receive?

11–15. Rex Corporation accepted a $5,000, 8%, 120-day note dated August 8 from Regis Company in settlement of a past bill. On October 11, Rex discounted the note at Park Bank at 9%. What are the note’s maturity value, discount period, and bank discount? What proceeds does Rex receive?

11–16. On May 12, Scott Rinse accepted an $8,000, 12%, 90-day note for a time extension of a bill for goods bought by Ron Prentice. On June 12, Scott discounted the note at Able Bank at 10%. What proceeds does Scott receive?

11–17. Hafers, an electrical supply company, sold $4,800 of equipment to Jim Coates Wiring, Inc. Coates signed a promissory note May 12 with 4.5% interest. The due date was August 10. Short of funds, Hafers contacted Charter One Bank on July 20; the bank agreed to take over the note at a 6.2% discount. What proceeds will Hafers receive?

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CHALLENGE PROBLEMS 11–18. Market News Publishing reported on the sale of a promissory note. ZTEST Electronics announced that it agreed to sell a promissory note (the “Note”) in the principal amount of $318,019.95 owed to them by Parmatech Electronic Corporation. The note, negotiated on March 15, is a 360-day note with 8.5% interest per annum. Halfway through the life of the note, Alpha Bank offered to purchase the note at 8.75%. Baker Bank offered to purchase the note at 9.0%. (a) What proceeds will ZTEST receive from Alpha Bank? (b) What proceeds will ZTEST receive from Baker Bank? (c) How much more will ZTEST receive from Alpha Bank? Round to the nearest cent.

11–19. Tina Mier must pay a $2,000 furniture bill. A finance company will loan Tina $2,000 for 8 months at a 9% discount rate. The finance company told Tina that if she wants to receive exactly $2,000, she must borrow more than $2,000. The finance company gave Tina the following formula: What to ask for

Amount in cash to be received 1 (Discount Time of loan)

Calculate Tina’s loan request and the effective rate of interest to nearest hundredth percent.

DVD SUMMARY PRACTICE TEST 1.

On December 12, Lowell Corporation accepted a $160,000, 120-day, noninterest-bearing note from Able.com. What is the maturity value of the note? (p. 279)

2.

The face value of a simple discount note is $17,000. The discount is 4% for 160 days. Calculate the following. (p. 280) a.

Amount of interest charged for each note.

b. Amount borrower would receive. c.

Amount payee would receive at maturity.

d. Effective rate (to the nearest tenth percent).

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3.

On July 14, Gracie Paul accepted a $60,000, 6%, 160-day note from Mike Lang. On November 12, Gracie discounted the note at Lend Bank at 7%. What proceeds did Gracie receive? (p. 282)

4.

Lee.com accepted a $70,000, 634%, 120-day note on July 26. Lee discounts the note on October 28 at LB Bank at 6%. What proceeds did Lee receive? (p. 282)

5.

The owner of Lease.com signed a $60,000 note at Reese Bank. Reese charges a 714% discount rate. If the loan is for 210 days, find (a) the proceeds and (b) the effective rate charged by the bank (to the nearest tenth percent). (p. 280)

6.

Sam Slater buys a $10,000, 13-week Treasury bill at 512%. What is the effective rate? Round to the nearest hundredth percent. (p. 281)

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Personal Finance A KIPLINGER APPROACH

M L H A R R I S /G E T T Y I M A G E S

can apply for a PLUS now and consolidate to lock in this year’s rate, says Mark Brenner, of College Loan Corp. (www.collegeloan.com), which makes such loans. Ask your school’s financialaid office for details.

Last chance to LOCK in

Other options. After July 1, parents choosing between a PLUS loan with an 8.5% fixed rate and a variable-rate home-equity line of credit should take a closer look at the latter, says Carpenter. The average rate for equity lines was recently 7.67%, and interest is deductible. With rates fixed on Stafford loans, private loans, which are issued at variable rates, could someday end up costing less than Staffords. Sallie Mae (www.salliemae.com), the largest of the student-loan companies, offers private loans at the prime rate—lately 7.5%— with no fees for borrowers who have a good credit history. Even if rates head south, borrowers “should exhaust federal loans first,” says Sallie Mae spokeswoman Martha Holler. Unlike private loans, payments on those loans can be extended, deferred or forgiven in certain cases.

t s eems li ke only yesterday that student-loan rates were sinking faster than a December sun. Alas, the days of magically vanishing—or modestly rising—rates are about to end. Starting July 1, the Deficit Reduction Act of 2005 will set a fixed rate of 6.8% on new Stafford loans, about two percentage points above this past year’s lowest rate. Similarly, PLUS loans for parent borrowers will be fixed at 8.5%, up from the current 6.1%. But the fixed rates won’t apply to outstanding Stafford and PLUS loans. On those loans, rates will continue to change each July 1 based on the 91-day Treasury-bill yield set the last Thursday in May. The T-bill rate is expected to rise, so it pays to consolidate your loans and lock in the lower rate. Things get a little tricky if you con-

A mixed bag. As for the other provisions of the Deficit Reduction Act, they represent “a mixed bag” for undergraduates, says Brenner. For Stafford loans, the law boosts the maximum amount you can borrow in each of the first two years of college (the total amount remains the same), phases out origination fees and expands Pell Grants for math and science students. Married couples will no longer be able to consolidate loans taken out separately into a single loan. And, as of July 1, students can no longer consolidate Staffords while they’re still in school. But Brenner says the changes “should in no way discourage American families from applying for the college of their choice.” There’s plenty of money for students who need it, he says, and federally sponsored loans remain “a hell of a deal.”

G Students who take

out Stafford loans after July 1 will pay a fixed interest rate of 6.8%.

CO L L EG E

| To save on student-loan interest rates,

consolidate your debt by July 1. By Jane Bennett Clark

I

solidated last spring to take advantage of bottom-cruising rates (as low as 2.87% for Stafford loans and 4.17% for PLUS loans) and have since taken out new loans. You can consolidate the new loans, but you’ll want to keep the two consolidations separate, says Gary Carpenter, executive director of the National Institute of Certified College Planners (www.niccp.com). “If you roll an old consolidation into a new one, you get a blended rate—the lower rate is lost,” says Carpenter. And you may have to shop for a lender; some balk at consolidating loans of less than $7,500. Although financial-aid packages were calculated this spring, next fall’s freshmen will pay the post-July, fixed rate on Staffords; likewise, PLUS loans for parents of incoming freshmen will carry the new fixed rate. However, parents of currently enrolled students

BUSINESS MATH ISSUE Kiplingers © 2006

The Deficit Reduction Act of 2005 is too complicated for students needing loans. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A Go to www.treasurydirect. gov and find out the latest rates for Treasury bills.

2005 Wall Street Journal ©

b site text We he e e S : s t T t Projec /slater9e) and Guide. Interne m r sou ce he.co e h R t .m e w n r (ww Inte ss Math Busine

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Video Case ONLINE BANKING

Online banking is very cost effective for the banking industry. Many customers enjoy the convenience; others, however, have doubts. For these individuals, online banking is a different way of thinking. Banks want customers flocking online because it costs less after initial startup fees. A teller transaction typically costs a bank on average $1 to $1.50, while Internet transactions cost less than 5 cents. Less cost means more profit. The Gartner Group, a research firm, says that 27 million Americans—one in 10—now do at least some of their banking online, up from 9 million a year earlier. According to a new Gallup poll, online banking services soared by 60% in the year 2000. CyberDialogue, an Internet consulting firm, predicted online banking will rise to 50.9 million customers by 2005. Most sites allow customers to view account information, transfer money, and pay bills online; some sites offer investment account data and transactions. Other applications are coming, including the ability to view and print account statements and canceled checks.

PROBLEM

1

In 2000, the number of households accessing their accounts through a computer increased to 12.5 million, an 81.42% increase from a year earlier. These numbers support the push for online banking. What was the number of online users last year? Round to the nearest million. PROBLEM

2

Jupiter Media Metrix, an online research firm, estimated that banking online will increase from 12.5 million to about 43.3 million in 2005. CyberDialogue, an Internet consulting firm, predicted that by the end of 2000, 24.6 million people would bank online and by 2005, the number would rise to 50.9 million. (a) What percent increase is Jupiter Media Metrix forecasting? (b) What percent increase is CyberDialogue forecasting? Round to the nearest hundredth percent. PROBLEM

3

E*Trade Bank pays at least 3.1% on checking accounts with balances of $1,000 or more. The national average is 0.78% for interest-bearing checking. If you have $2,300 in your account and bank at E*Trade based on simple interest: (a) How much interest would you earn at the end of 30 days (ordinary interest)? (b) How much interest would you earn at a nononline bank? PROBLEM

4

Online banking users—people who do basic banking tasks such as occasionally transferring money between accounts online—jumped to an estimated 20 million in December 2000 from 15.9 million in September 2000. What was the percent increase? Round to the nearest hundredth percent.

294

Pundits wrote off most Web banking because of all the things customers couldn’t do—close on a loan, sign for a mortgage, or withdraw cash. The startups are applying increasingly innovative strategies to clear these hurdles. Security was, and still is, an issue for many people. According to a recent study, 85% of information technology staffs at corporations and government agencies had detected a computer security breach in the past 12 months, and 64% acknowledged financial losses as a result. Measures are being taken to improve security. In addition to the usual conveniences of online banking, online banks can pay higher rates on deposits than branchbased banks. However, problems do exist in online banking, such as you can rack up late fees for bill paying and not even know it. When picking an online banking service, look for the following: (1) 128-bit encryption, the standard in the industry; (2) written guarantees to protect from losses in case of online fraud or bank error; (3) automatic lockout if you wrongly enter your password more than three or four times; and (4) evidence that the bank is FDIC insured.

PROBLEM

5

On January 9, 2001, Bank of America Corporation announced that it had more than 3 million online banking customers. If 130,000 customers are added in a month, what is the percent increase? Round to the nearest hundredth percent. PROBLEM

6

The E*Trade Bank is an Internet bank in Menlo Park, California, owned by Internet brokerage company E*Trade Group. On January 4, 2001, E*Trade Bank said it had added more than $1 billion in net new deposits in its fourth quarter of 2000, bringing its total deposits to more than $5.7 billion. E*Trade had a total of $1.1 billion in deposits at the end of 1998. What is the percent increase in net deposits in the year 2000 compared to 1998? Round to the nearest hundredth percent. PROBLEM

7

The research firm The Gartner Group says that in 2001, 27 million Americans—one in 10—do at least some of their banking online, up from 9 million a year earlier. (a) How many were banking online last year? (b) What was the percent increase in online banking in 2001? Round to nearest hundredth percent. PROBLEM

8

Industry experts expect that online banking and bill payment, like other forms of e-commerce, will continue to grow at a rapid pace. According to Killen & Associates, the number of bills paid online will rise to 11.7 billion by 2001, a 77% increase. What had been the amount of users in 2000? Round to the nearest tenth.

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CHAPTER

12

Compound Interest and Present Value

LEARNING UNIT OBJECTIVES Note: A complete set of plastic overlays showing the concepts of compound interest and present value is found in Chapter 13. LU 12–1: Compound Interest (Future Value)—The Big Picture • Compare simple interest with compound interest (pp. 296–298 ). • Calculate the compound amount and interest manually and by table lookup (pp. 298–301). • Explain and compute the effective rate (APY) (p. 301).

LU 12–2: Present Value—The Big Picture • Compare present value (PV) with compound interest (FV) (p. 303 ). • Compute present value by table lookup (pp. 304–306 ). • Check the present value answer by compounding (p. 306 ).

urnal © 2006 Wall Street Jo

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Chapter 12 Compound Interest and Present Value

Would you like to save a million dollars? We omitted the beginning of the Wall Street Journal clipping “How Math Fattens Your Wallet” because it explained the years involved in recouping losses when interest is only char ged on the principal. The clipping contrasts this extended time by introducing compound interest, which means that interest is added to the principal and then additional interest is paid on both the old principal and its interest. This compounding can make it possible for you to save a million dollars. In this chapter we look at the power of compounding—interest paid on earned interest. Let’s begin by studying Learning Unit 12–1, which shows you how to calculate compound interest. Wall Street Journal © 2005

Learning Unit 12–1: Compound Interest (Future Value)—The Big Picture Check out the plastic overlays that appear within Chapter 13 to review these concepts.

FIGURE

So far we have discussed only simple interest, which is interest on the principal alone. Simple interest is either paid at the end of the loan period or deducted in advance. From the chapter introduction, you know that interest can also be compounded. Compounding involves the calculation of interest periodically over the life of the loan (or investment). After each calculation, the interest is added to the principal. Future calculations are on the adjusted principal (old principal plus interest). Compound interest, then, is the interest on the principal plus the interest of prior periods. Future value (FV), or the compound amount, is the final amount of the loan or investment at the end of the last period. In the beginning of this unit, do not be concerned with how to calculate compounding but try to understand the meaning of compounding. Figure 12.1 shows how $1 will grow if it is calculated for 4 years at 8% annually . This means that the interest is calculated on the balance once a year . In Figure 12.1, we start with $1, which is the present value (PV). After year 1, the dollar with interest is worth $1.08. At the end of year 2, the dollar is worth $1.17. By the end of year 4, the dollar is worth $1.36 . Note how we start with the present and look to see what the dollar will be worth in the future. Compounding goes from present value to future value.

Compounding goes from present value to future value

12.1

Future value of $1 at 8% for four periods

$5.00 4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 .50 .00

Present value $1.00

After 1 period, $1 is worth $1.08 $1.08

After 2 periods, $1 is worth $1.17 $1.1664

After 3 periods, $1 is worth $1.26 $1.2597

0

1

2 Number of periods

3

Future value After 4 periods, $1 is worth $1.36 $1.3605

4

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Learning Unit 12–1

297

Before you learn how to calculate compound interest and compare it to simple interest, you must understand the terms that follow . These terms are also used in Chapter 13. •

Compounded annually: Interest calculated on the balance once a year .

•

Compounded semiannually: Interest calculated on the balance every 6 months or every 1 2 year. Compounded quarterly: Interest calculated on the balance every 3 months or every 1 4 year. Compounded monthly: Interest calculated on the balance each month. Compounded daily: Interest calculated on the balance each day. Number of periods:1 Number of years multiplied by the number of times the interest is compounded per year . For example, if you compound $1 for 4 years at 8% annually , semiannually, or quarterly, the following periods will result: Annually: 4 years 1 4 periods Semiannually: 4 years 2 8 periods Quarterly: 4 years 4 16 periods Rate for each period:2 Annual interest rate divided by the number of times the interest is compounded per year. Compounding changes the interest rate for annual, semiannual, and quarterly periods as follows: Annually: 8% 1 8% Semiannually: 8% 2 4% Quarterly: 8% 4 2% Note that both the number of periods (4) and the rate (8%) for the annual example did not change. You will see later that rate and periods (not years) will always change unless interest is compounded yearly.

• • • •

•

Now you are ready to learn the dif interest.

ference between simple interest and compound

Simple versus Compound Interest Did you know that money invested at 6% will double in 12 years? The following Wall Street Journal clipping “Confused by Investing?” shows how to calculate the number of years it takes for your investment to double.

Reprinted by permission of The Wall Street Journal, © 2003 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

1

Periods are often expressed with the letter N for number of periods.

2

Rate is often expressed with the letter i for interest.

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Chapter 12 Compound Interest and Present Value

The following three situations of Bill Smith will clarify the dif ference between simple interest and compound interest. Situation 1: Calculating Simple Interest and Maturity Value EXAMPLE Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s simple interest? To calculate simple interest, we use the following simple interest formula: Interest (I ) Principal (P ) Rate (R ) Time (T ) $25.60

$80

.08

4

In 4 years Bill receives a total of $105.60 ($80.00 $25.60)—principal plus simple interest. Now let’s look at the interest Bill would earn if the bank compounded Bill’ s interest on his savings. Situation 2: Calculating Compound Amount and Interest without Tables3 You can use the following steps to calculate the compound amount and the interest manually: CALCULATING COMPOUND AMOUNT AND INTEREST MANUALLY Step 1.

Calculate the simple interest and add it to the principal. Use this total to figure next year’s interest.

Step 2.

Repeat for the total number of periods.

Step 3.

Compound amount Principal Compound interest.

Bill Smith deposited $80 in a savings account for 4 years at an annual compounded rate of 8%. What are Bill’s compound amount and interest? The following shows how the compounded rate af fects Bill’s interest:

EXAMPLE

Interest Beginning balance Amount at year-end

Year 1 $80.00 .08

Year 2 $86.40 .08

Year 3 $ 93.31 .08

Year 4 $100.77 .08

$ 6.40 80.00

$ 6.91 86.40

$ 7.46 93.31

$ 8.06 100.77

$86.40

$93.31

$100.77

$108.83

Note that the beginning year 2 interest is the result of the interest of year 1 added to the principal. At the end of each interest period, we add on the period’ s interest. This interest becomes part of the principal we use for the calculation of the next period’ s interest. We can determine Bill’ s compound interest as follows: 4 Compound amount Principal

$108.83 80.00

Compound interest

$ 28.83

Note: In Situation 1 the interest was $25.60.

We could have used the following simplified process to calculate the compound amount and interest:

3

For simplicity of presentation, round each calculation to nearest cent before continuing the compounding process. The compound amount will be off by 1 cent. 4

The formula for compounding is A P(1 i)N, where A equals compound amount, P equals the principal, i equals interest per periA od, and N equals number of periods. The calculator sequence would be as follows for Bill Smith: 1 .08 yx 4 80 108.84. Financial Calculator Guide booklet is available that shows how to operate HP 10BII and TI BA II Plus.

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Learning Unit 12–1

Year 1 $80.00 1.08

Year 2 $86.40 1.08

Year 3 $ 93.31 1.08

$86.40

$93.31

$100.77

299

Year 4 $100.77 1.08 $108.83 5

Future value

When using this simplification, you do not have to add the new interest to the previous balance. Remember that compounding results in higher interest than simple interest. Compounding is the sum of principal and interest multiplied by the interest rate we use to calculate interest for the next period. So, 1.08 above is 108%, with 100% as the base and 8% as the interest. Situation 3: Calculating Compound Amount by Table Lookup To calculate the compound amount with a future value table, use the following steps: CALCULATING COMPOUND AMOUNT BY TABLE LOOKUP

Four Periods No. of times compounded No. of years in 1 year 1 4

Step 1.

Find the periods: Years multiplied by number of times interest is compounded in 1 year.

Step 2.

Find the rate: Annual rate divided by number of times interest is compounded in 1 year.

Step 3.

Go down the Period column of the table to the number of periods desired; look across the row to find the rate. At the intersection of the two columns is the table factor for the compound amount of $1.

Step 4.

Multiply the table factor by the amount of the loan. This gives the compound amount.

In Situation 2, Bill deposited $80 into a savings account for 4 years at an interest rate of 8% compounded annually . Bill heard that he could calculate the compound amount and interest by using tables. In Situation 3, Bill learns how to do this. Again, Bill wants to know the value of $80 in 4 years at 8%. He begins by using Table 12.1 (p. 300). Looking at Table 12.1, Bill goes down the Period column to period 4, then across the row to the 8% column. At the intersection, Bill sees the number 1.3605. The marginal notes show how Bill arrived at the periods and rate. The 1.3605 table number means that $1 compounded at this rate will increase in value in 4 years to about $1.36. Do you recognize the $1.36? Figure 12.1 showed how $1 grew to $1.36. Since Bill wants to know the value of $80, he multiplies the dollar amount by the table factor as follows: $80.00

1.3605

$108.84

Principal Table factor Compound amount (future value) 8% Rate 8% 8% rate 1

Annual rate No. of times compounded in 1 year

Figure 12.2 (p. 300) illustrates this compounding procedure. We can say that compounding is a future value (FV) since we are looking into the future. Thus, $108.84 $80.00 $28.84 interest for 4 years at 8% compounded annually on $80.00 Now let’s look at two examples that illustrate compounding more than once a year Find the interest on $6,000 at 10% compounded semiannually for 5 years. calculate the interest as follows:

EXAMPLE

Periods 2 5 years 10 Rate 10% 2 5% 10 periods, 5%, in Table 12.1 1.6289 (table factor)

5

Off 1 cent due to rounding.

$6,000 1.6289

. We

$9,773.40 6,000.00 $3,773.40 interest

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Chapter 12 Compound Interest and Present Value

TABLE

12.1

Future value of $1 at compound interest

1%

112%

2%

3%

4%

5%

6%

7%

8%

1

1.0100

1.0150

1.0200

1.0300

1.0400

1.0500

1.0600

1.0700

1.0800

1.0900

1.1000

2

1.0201

1.0302

1.0404

1.0609

1.0816

1.1025

1.1236

1.1449

1.1664

1.1881

1.2100

3

1.0303

1.0457

1.0612

1.0927

1.1249

1.1576

1.1910

1.2250

1.2597

1.2950

1.3310

4

1.0406

1.0614

1.0824

1.1255

1.1699

1.2155

1.2625

1.3108

1.3605

1.4116

1.4641

5

1.0510

1.0773

1.1041

1.1593

1.2167

1.2763

1.3382

1.4026

1.4693

1.5386

1.6105

6

1.0615

1.0934

1.1262

1.1941

1.2653

1.3401

1.4185

1.5007

1.5869

1.6771

1.7716

7

1.0721

1.1098

1.1487

1.2299

1.3159

1.4071

1.5036

1.6058

1.7138

1.8280

1.9487

8

1.0829

1.1265

1.1717

1.2668

1.3686

1.4775

1.5938

1.7182

1.8509

1.9926

2.1436

9

1.0937

1.1434

1.1951

1.3048

1.4233

1.5513

1.6895

1.8385

1.9990

2.1719

2.3579

10

1.1046

1.1605

1.2190

1.3439

1.4802

1.6289

1.7908

1.9672

2.1589

2.3674

2.5937

11

1.1157

1.1780

1.2434

1.3842

1.5395

1.7103

1.8983

2.1049

2.3316

2.5804

2.8531

12

1.1268

1.1960

1.2682

1.4258

1.6010

1.7959

2.0122

2.2522

2.5182

2.8127

3.1384

13

1.1381

1.2135

1.2936

1.4685

1.6651

1.8856

2.1329

2.4098

2.7196

3.0658

3.4523

14

1.1495

1.2318

1.3195

1.5126

1.7317

1.9799

2.2609

2.5785

2.9372

3.3417

3.7975

15

1.1610

1.2502

1.3459

1.5580

1.8009

2.0789

2.3966

2.7590

3.1722

3.6425

4.1772

16

1.1726

1.2690

1.3728

1.6047

1.8730

2.1829

2.5404

2.9522

3.4259

3.9703

4.5950

17

1.1843

1.2880

1.4002

1.6528

1.9479

2.2920

2.6928

3.1588

3.7000

4.3276

5.0545

18

1.1961

1.3073

1.4282

1.7024

2.0258

2.4066

2.8543

3.3799

3.9960

4.7171

5.5599

19

1.2081

1.3270

1.4568

1.7535

2.1068

2.5270

3.0256

3.6165

4.3157

5.1417

6.1159

20

1.2202

1.3469

1.4859

1.8061

2.1911

2.6533

3.2071

3.8697

4.6610

5.6044

6.7275

21

1.2324

1.3671

1.5157

1.8603

2.2788

2.7860

3.3996

4.1406

5.0338

6.1088

7.4002

22

1.2447

1.3876

1.5460

1.9161

2.3699

2.9253

3.6035

4.4304

5.4365

6.6586

8.1403

23

1.2572

1.4084

1.5769

1.9736

2.4647

3.0715

3.8197

4.7405

5.8715

7.2579

8.9543

24

1.2697

1.4295

1.6084

2.0328

2.5633

3.2251

4.0489

5.0724

6.3412

7.9111

9.8497

25

1.2824

1.4510

1.6406

2.0938

2.6658

3.3864

4.2919

5.4274

6.8485

8.6231

10.8347

26

1.2953

1.4727

1.6734

2.1566

2.7725

3.5557

4.5494

5.8074

7.3964

9.3992

11.9182

27

1.3082

1.4948

1.7069

2.2213

2.8834

3.7335

4.8223

6.2139

7.9881

10.2451

13.1100

28

1.3213

1.5172

1.7410

2.2879

2.9987

3.9201

5.1117

6.6488

8.6271

11.1672

14.4210

29

1.3345

1.5400

1.7758

2.3566

3.1187

4.1161

5.4184

7.1143

9.3173

12.1722

15.8631

30

1.3478

1.5631

1.8114

2.4273

3.2434

4.3219

5.7435

7.6123

10.0627

13.2677

17.4494

Period

9%

10%

Note: For more detailed tables, see your reference booklet, the Business Math Handbook.

EXAMPLE Pam Donahue deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will be the balance of her account at the end of 5 years?

Periods ⫽ 4 ⫻ 5 years ⫽ 20 Rate ⫽ 6% ⫼ 4 ⫽ 112%

FIGURE

Compounding starts with the present and looks to the future

12.2 $

Compounding (FV) $80 Present value

$108.84 Future value

8% interest

0

1

2 Number of periods

3

4

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Learning Unit 12–1

301

20 periods, 112%, in Table 12.1 ⫽ 1.3469 (table factor) $8,000 ⫻ 1.3469 ⫽ $10,775.20 Next, let’s look at bank rates and how they af fect interest.

Bank Rates—Nominal versus Effective Rates (Annual Percentage Yield, or APY)

Interest

Portion Base ⫻ Rate ? Principal

Effective Rate

Banks often advertise their annual (nominal) interest rates and not their true or ef fective rate (annual percentage yield, or APY). This has made it difficult for investors and depositors to determine the actual rates of interest they were receiving. The Truth in Savings law forced savings institutions to reveal their actual rate of interest. The APY is defined in the Truth in Savings law as the percentage rate expressing the total amount of interest that would be received on a $100 deposit based on the annual rate and frequency of compounding for a 365-day period. As you can see from the advertisement on the left, banks now refer to the effective rate of interest as the annual percentage yield. Let’s study the rates of two banks to see which bank has the better return for the investor. Blue Bank pays 8% interest compounded quarterly on $8,000. Sun Bank offers 8% interest compounded semiannually on $8,000. The 8% rate is the nominal rate, or stated rate, on which the bank calculates the interest. To calculate the effective rate (annual percentage yield, or APY), however, we can use the following formula: Effective rate (APY)6 ⫽

Interest for 1 year Principal

Now let’s calculate the ef fective rate (APY) for Blue Bank and Sun Bank. Note the effective rates (APY) can be seen from Table 12.1 for $1: 1.0824 4 periods, 2% 1.0816 2 periods, 4%

Blue, 8% compounded quarterly

Sun, 8% compounded semiannually

Periods ⫽ 4 (4 ⫻ 1)

Periods ⫽ 2 (2 ⫻ 1)

Percent ⫽

8% ⫽ 2% 4

Percent ⫽

8% ⫽ 4% 2

Principal ⫽ $8,000

Principal ⫽ $8,000

Table 12.1 lookup: 4 periods, 2%

Table 12.1 lookup: 2 periods, 4% 1.0816 ⫻ $8,000 $8,652.80 ⫺ 8,000.00 $ 652.80

1.0824 ⫻ $8,000 Less $8,659.20 principal ⫺ 8,000.00 $ 659.20 Effective rate (APY) ⫽

$659.20 ⫽ .0824 $8,000 ⫽ 8.24%

$652.80 ⫽ .0816 $8,000 ⫽ 8.16%

Figure 12.3 (p. 302) illustrates a comparison of nominal and ef fective rates (APY) of interest. This comparison should make you question any advertisement of interest rates before depositing your money . Before concluding this unit, we briefly discuss compounding interest daily .

Compounding Interest Daily Although many banks add interest to each account quarterly , some banks pay interest that is compounded daily, and other banks use continuous compounding. Remember that 6

Round to the nearest hundredth percent as needed. In practice, the rate is often rounded to the nearest thousandth.

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Chapter 12 Compound Interest and Present Value

FIGURE

Beginning balance

12.3

Nominal rate of interest

Nominal and effective rates (APY) of interest compared $1,000

Compounding period

End balance

Effective rate (APY) of interest

Annual

$1,060.00

6.00%

Semiannual

$1,060.90

6.09%

Quarterly

$1,061.36

6.14%

Daily

$1,062.70

6.18%

+ 6%

continuous compounding sounds great, but in fact, it yields only a fraction of a percent more interest over a year than daily compounding. Today, computers perform these calculations. Table 12.2 is a partial table showing what $1 will grow to in the future by daily compounded interest, 360-day basis. For example, we can calculate interest compounded daily on $900 at 6% per year for 25 years as follows: $900 ⫻ 4.4811 ⫽ $4,032.99 daily compounding Now it’s time to check your progress with the following Practice Quiz.

LU 12–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1.

Complete the following without a table (round each calculation to the nearest cent as needed): Rate of Number of compound periods to be Total Total Principal Time interest Compounded compounded amount interest $200 1 year 8% Quarterly a. b. c.

2. 3.

Solve the previous problem by using compound value (FV) in Table 12.1. Lionel Rodgers deposits $6,000 in Victory Bank, which pays 3% interest compounded semiannually. How much will Lionel have in his account at the end of 8 years? Find the ef fective rate (APY) for the year: principal, $7,000; interest rate, 12%; and compounded quarterly. Calculate by Table 12.2 what $1,500 compounded daily for 5 years will grow to at 7%.

DVD

4. 5.

TABLE

12.2

Interest on a $1 deposit compounded daily—360-day basis

Number of years

6.00%

6.50%

7.00%

7.50%

8.00%

8.50%

9.00%

9.50%

1

1.0618

1.0672

1.0725

1.0779

1.0833

1.0887

1.0942

1.0996

1.1052

2

1.1275

1.1388

1.1503

1.1618

1.1735

1.1853

1.1972

1.2092

1.2214

3

1.1972

1.2153

1.2337

1.2523

1.2712

1.2904

1.3099

1.3297

1.3498

4

1.2712

1.2969

1.3231

1.3498

1.3771

1.4049

1.4333

1.4622

1.4917

5

1.3498

1.3840

1.4190

1.4549

1.4917

1.5295

1.5682

1.6079

1.6486

6

1.4333

1.4769

1.5219

1.5682

1.6160

1.6652

1.7159

1.7681

1.8220

7

1.5219

1.5761

1.6322

1.6904

1.7506

1.8129

1.8775

1.9443

2.0136

8

1.6160

1.6819

1.7506

1.8220

1.8963

1.9737

2.0543

2.1381

2.2253

10.00%

9

1.7159

1.7949

1.8775

1.9639

2.0543

2.1488

2.2477

2.3511

2.4593

10

1.8220

1.9154

2.0136

2.1168

2.2253

2.3394

2.4593

2.5854

2.7179

15

2.4594

2.6509

2.8574

3.0799

3.3197

3.5782

3.8568

4.1571

4.4808

20

3.3198

3.6689

4.0546

4.4810

4.9522

5.4728

6.0482

6.6842

7.3870

25

4.4811

5.0777

5.7536

6.5195

7.3874

8.3708

9.4851

10.7477

12.1782

30

6.0487

7.0275

8.1645

9.4855

11.0202

12.8032

14.8747

17.2813

20.0772

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Learning Unit 12–2

✓

Solutions

2. 3. 4.

a. 4 (4 ⫻ 1) b. $216.48 c. $16.48 ($216.48 ⫺ $200) $200 ⫻ 1.02 ⫽ $204 ⫻ 1.02 ⫽ $208.08 ⫻ 1.02 ⫽ $212.24 ⫻ 1.02 ⫽ $216.48 $200 ⫻ 1.0824 ⫽ $216.48 (4 periods, 2%) 16 periods, 1 12%, $6,000 ⫻ 1.2690 ⫽ $7,614 4 periods, 3%, $7,000 ⫻ 1.1255 ⫽ $7,878.50 $878.50 ⫽ 12.55% ⫺ 7,000.00 $7,000.00 $ 878.50

5.

$1,500 ⫻ 1.4190 ⫽ $2,128.50

1.

Check out the plastic overlays that appear within Chapter 13 to review these concepts.

LU 12–1a

303

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 308)

1.

Complete the following without a table (round each calculation to the nearest cent as needed): Rate of Number of compound periods to be Total Total Principal Time interest Compounded compounded amount interest $500 1 year 8% Quarterly a. b. c.

2. 3.

Solve the previous problem by using compound value (FV). See Table 12.1. Lionel Rodgers deposits $7,000 in Victory Bank, which pays 4% interest compounded semiannually. How much will Lionel have in his account at the end of 8 years? Find the effective rate (APY) for the year: principal, $8,000; interest rate, 6%; and compounded quarterly. Round to the nearest hundredth percent. Calculate by Table 12.2 what $1,800 compounded daily for 5 years will grow to at 6%.

4. 5.

Learning Unit 12–2: Present Value—The Big Picture Figure 12.1 (p. 296) in Learning Unit 12–1 showed how by compounding, the future value of $1 became $1.36. This learning unit discusses present value. Before we look at specific calculations involving present value, let’ s look at the concept of present value. Figure 12.4 shows that if we invested 74 cents today , compounding would cause the 74 cents to grow to $1 in the future. For example, let’ s assume you ask this question: “If I need $1 in 4 years in the future, how much must I put in the bank today (assume an 8% annual interest)?” To answer this question, you must know the present value of that $1 today. From Figure 12.4, you can see that the present value of $1 is .7350. Remember that the $1 is only worth 74 cents if you wait 4 periods to receive it. This is one reason why so many athletes get such big contracts—much of the money is paid in later years when it is not worth as much. FIGURE

Present value goes from the future value to the present value

12.4

Present value of $1 at 8% for four periods

$ 1.20 1.10 1.00 .90 .80 .70 .60 .50 .40 .30 .20 .10 .00

Present value

$.9259

$.7350

$.7938

0

1

Future value $1.0000

$.8573

2 Number of periods

3

4

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Chapter 12 Compound Interest and Present Value

Present value starts with the future and looks to the present

12.5

FIGURE

$

Present value $80 Present value

8% interest

0

1

$108.84 Future value

2 3 Number of periods

4

Relationship of Compounding (FV) to Present Value (PV)—The Bill Smith Example Continued In Learning Unit 12–1, our consideration of compounding started in the present ($80) and looked to find the future amount of $108.84. Present value (PV) starts with the future and tries to calculate its worth in the present ($80). For example, in Figure 12.5, we assume Bill Smith knew that in 4 years he wanted to buy a bike that cost $108.84 (future). Bill’s bank pays 8% interest compounded annually . How much money must Bill put in the bank today (present) to have $108.84 in 4 years? To work from the future to the present, we can use a present value (PV) table. In the next section you will learn how to use this table. RF/Corbis

How to Use a Present Value (PV) Table7 To calculate present value with a present value table, use the following steps: CALCULATING PRESENT VALUE BY TABLE LOOKUP

Periods 4 ⫻ No. of years

1

⫽

No. of times compounded in 1 year

4

Step 1.

Find the periods: Years multiplied by number of times interest is compounded in 1 year.

Step 2.

Find the rate: Annual rate divided by numbers of times interest is compounded in 1 year.

Step 3.

Go down the Period column of the table to the number of periods desired; look across the row to find the rate. At the intersection of the two columns is the table factor for the compound value of $1.

Step 4.

Multiply the table factor times the future value. This gives the present value.

Table 12.3 is a present value (PV) table that tells you what $1 is worth today at different interest rates. To continue our Bill Smith example, go down the Period column in Table 12.3 to 4. Then go across to the 8% column. At 8% for 4 periods, we see a table factor of .7350. This means that $1 in the future is worth approximately 74 cents today . If Bill invested 74 cents today at 8% for 4 periods, Bill would have $1. Since Bill knows the bike will cost $108.84 in the future, he completes the following calculation: $108.84 ⫻ .7350 ⫽ $80.00 This means that $108.84 in today’ s dollars is worth $80.00. Now let’ s check this.

A , where A equals future amount (compound amount), N equals number of com(1 ⫹ i ) N pounding periods, and i equals interest rate per compounding period. The calculator sequence for Bill Smith would be as follows: 1 ⫹ .08 yx 4 ⫽ M⫹ 108.84 ⫼ MR ⫽ 80.03. 7

The formula for present value is PV ⫽

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Learning Unit 12–2

TABLE

12.3

305

Present value of $1 at end period

1%

112%

2%

3%

4%

5%

6%

7%

8%

9%

10%

1

.9901

.9852

.9804

.9709

.9615

.9524

.9434

.9346

.9259

.9174

.9091

2

.9803

.9707

.9612

.9426

.9246

.9070

.8900

.8734

.8573

.8417

.8264

3

.9706

.9563

.9423

.9151

.8890

.8638

.8396

.8163

.7938

.7722

.7513

4

.9610

.9422

.9238

.8885

.8548

.8227

.7921

.7629

.7350

.7084

.6830

5

.9515

.9283

.9057

.8626

.8219

.7835

.7473

.7130

.6806

.6499

.6209

6

.9420

.9145

.8880

.8375

.7903

.7462

.7050

.6663

.6302

.5963

.5645

7

.9327

.9010

.8706

.8131

.7599

.7107

.6651

.6227

.5835

.5470

.5132

8

.9235

.8877

.8535

.7894

.7307

.6768

.6274

.5820

.5403

.5019

.4665

9

.9143

.8746

.8368

.7664

.7026

.6446

.5919

.5439

.5002

.4604

.4241

10

.9053

.8617

.8203

.7441

.6756

.6139

.5584

.5083

.4632

.4224

.3855

11

.8963

.8489

.8043

.7224

.6496

.5847

.5268

.4751

.4289

.3875

.3505

12

.8874

.8364

.7885

.7014

.6246

.5568

.4970

.4440

.3971

.3555

.3186

13

.8787

.8240

.7730

.6810

.6006

.5303

.4688

.4150

.3677

.3262

.2897

14

.8700

.8119

.7579

.6611

.5775

.5051

.4423

.3878

.3405

.2992

.2633

15

.8613

.7999

.7430

.6419

.5553

.4810

.4173

.3624

.3152

.2745

.2394

16

.8528

.7880

.7284

.6232

.5339

.4581

.3936

.3387

.2919

.2519

.2176

17

.8444

.7764

.7142

.6050

.5134

.4363

.3714

.3166

.2703

.2311

.1978

18

.8360

.7649

.7002

.5874

.4936

.4155

.3503

.2959

.2502

.2120

.1799

19

.8277

.7536

.6864

.5703

.4746

.3957

.3305

.2765

.2317

.1945

.1635

20

.8195

.7425

.6730

.5537

.4564

.3769

.3118

.2584

.2145

.1784

.1486

21

.8114

.7315

.6598

.5375

.4388

.3589

.2942

.2415

.1987

.1637

.1351

22

.8034

.7207

.6468

.5219

.4220

.3418

.2775

.2257

.1839

.1502

.1228

23

.7954

.7100

.6342

.5067

.4057

.3256

.2618

.2109

.1703

.1378

.1117

24

.7876

.6995

.6217

.4919

.3901

.3101

.2470

.1971

.1577

.1264

.1015

25

.7798

.6892

.6095

.4776

.3751

.2953

.2330

.1842

.1460

.1160

.0923

26

.7720

.6790

.5976

.4637

.3607

.2812

.2198

.1722

.1352

.1064

.0839

27

.7644

.6690

.5859

.4502

.3468

.2678

.2074

.1609

.1252

.0976

.0763

28

.7568

.6591

.5744

.4371

.3335

.2551

.1956

.1504

.1159

.0895

.0693

29

.7493

.6494

.5631

.4243

.3207

.2429

.1846

.1406

.1073

.0822

.0630

30

.7419

.6398

.5521

.4120

.3083

.2314

.1741

.1314

.0994

.0754

.0573

35

.7059

.5939

.5000

.3554

.2534

.1813

.1301

.0937

.0676

.0490

.0356

40

.6717

.5513

.4529

.3066

.2083

.1420

.0972

.0668

.0460

.0318

.0221

Period

Note: For more detailed tables, see your booklet, the Business Math Handbook.

Comparing Compound Interest (FV) Table 12.1 with Present Value (PV) Table 12.3 We know from our calculations that Bill needs to invest $80 for 4 years at 8% compound interest annually to buy his bike. We can check this by going back to Table 12.1 and comparing it with Table 12.3. Let’ s do this now .

Compound value Table 12.1 Present value

Table 12.1 1.3605

$80.00

Present value Table 12.3 Future value

Table 12.3

$108.84

.7350

Future value

$108.84

Present value

(4 per., 8%)

(4 per., 8%)

We know the present dollar amount and find what the dollar amount is worth in the future.

We know the future dollar amount and find what the dollar amount is worth in the present.

$80.00

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Chapter 12 Compound Interest and Present Value

FIGURE

The present value is what we need now to have $20,000 in the future

12.6

$

Present value $14,568 Present value

$20,000 Future value

0

2 3 Number of years

1

4

Note that the table factor for compounding is over 1 (1.3605) and the table factor for present value is less than 1 (.7350). The compound value table starts with the present and goes to the future. The present value table starts with the future and goes to the present. Let’s look at another example before trying the Practice Quiz. EXAMPLE Rene Weaver needs $20,000 for college in 4 years. She can earn 8% compounded quarterly at her bank. How much must Rene deposit at the beginning of the year to have $20,000 in 4 years? Remember that in this example the bank compounds the interest quarterly. Let’s first determine the period and rate on a quarterly basis:

Periods ⫽ 4 ⫻ 4 years ⫽ 16 periods

Rate ⫽

8% ⫽ 2% 4

Now we go to Table 12.3 and find 16 under the Period column. We then move across to the 2% column and find the .7284 table factor. $20,000 ⫻ .7284 ⫽ $14,568 (future value)

(present value)

We illustrate this in Figure 12.6. We can check the $14,568 present value by using the compound value

Table 12.1:

8

16 periods, 2% column ⫽ 1.3728 ⫻ $14,568 ⫽ $19,998.95

Let’s test your understanding of this unit with the Practice Quiz.

LU 12–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Use the present value Table 12.3 to complete:

1. 2. 3. 4.

✓

Future amount Length Rate Table Rate PV PV desired of time compounded period used factor amount $ 7,000 6 years 6% semiannually _____ _____ _____ _____ $15,000 20 years 10% annually _____ _____ _____ _____ Bill Blum needs $20,000 6 years from today to attend V.P.R. Tech. How much must Bill put in the bank today (12% quarterly) to reach his goal? Bob Fry wants to buy his grandson a Ford Taurus in 4 years. The cost of a car will be $24,000. Assuming a bank rate of 8% compounded quarterly , how much must Bob put in the bank today?

Solutions

1. 2.

12 periods (6 years ⫻ 2) 20 periods (20 years ⫻ 1)

3.

6 years ⫻ 4 ⫽ 24 periods

4.

4 ⫻ 4 years ⫽ 16 periods

8

3% (6% ⫼ 2) 10% (10% ⫼ 1) 12% ⫽ 3% 4 8% ⫽ 2% 4

Not quite $20,000 due to rounding of table factors.

.7014 $4,909.80 ($7,000 ⫻ .7014) .1486 $2,229.00 ($15,000 ⫻ .1486) .4919 ⫻ $20,000 ⫽ $9,838 .7284 ⫻ $24,000 ⫽ $17,481.60

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

LU 12–2a

307

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 308)

Use the Business Math Handbook to complete:

1. 2. 3. 4.

Future amount Length Rate Table Rate PV PV desired of time compounded period used factor amount $ 9,000 7 years 5% semiannually _____ _____ _____ _____ $20,000 20 years 4% annually _____ _____ _____ _____ Bill Blum needs $40,000 6 years from today to attend V.P.R. Tech. How much must Bill put in the bank today (8% quarterly) to reach his goal? Bob Fry wants to buy his grandson a Ford Taurus in 4 years. The cost of a car will be $28,000. Assuming a bank rate of 4% compounded quarterly , how much must Bob put in the bank today?

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Calculating compound amount without tables (future value),* p. 298

Determine new amount by multiplying rate times new balance (that includes interest added on). Start in present and look to future. Compound Compound ⫽ ⫺ Principal interest amount Compounding PV FV

$100 in savings account, compounded annually for 2 years at 8%: $108 $100 ⫻ 1.08 ⫻ 1.08 $108 $116.64 (future value)

Calculating compound amount (future value) by table lookup, p. 299

Periods ⫽ Rate ⫽

Number of times Years of compounded ⫻ loan per year

Annual rate Number of times compounded per year

Multiply table factor (intersection of period and rate) times amount of principal. Effective rate (APY), p. 301

Effective rate (APY) ⫽

Interest for 1 year Principal

or Rate can be seen in Table 12.1 factor.

*A ⫽ P (1 ⫹ i ) N.

Example: $2,000 @ 12% 5 years compounded quarterly: Periods ⫽ 4 ⫻ 5 years ⫽ 20 12% Rate ⫽ ⫽ 3% 4 20 periods, 3% ⫽ 1.8061 (table factor) $2,000 ⫻ 1.8061 ⫽ $3,612.20 (future value) $1,000 at 10% compounded semiannually for 1 year. By Table 12.1: 2 periods, 5% 1.1025 means at end of year investor has earned 110.25% of original principal. Thus the interest is 10.25%. $1,000 ⫻ 1.1025 ⫽ $1,102.50 ⫺ 1,000.00 $ 102.50 $102.50 ⫽ 10.25% $1,000 effective rate (APY)

(continues)

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Chapter 12 Compound Interest and Present Value

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Calculating present value (PV) with table lookup*, p. 304

Start with future and calculate worth in the present. Periods and rate computed like in compound interest. Present value PV FV Find periods and rate. Multiply table factor (intersection of period and rate) times amount of loan.

Example: Want $3,612.20 after 5 years with rate of 12% compounded quarterly:

KEY TERMS

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

Annual percentage yield (APY), p. 301 Compound amount, p. 296 Compounded annually, p. 297 Compounded daily, p. 297 Compounded monthly, p. 297

1. 2. 3. 4. 5.

Periods ⫽ 4 ⫻ 5 ⫽ 20; % ⫽ 3% By Table 12.3: 20 periods, 3% ⫽ .5537 $3,612.20 ⫻ .5537 ⫽ $2,000.08 Invested today will yield desired amount in future

Compounded quarterly, p. 297 Compounded semiannually, p. 297 Compounding, p. 296 Compound interest, p. 296 Effective rate, p. 301

LU 12–1a (p. 303) 4 periods; Int. ⫽ $41.22; $541.21 $541.20 $9,609.60 6.14% $2,429.64

Future value (FV), p. 296 Nominal rate, p. 301 Number of periods, p. 297 Present value (PV), p. 296 Rate for each period, p. 297

LU 12–2a (p. 307) 1. 2. 3. 4.

$6,369.30 $9,128 $24,868 $23,878.40

A if table not used. (1 ⫹ i ) N

*

Critical Thinking Discussion Questions 1. Explain how periods and rates are calculated in compounding problems. Compare simple interest to compound interest. 2. What are the steps to calculate the compound amount by table? Why is the compound table factor greater than $1?

3. What is the effective rate (APY)? Why can the effective rate be seen directly from the table factor? 4. Explain the dif ference between compounding and present value. Why is the present value table factor less than $1?

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Complete the following without using Table 12.1 (round to the nearest cent for each calculation) and then check by (check will be off due to rounding).

Principal 12–1. $1,400

Time (years)

Rate of compound interest

Compounded

2

4%

Semiannually

Periods

Rate

Total amount

Table 12.1

Total interest

Complete the following using compound future value Table 12.1: Time

Principal

Rate

Compounded

12–2. 9 years

$10,000

3%

Annually

12–3. 6 months

$10,000

8%

Quarterly

12–4. 3 years

$2,000

12%

Semiannually

Amount

Interest

Calculate the effective rate (APY) of interest for 1 year . 12–5. Principal: $15,500 Interest rate: 12% Compounded quarterly Effective rate (APY): 12–6. Using Table 12.2, calculate what $700 would grow to at 6 12% per year compounded daily for 7 years. Complete the following using present value of Table 12.3 or Business Math Handbook Table. Amount desired at end of period

On PV Table 12.3 Length of time

Rate

Compounded

12–7. $4,500

7 years

2%

Semiannually

12–8. $8,900

4 years

6%

Monthly

12–9. $17,600

7 years

12%

Quarterly

12–10. $20,000

20 years

8%

Annually

Period used

Rate used

PV factor used

PV of amount desired at end of period

309

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12–11. Check your answer in Problem 12–9 by the compound value Table 12.1. The answer will be off due to rounding.

WORD PROBLEMS 12–12. Savings plans and the cost of college attendance were discussed in the September 18, 2006 issue of U.S. News & World Report. Greg Lawrence anticipates he will need approximately $218,000 in 15 years to cover his 3 year old daughter ’s college bills for a 4 year degree. How much would he have to invest today , at an interest rate of 8 percent compounded semiannually?

12–13. Jennifer Toby, owner of a local Subway shop, loaned $25,000 to Mike Roy to help him open a Subway franchise. Mike plans to repay Jennifer at the end of 7 years with 4% interest compounded semiannually . How much will Jennifer receive at the end of 7 years?

12–14. Molly Slate deposited $35,000 at Quazi Bank at 6% interest compounded quarterly . What is the effective rate (APY) to the nearest hundredth percent?

12–15. Melvin Indecision has difficulty deciding whether to put his savings in Mystic Bank or Four Rivers Bank. Mystic of fers 10% interest compounded semiannually. Four Rivers offers 8% interest compounded quarterly. Melvin has $10,000 to invest. He expects to withdraw the money at the end of 4 years. Which bank gives Melvin the better deal? Check your answer.

12–16. Brian Costa deposited $20,000 in a new savings account at 12% interest compounded semiannually . At the beginning of year 4, Brian deposits an additional $30,000 at 12% interest compounded semiannually . At the end of 6 years, what is the balance in Brian’s account?

12–17. Lee Wills loaned Audrey Chin $16,000 to open a hair salon. After 6 years, Audrey will repay Lee with 8% interest compounded quarterly. How much will Lee receive at the end of 6 years?

12–18. The Dallas Morning News on June 12, 2006, reported on saving for retirement. Carl Hendrik is 56 years old and has worked for Texas Instruments Inc for 35 years. He has amassed a plump nest egg of $700,000. His bank compounds interest semiannually, at 6%. Carl plans to retire at 65, if he places his money in the bank, how much will his investment be worth at retirement?

310

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Page 311

12–19. John Roe, an employee of The Gap, loans $3,000 to another employee at the store. He will be repaid at the end of 4 years with interest at 6% compounded quarterly. How much will John be repaid? 12–20. On September 14, 2006 USA Today ran a story on funding for retirement. The average 65 year old woman can expect to live to nearly 87 according to the American Academy of Actuaries. Mary Tully is 40 years old. She expects to need at least $420,000 when she retires at age 65. How much money must she invest today , in an account paying 6% interest compounded annually, to have the amount of money she needs?

12–21. Security National Bank is quoting 1-year certificates of deposits with an interest rate of 5% compounded semiannually . Joe Saver purchased a $5,000 CD. What is the CD’s effective rate (APY) to the nearest hundredth percent? Use tables in the Business Math Handbook.

12–22. Jim Jones, an owner of a Bur ger King restaurant, assumes that his restaurant will need a new roof in 7 years. He estimates the roof will cost him $9,000 at that time. What amount should Jim invest today at 6% compounded quarterly to be able to pay for the roof? Check your answer .

12–23. Tony Ring wants to attend Northeast College. He will need $60,000 4 years from today . Assume Tony’s bank pays 12% interest compounded semiannually. What must Tony deposit today so he will have $60,000 in 4 years?

12–24. Could you check your answer (to the nearest dollar) in Problem 12–23 by using the compound value Table 12.1? The answer will be slightly off due to rounding. 12–25. Pete Air wants to buy a used Jeep in 5 years. He estimates the Jeep will cost $15,000. Assume Pete invests $10,000 now at 12% interest compounded semiannually. Will Pete have enough money to buy his Jeep at the end of 5 years?

12–26. Lance Jackson deposited $5,000 at Basil Bank at 9% interest compounded daily . What is Lance’s investment at the end of 4 years? 12–27. Paul Havlik promised his grandson Jamie that he would give him $6,000 8 years from today for graduating from high school. Assume money is worth 6% interest compounded semiannually . What is the present value of this $6,000?

12–28. Earl Ezekiel wants to retire in San Diego when he is 65 years old. Earl is now 50. He believes he will need $300,000 to retire comfortably. To date, Earl has set aside no retirement money . Assume Earl gets 6% interest compounded semiannually. How much must Earl invest today to meet his $300,000 goal?

311

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12–29. Lorna Evenson would like to buy a $19,000 car in 4 years. Lorna wants to put the money aside now . Lorna’s bank offers 8% interest compounded semiannually. How much must Lorna invest today?

12–30. John Smith saw the following advertisement. Could you show him how $88.77 was calculated?

9-Month CD

6.05

%Annual* Percentage Yield

*As of January 31, 200X, and subject to change. Interest on the 9-month CD is credited on the maturity date and is not compounded. For example, a $2,000, 9-month CD on deposit for an interest rate of 6.00% (6.05% APY) will earn $88.77 at maturity. Withdrawals prior to maturity require the consent of the bank and are subject to a substantial penalty. There is $500 minimum deposit for IRA, SEP IRA, and Keogh CDs (except for 9-month CD for which the minimum deposit is $1,000). There is $1,000 minimum deposit for all personal CDs (except for 9-month CD for which the minimum deposit is $2,000). Offer not valid on jumbo CDs.

CHALLENGE PROBLEMS 12–31. Mary started her first job at 22. She began saving money immediately but stopped after five years. Mary invested $2,500 each year until age 27. She receives 10% interest compounded annually and plans to retire at 62. (a) What amount will Mary have when she reaches retirement age? Use the tables in the Business Math Handbook. (b) What is the total amount of interest she will have received?

12–32. You are the financial planner for Johnson Controls. Last year ’s profits were $700,000. The board of directors decided to forgo dividends to stockholders and retire high-interest outstanding bonds that were issued 5 years ago at a face value of $1,250,000. You have been asked to invest the profits in a bank. The board must know how much money you will need from the profits earned to retire the bonds in 10 years. Bank A pays 6% compounded quarterly, and Bank B pays 6 12% compounded annually. Which bank would you recommend, and how much of the company’ s profit should be placed in the bank? If you recommended that the remaining money not be distributed to stockholders but be placed in Bank B, how much would the remaining money be worth in 10 years? Use tables in the Business Math Handbook.* Round final answer to nearest dollar.

*Check glossary for unfamiliar terms.

312

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DVD SUMMARY PRACTICE TEST 1.

Mia Kaminsky, owner of a Starbucks franchise, loaned $40,000 to Lee Reese to help him open a new flower shop online. Lee plans to repay Mia at the end of 5 years with 4% interest compounded semiannually . How much will Mia receive at the end of 5 years? (p. 299)

2.

Joe Beary wants to attend Riverside College. Eight years from today he will need $50,000. If Joe’ s bank pays 6% interest compounded semiannually, what must Joe deposit today to have $50,000 in 8 years? (p. 304)

3.

Shelley Katz deposited $30,000 in a savings account at 5% interest compounded semiannually . At the beginning of year 4, Shelley deposits an additional $80,000 at 5% interest compounded semiannually . At the end of 6 years, what is the balance in Shelley’s account? (p. 299)

4.

Earl Miller, owner of a Papa Gino’s franchise, wants to buy a new delivery truck in 6 years. He estimates the truck will cost $30,000. If Earl invests $20,000 now at 5% interest compounded semiannually , will Earl have enough money to buy his delivery truck at the end of 6 years? (pp. 299, 304)

5.

Minnie Rose deposited $16,000 in Street Bank at 6% interest compounded quarterly . What was the effective rate (APY)? Round to the nearest hundredth percent. (p. 301)

6.

Lou Ling, owner of Lou’s Lube, estimates that he will need $70,000 for new equipment in 7 years. Lou decided to put aside money today so it will be available in 7 years. Reel Bank of fers Lou 6% interest compounded quarterly. How much must Lou invest to have $70,000 in 7 years? (p. 304)

7.

Bernie Long wants to retire to California when she is 60 years of age. Bernie is now 40. She believes that she will need $900,000 to retire comfortably. To date, Bernie has set aside no retirement money . If Bernie gets 8% compounded semiannually, how much must Bernie invest today to meet her $900,000 goal? (p. 304)

8.

Sam Slater deposited $19,000 in a savings account at 7% interest compounded daily . At the end of 6 years, what is the balance in Sam’s account? (p. 301)

313

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Personal Finance A KIPLINGER APPROACH

FA Q

CREDIT

| Some card issuers offer help to the shopping-addicted and the savings-impaired.

Keep the CHANGE

A

s a nat i o n, we’re big spenders, not savers. So it figures that banks would invent a way for us to do both at once. Buy something using one of the new cards from American Express, Bank of America and a handful of other issuers, and the banks will stash a cash rebate into a savings account. Shopping and traveling won’t replace your IRA contributions, but if you use plastic for gas and groceries, the money can add up. American Express’s One card funnels 1% of all purchases into a savings account that now pays 3.5%. At that rate, if you charge $2,000 a month, you’ll have $6,000 in 18 years—not enough for your child’s college tuition, but maybe enough for books. Amex will waive the $35 annual fee the first year and seed your account with $25. Bank of America effec-

tively puts your pocket change into an electronic piggy bank. Sign up for its Keep the Change program and the bank rounds up all purchases on your debit card to the nearest dollar and moves the difference into a savings account. For three months, the bank matches your deposits 100%. After that, it matches 5% per year up to $250. That’s no windfall. But look at it this way: A penny spent becomes a penny saved. —JOAN GOLDWASSER

| Should I buy INSURANCE that pays

I A N T S T E P I N C /G E T T Y I M A G E S

off my mortgage if I die or I’m disabled?

Kiplinger © 2006

No. It may be tempting if you are stretching to make the mortgage payment, but refrain. You’ll pay less for term life insurance, which lets beneficiaries choose how to use the money. They may not want to pay off theISSUE mortgage right away. Likewise, long-term-disability insurance BUSINESS MATH makes more sense It helps cover all your bills not just one Keep the change is a gimmick that banks are using just to get new customers. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

314

ALISON SEIFFER

S C O G I N M AYO

if they pay several thousand metal roof, and walls that were built by injecting condollars more—an extra $50 crete into Styrofoam forms. a month on the mortgage— The new house earned a they save $100 on utilities. five-star rating from Austin Other builders are hopEnergy’s Green Building ping on this bandwagon, Program. Last fall, when a partly to meet popular heat wave hit and the city set power-demand records, Moore used half as much electricity as the typical Austin-area homeowner. High electricity and heating costs have revived interest in energy efficiency. But the movement toward green homes is also spreading because builders can put up normal-looking homes and still G Glenn Moore’s “green” home fits the neighborhood. incorporate energy-saving features, such as demand and partly to forehome-monitoring technolostall being forced by law to gy and on-demand water build more-energy-efficient heaters. Jim Petersen, direc- houses. National Association tor of research and developof Home Builders spokesment for Pulte Homes, says man John Loyer says in ten green is hot in the Southto 15 years, building green west. Buyers accept higher may just be called building. —PAT MERTZ ESSWEIN costs, he adds, because even

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A Go to Web and find out the latest rates for 6 months, 1 year, and 5-year CDs along with the current rates for markets.

2006 Wall Street Journal ©

eb site e text W The e S : s t t Projec /slater9e) and Guide. Interne he.com rnet Resource h .m w (ww Inte ss Math Busine

315

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Chapter

Page 316

13

Annuities and Sinking Funds

LEARNING UNIT OBJECTIVES Note: A complete set of plastic overlays showing the concept of annuities is found at the end of the chapter (p. 336A). LU 13–1: Annuities: Ordinary Annuity and Annuity Due (Find Future Value) • Differentiate between contingent annuities and annuities certain (p. 318). • Calculate the future value of an ordinary annuity and an annuity due manually and by table lookup (pp. 319–323).

LU 13–2: Present Value of an Ordinary Annuity (Find Present Value) • Calculate the present value of an ordinary annuity by table lookup and manually check the calculation (pp. 323–325). • Compare the calculation of the present value of one lump sum versus the present value of an ordinary annuity (p. 325).

Wall Street Jo urnal © 2005

LU 13–3: Sinking Funds (Find Periodic Payments) • Calculate the payment made at the end of each period by table lookup (pp. 326–327). • Check table lookup by using ordinary annuity table (p. 327 ).

Wall Street Jo urnal © 2005

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317

$1

,2

87

Learning Unit 13–1

INVESTING YOUR SAVINGS Assuming the price of coffee remains the same, we added up what you would save if you gave up coffee over 30 years and what you would save if you made coffee at home instead of buying it. We then invested the savings. We compounded each amount weekly at annual rates: 0 percent, which means you did nothing with the money; at 6 percent, which is an average expected rate of return on a stock portfolio, and at 10 percent, an aggressive expected rate of return.

$1,

209

$

In 30 years... 0% annual returns

$

$

$

6% annual returns

$

$

$108,000

$

$

10% annual returns

$

$

$

$36,270 $101,600 $230,000

$

$

$

$38,610

$

$

$245,000

Boston Sunday Globe © 2004

Lisa Poole/AP Wide World

A Boston Globe article entitled “Cost of Living: A Cup a Day” states at the beginning of the clipping that each month the Globe runs a feature on an everyday expense to see how much it costs an average person. Since many people are cof fee drinkers, the Globe assumed that a person drank 3 cups a day of Dunkin’ Donuts coffee at the cost of $1.65 a cup. For a five-day week, the person would spend $1,287 annually (52 weeks). If the person brewed the cof fee at home, the cost of the beans per cup would be $0.10 a cup with an annual expense of $78, saving $1,209 over the Dunkin’ Donuts coffee. If a person gave up drinking cof fee, the person would save $1,287. The clipping continued with the discussion on “Investing Your Savings” shown above. Note how much you would have in 30 years if you invested your money in 0%, 6%, and 10% annual returns. Using the magic of compounding, if you saved $1,287 a year , your money could grow to a quarter of a million dollars. This chapter shows how to compute compound interest that results from a stream of payments, or an annuity. Chapter 12 showed how to calculate compound interest on a lumpsum payment deposited at the beginning of a particular time. Knowing how to calculate interest compounding on a lump sum will make the calculation of interest compounding on annuities easier to understand. We begin the chapter by explaining the dif ference between calculating the future value of an ordinary annuity and an annuity due. Then you learn how to find the present value of an ordinary annuity . The chapter ends with a discussion of sinking funds.

Learning Unit 13–1: Annuities: Ordinary Annuity and Annuity Due (Find Future Value) Many parents of small children are concerned about being able to af ford to pay for their children’s college educations. Some parents deposit a lump sum in a financial institution when the child is in diapers. The interest on this sum is compounded until the child is 18, when the parents withdraw the money for college expenses. Parents could also fund their children’s educations with annuities by depositing a series of payments for a certain time. The concept of annuities is the first topic in this learning unit.

Concept of an Annuity—The Big Picture All of us would probably like to win $1 million in a state lottery . What happens when you have the winning ticket? You take it to the lottery headquarters. When you turn in the ticket, do you immediately receive a check for $1 million? No. Lottery payof fs are not usually made in lump sums. Lottery winners receive a series of payments over a period of time—usually years. This stream of payments is an annuity. By paying the winners an annuity , lotteries do not actually spend $1 million. The lottery deposits a sum of money in a financial institution.

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Chapter 13 Annuities and Sinking Funds

FIGURE

13.1

Future value of an annuity of $1 at 8%

$3.50 3.00 2.50 2.00 1.50 1.00 .50

$3.2464

$2.0800 $1.00

1

2 End of period

3

The continual growth of this sum through compound interest provides the lottery winner with a series of payments. When we calculated the maturity value of a lump-sum payment in Chapter 12, the maturity value was the principal and its interest. Now we are looking not at lumpsum payments but at a series of payments (usually of equal amounts over regular payment periods) plus the interest that accumulates. So the future value of an annuity is the future dollar amount of a series of payments plus interest. 1 The term of the annuity is the time from the beginning of the first payment period to the end of the last payment period. The concept of the future value of an annuity is illustrated in Figure 13.1. Do not be concerned about the calculations (we will do them soon). Let’ s first focus on the big picture of annuities. In Figure 13.1 we see the following: At end of period 1: At end of period 2:

Sharon Hoogstraten

At end of period 3:

The $1 is still worth $1 because it was invested at the end of the period. An additional $1 is invested. The $2.00 is now worth $2.08. Note the $1 from period 1 earns interest but not the $1 invested at the end of period 2. An additional $1 is invested. The $3.00 is now worth $3.25 . Remember that the last dollar invested earns no interest.

Before learning how to calculate annuities, you should understand the two classifications of annuities.

How Annuities Are Classified Annuities have many uses in addition to lottery payof fs. Some of these uses are insurance companies’ pension installments, Social Security payments, home mortgages, businesses paying off notes, bond interest, and savings for a vacation trip or college education. Annuities are classified into two major groups: contingent annuities and annuities certain. Contingent annuities have no fixed number of payments but depend on an uncertain event (e.g., life insurance payments that cease when the insured dies). Annuities certain have a specific stated number of payments (e.g., mortgage payments on a home). Based on the time of the payment, we can divide each of these two major annuity groups into the following: 1.

2.

Ordinary annuity—regular deposits (payments) made at the end of the period. Periods could be months, quarters, years, and so on. An ordinary annuity could be salaries, stock dividends, and so on. Annuity due—regular deposits (payments) made at the beginning of the period, such as rent or life insurance premiums.

The remainder of this unit shows you how to calculate and check ordinary annuities and annuities due. Remember that you are calculating the dollar amount of the annuity at the end of the annuity term or at the end of the last period.

1

The term amount of an annuity has the same meaning as future value of an annuity.

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Page 319

Learning Unit 13–1

319

Ordinary Annuities: Money Invested at End of Period (Find Future Value) Before we explain how to use a table that simplifies calculating ordinary annuities, let’ first determine how to calculate the future value of an ordinary annuity manually .

s

Calculating Future Value of Ordinary Annuities Manually Remember that an ordinary annuity invests money at the end of each year (period). After we calculate ordinary annuities manually , you will see that the total value of the investment comes from the stream of yearly investments and the buildup of interest on the current balance. Check out the plastic overlays that appear in Chapter 13, p. 336A, to review these concepts.

CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY MANUALLY Step 1.

For period 1, no interest calculation is necessary, since money is invested at the end of the period.

Step 2.

For period 2, calculate interest on the balance and add the interest to the previous balance.

Step 3.

Add the additional investment at the end of period 2 to the new balance.

Step 4.

Repeat Steps 2 and 3 until the end of the desired period is reached.

EXAMPLE Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.

We calculate this manually as follows:

Step 1.

End of year 1:

$3,000.00

Year 2:

$3,000.00

Step 2.

240.00

$3,240.00 Step 3.

End of year 2: 3,000.00 Year 3: $6,240.00 499.20

Step 4.

$6,739.20 3,000.00 End of year 3:

$9,739.20

Early years 1

2

3

No interest, since this is put in at end of year 1. (Remember, payment is made at end of period.) Value of investment before investment at end of year 2. Interest (.08 $3,000) for year 2. Value of investment at end of year 2 before second investment. Second investment at end of year 2. Investment balance going into year 3. Interest for year 3 (.08 $6,240). Value before investment at end of year 3. Investment at end of year 3. Total value of investment after investment at end of year 3. Note: We totally invested $9,000 over three different periods. It is now worth $9,739.20

$3,000 $3,000 $3,000

When you deposit $3,000 at the end of each year at an annual rate of 8%, the total value of the annuity is $9,739.20 . What we called maturity value in compounding is now called the future value of the annuity. Remember that Interest Principal Rate Time, with the principal changing because of the interest payments and the additional deposits. We can make this calculation easier by using Table 13.1 (p. 320).

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Page 320

Chapter 13 Annuities and Sinking Funds

13.1

TABLE Period

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Ordinary annuity table: Compound sum of an annuity of $1

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

13%

1

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

2

2.0200

2.0300

2.0400

2.0500

2.0600

2.0700

3

3.0604

3.0909

3.1216

3.1525

3.1836

3.2149

2.0800

2.0900

2.1000

2.1100

2.1200

2.1300

3.2464

3.2781

3.3100

3.3421

3.3744

3.4069

4

4.1216

4.1836

4.2465

4.3101

4.3746

5

5.2040

5.3091

5.4163

5.5256

5.6371

4.4399

4.5061

4.5731

4.6410

4.7097

4.7793

4.8498

5.7507

5.8666

5.9847

6.1051

6.2278

6.3528

6.4803

6

6.3081

6.4684

6.6330

6.8019

7

7.4343

7.6625

7.8983

8.1420

6.9753

7.1533

7.3359

7.5233

7.7156

7.9129

8.1152

8.3227

8.3938

8.6540

8.9228

9.2004

9.4872

9.7833

10.0890

10.4047

8

8.5829

8.8923

9.2142

9.5491

9.8975

10.2598

10.6366

11.0285

11.4359

11.8594

12.2997

12.7573

9

9.7546

10.1591

10.5828

11.0265

11.4913

11.9780

12.4876

13.0210

13.5795

14.1640

14.7757

15.4157

10

10.9497

11.4639

12.0061

12.5779

13.1808

13.8164

14.4866

15.1929

15.9374

16.7220

17.5487

18.4197

11

12.1687

12.8078

13.4863

14.2068

14.9716

15.7836

16.6455

17.5603

18.5312

19.5614

20.6546

21.8143

12

13.4120

14.1920

15.0258

15.9171

16.8699

17.8884

18.9771

20.1407

21.3843

22.7132

24.1331

25.6502

13

14.6803

15.6178

16.6268

17.7129

18.8821

20.1406

21.4953

22.9534

24.5227

26.2116

28.0291

29.9847

14

15.9739

17.0863

18.2919

19.5986

21.0150

22.5505

24.2149

26.0192

27.9750

30.0949

32.3926

34.8827

15

17.2934

18.5989

20.0236

21.5785

23.2759

25.1290

27.1521

29.3609

31.7725

34.4054

37.2797

40.4174

16

18.6392

20.1569

21.8245

23.6574

25.6725

27.8880

30.3243

33.0034

35.9497

39.1899

42.7533

46.6717

17

20.0120

21.7616

23.6975

25.8403

28.2128

30.8402

33.7503

36.9737

40.5447

44.5008

48.8837

53.7390

18

21.4122

23.4144

25.6454

28.1323

30.9056

33.9990

37.4503

41.3014

45.5992

50.3959

55.7497

61.7251

19

22.8405

25.1169

27.6712

30.5389

33.7599

37.3789

41.4463

46.0185

51.1591

56.9395

63.4397

70.7494

20

24.2973

26.8704

29.7781

33.0659

36.7855

40.9954

45.7620

51.1602

57.2750

64.2028

72.0524

80.9468

25

32.0302

36.4593

41.6459

47.7270

54.8644

63.2489

73.1060

84.7010

98.3471

114.4133

133.3338

155.6194

30

40.5679

47.5754

56.0849

66.4386

79.0580

94.4606 113.2833 136.3077

164.4941

199.0209

241.3327

293.1989

40

60.4017

75.4012

95.0254 120.7993 154.7616 199.6346 259.0569 337.8831

442.5928

581.8260

767.0913 1013.7030

50

84.5790 112.7968 152.6669 209.3470 290.3351 406.5277 573.7711 815.0853 1163.9090 1668.7710 2400.0180 3459.5010

Note: This is only a sampling of tables available. The Business Math Handbook shows tables from 12 % to 15%.

Calculating Future Value of Ordinary Annuities by Table Lookup Use the following steps to calculate the future value of an ordinary annuity by table lookup. 2 CALCULATING FUTURE VALUE OF AN ORDINARY ANNUITY BY TABLE LOOKUP Step 1.

Calculate the number of periods and rate per period.

Step 2.

Look up the periods and rate in an ordinary annuity table. The intersection gives the table factor for the future value of $1.

Step 3.

Multiply the payment each period by the table factor. This gives the future value of the annuity. Future value of Annuity payment Ordinary annuity ordinary annuity each period table factor

EXAMPLE Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8% (see p. 321).

The formula for an ordinary annuity is A Pmt 3 (1 i)i 1 4 where A equals future value of an ordinary annuity, Pmt equals annuity payment, i equals interest, and n equals number of periods. The calculator sequence for this example is: 1 .08 y x 3 1 .08 3,000 9,739.20. A Financial Calculator Guide booklet is available that shows how to operate HP 10BII and TI BA II Plus. 2

1

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321

Learning Unit 13–1

Step 1.

Periods 3 years 1 3

Rate

8% 8% Annually

Go to Table 13.1, an ordinary annuity table. Look for 3 under the Period column. Go across to 8%. At the intersection is the table factor , 3.2464. (This was the example we showed in Figure 13.1.) Step 3. Multiply $3,000 3.2464 $9,739.20 (the same figure we calculated manually). Step 2.

Annuities Due: Money Invested at Beginning of Period (Find Future Value) In this section we look at what the dif ference in the total investment would be for an annuity due. As in the previous section, we will first make the calculation manually and then use the table lookup. Calculating Future Value of Annuities Due Manually Use the steps that follow to calculate the future value of an annuity due manually

.

CALCULATING FUTURE VALUE OF AN ANNUITY DUE MANUALLY Step 1.

Calculate the interest on the balance for the period and add it to the previous balance.

Step 2.

Add additional investment at the beginning of the period to the new balance.

Step 3.

Repeat Steps 1 and 2 until the end of the desired period is reached.

Remember that in an annuity due, we deposit the money at the beginning of the year and gain more interest. Common sense should tell us that the annuity due will give a higher final value. We will use the same example that we used before. Find the value of an investment after 3 years for a $3,000 annuity due at 8%. We calculate this manually as follows:

EXAMPLE

Beginning year 1:

$3,000.00

Step 1.

240.00

First investment (will earn interest for 3 years). Interest (.08 $3,000).

$3,240.00

Value of investment at end of year 1.

Step 2.

Year 2: 3,000.00

Second investment (will earn interest for 2 years).

Step 3.

$6,240.00 499.20 $6,739.20 Year 3: 3,000.00 $9,739.20

779.14

End of year 3: $10,518.34

Interest for year 2 (.08 $6,240). Value of investment at end of year 2. Third investment (will earn interest for 1 year). Interest (.08 $9,739.20). At the end of year 3, final value.

Beginning of years 1

2

$3,000 $3,000 $3,000

3

Note: Our total investment of $9,000 is worth $10,518.34 . For an ordinary annuity , our total investment was only worth $9,739.20.

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Chapter 13 Annuities and Sinking Funds

Calculating Future Value of Annuities Due by Table Lookup To calculate the future value of an annuity due with a table lookup, use the steps that follow . CALCULATING FUTURE VALUE OF AN ANNUITY DUE BY TABLE LOOKUP3 Step 1.

Calculate the number of periods and the rate per period. Add one extra period.

Step 2.

Look up in an ordinary annuity table the periods and rate. The intersection gives the table factor for future value of $1.

Step 3.

Multiply payment each period by the table factor.

Step 4.

Subtract 1 payment from Step 3. Annuity Ordinary* Future value of ° payment annuity ¢ 1 Payment an annuity due each period table factor *Add 1 period.

Let’s check the $10,518.34 by table lookup. Step 1.

Periods 3 years 1

Step 2.

Table factor, 4.5061 $3,000 4.5061

Step 3. Step 4.

Rate

3 1 extra 4

$13,518.30 3,000.00

$10,518.30

8% 8% Annually

Be sure to subtract 1 payment. (off 4 cents due to rounding)

Note that the annuity due shows an ending value of $10,518.30, while the ending value of ordinary annuity was $9,739.20. We had a higher ending value with the annuity due because the investment took place at the beginning of each period. Annuity payments do not have to be made yearly . They could be made semiannually , monthly, quarterly, and so on. Let’ s look at one more example with a dif ferent number of periods and rate. Different Number of Periods and Rates By using a dif ferent number of periods and rates, we will contrast an ordinary annuity with an annuity due in the following example: Using Table 13.1 (p. 320), find the value of a $3,000 investment after 3 years made quarterly at 8%. In the annuity due calculation, be sure to add one period and subtract one payment from the total value.

EXAMPLE

Ordinary annuity

Step 1. Periods 3 years 4 12

Rate 8% 4 2% Step 2. Table 13.1: 12 periods, 2% 13.4120 Step 3. $3,000 13.4120 $40,236

Annuity due Periods 3 years 4 12 Rate 8% 4 2% Table 13.1: 13 periods, 2% 14.6803 $3,000 14.6803 $44,040.90 3,000.00

Step 1 Step 2 Step 3 Step 4

$41,040.90 Again, note that with annuity due, the total value is greater since you invest the money at the beginning of each period. Now check your progress with the Practice Quiz. 3

The formula for an annuity due is A Pmt (1 i)i 1 (1 i), where A equals future value of annuity due, Pmt equals annuity payment, i equals interest, and n equals number of periods. This formula is the same as that in footnote 2 except we multiply the future value of annuity by 1 i since payments are made at the beginning of the period. The calculator sequence for this example is: 1 .08 9,739.20 10,518.34. n

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Learning Unit 13–2

LU 13–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

Using Table 13.1, (a) find the value of an investment after 4 years on an ordinary annuity of $4,000 made semiannually at 10%; and (b) recalculate, assuming an annuity due. Wally Beaver won a lottery and will receive a check for $4,000 at the beginning of each 6 months for the next 5 years. If Wally deposits each check into an account that pays 6%, how much will he have at the end of the 5 years?

1. 2.

DVD ✓

Solutions

1.

a. Step 1. Periods 4 years 2 8

Step 2. Step 3.

2.

Step 1.

Step 2. Step 3. Step 4.

LU 13–1a

323

10% 2 5% Factor 9.5491 $4,000 9.5491 $38,196.40

5 years 2

10 1 11 periods Table factor, 12.8078 $4,000 12.8078 $51,231.20 4,000.00 $47,231.20

b. Periods 4 years 2 819 10% 2 5% Factor 11.0265 $4,000 11.0265 $44,106 1 payment 4,000 6% 3% 2

Step 1

Step 2 Step 3 Step 4

$40,106

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 329)

1. 2.

Using Table 13.1, (a) find the value of an investment after 4 years on an ordinary annuity of $5,000 made semiannually at 4%; and (b) recalculate, assuming an annuity due. Wally Beaver won a lottery and will receive a check for $2,500 at the beginning of each 6 months for the next 6 years. If Wally deposits each check into an account that pays 6%, how much will he have at the end of the 6 years?

Learning Unit 13–2: Present Value of an Ordinary Annuity (Find Present Value)4 This unit begins by presenting the concept of present value of an ordinary annuity . Then you will learn how to use a table to calculate the present value of an ordinary annuity .

Concept of Present Value of an Ordinary Annuity— The Big Picture Let’s assume that we want to know how much money we need to invest today to receive a stream of payments for a given number of years in the future. This is called the present value of an ordinary annuity. In Figure 13.2 (p. 324) you can see that if you wanted to withdraw $1 at the end of one period, you would have to invest 93 cents today. If at the end of each period for three periods, you wanted to withdraw $1, you would have to put $2.58 in the bank today at 8% interest. (Note that we go from the future back to the present.) Now let’s look at how we could use tables to calculate the present value of annuities and then check our answer .

Calculating Present Value of an Ordinary Annuity by Table Lookup Use the steps on p. 324 to calculate by table lookup the present value of an ordinary annuity .5 4

For simplicity we omit a discussion of present value of annuity due that would require subtracting a period and adding a 1. n 5 The formula for the present value of an ordinary annuity is P Pmt 1 1 i (1 i) , where P equals present value of annuity, Pmt equals annuity payment, i equals interest, and n equals number of periods. The calculator sequence would be as follows for the John Fitch example: 1 .08 y x 3 M 1 MR .08 8,000 21,000.

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Chapter 13 Annuities and Sinking Funds

FIGURE

13.2

$3.50 3.00 2.50 2.00 1.50 1.00 .50

Present value of an annuity of $1 at 8%

$2.5771 $1.7833 $.9259

1

2 Number of periods

3

CALCULATING PRESENT VALUE OF AN ORDINARY ANNUITY BY TABLE LOOKUP Step 1.

Calculate the number of periods and rate per period.

Step 2.

Look up the periods and rate in the present value of an annuity table. The intersection gives the table factor for the present value of $1.

Step 3.

Multiply the withdrawal for each period by the table factor. This gives the present value of an ordinary annuity. Present value of Annuity Present value of ordinary annuity payment payment ordinary annuity table

TABLE Period

13.2 2%

Present value of an annuity of $1 3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

13%

1

0.9804

0.9709

0.9615

0.9524

0.9434

0.9346

0.9259

0.9174

0.9091

0.9009

0.8929

0.8850

2

1.9416

1.9135

1.8861

1.8594

1.8334

1.8080

1.7833

1.7591

1.7355

1.7125

1.6901

1.6681

3

2.8839

2.8286

2.7751

2.7232

2.6730

2.6243

2.5771

2.5313

2.4869

2.4437

2.4018

2.3612

4

3.8077

3.7171

3.6299

3.5459

3.4651

3.3872

3.3121

3.2397

3.1699

3.1024

3.0373

2.9745

5

4.7134

4.5797

4.4518

4.3295

4.2124

4.1002

3.9927

3.8897

3.7908

3.6959

3.6048

3.5172

6

5.6014

5.4172

5.2421

5.0757

4.9173

4.7665

4.6229

4.4859

4.3553

4.2305

4.1114

3.9975

7

6.4720

6.2303

6.0021

5.7864

5.5824

5.3893

5.2064

5.0330

4.8684

4.7122

4.5638

4.4226

8

7.3255

7.0197

6.7327

6.4632

6.2098

5.9713

5.7466

5.5348

5.3349

5.1461

4.9676

4.7988

9

8.1622

7.7861

7.4353

7.1078

6.8017

6.5152

6.2469

5.9952

5.7590

5.5370

5.3282

5.1317

10

8.9826

8.5302

8.1109

7.7217

7.3601

7.0236

6.7101

6.4177

6.1446

5.8892

5.6502

5.4262

11

9.7868

9.2526

8.7605

8.3064

7.8869

7.4987

7.1390

6.8052

6.4951

6.2065

5.9377

5.6869

12

10.5753

9.9540

9.3851

8.8632

8.3838

7.9427

7.5361

7.1607

6.8137

6.4924

6.1944

5.9176

13

11.3483 10.6350

9.9856

9.3936

8.8527

8.3576

7.9038

7.4869

7.1034

6.7499

6.4235

6.1218

14

12.1062 11.2961 10.5631

9.8986

9.2950

8.7455

8.2442

7.7862

7.3667

6.9819

6.6282

6.3025

15

12.8492 11.9379 11.1184

10.3796

9.7122

9.1079

8.5595

8.0607

7.6061

7.1909

6.8109

6.4624

16

13.5777 12.5611 11.6523

10.8378 10.1059

9.4466

8.8514

8.3126

7.8237

7.3792

6.9740

6.6039

17

14.2918 13.1661 12.1657

11.2741 10.4773

9.7632

9.1216

8.5436

8.0216

7.5488

7.1196

6.7291

18

14.9920 13.7535 12.6593

11.6896 10.8276

10.0591

9.3719

8.7556

8.2014

7.7016

7.2497

6.8399

19

15.6784 14.3238 13.1339

12.0853 11.1581

10.3356

9.6036

8.9501

8.3649

7.8393

7.3658

6.9380

20

16.3514 14.8775 13.5903

12.4622 11.4699

10.5940

9.8181

9.1285

8.5136

7.9633

7.4694

7.0248

25

19.5234 17.4131 15.6221

14.0939 12.7834

11.6536 10.6748

9.8226

9.0770

8.4217

7.8431

7.3300

30

22.3964 19.6004 17.2920

15.3724 13.7648

12.4090 11.2578 10.2737

9.4269

8.6938

8.0552

7.4957

40

27.3554 23.1148 19.7928

17.1591 15.0463

13.3317 11.9246 10.7574

9.7790

8.9511

8.2438

7.6344

50

31.4236 25.7298 21.4822

18.2559 15.7619

13.8007 12.2335 10.9617

9.9148

9.0417

8.3045

7.6752

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Learning Unit 13–2

325

John Fitch wants to receive an $8,000 annuity in 3 years. Interest on the annuity is 8% annually. John will make withdrawals at the end of each year. How much must John invest today to receive a stream of payments for 3 years? Use Table 13.2 (p. 324). Remember that interest could be earned semiannually, quarterly, and so on, as shown in the previous unit.

EXAMPLE

Step 1.

3 years 1 3 periods

8% 8% Annually

Table factor, 2.5771 (we saw this in Figure 13.2) Step 3. $8,000 2.5771 $20,616.80 Step 2.

If John wants to withdraw $8,000 at the end of each period for 3 years, he will have to deposit $20,616.80 in the bank today. $20,616.80 1,649.34 $22,266.14 8,000.00 $14,266.14 1,141.29 $15,407.43 8,000.00 $ 7,407.43 592.59 $ 8,000.02 8,000.00 .026

Interest at end of year 1 (.08 $20,616.80) First payment to John Interest at end of year 2 (.08 $14,266.14) Second payment to John Interest at end of year 3 (.08 $7,407.43) After end of year 3 John receives his last $8,000

Before we leave this unit, let’ s work out two examples that show the relationship of Chapter 13 to Chapter 12. Use the tables in your Business Math Handbook.

Lump Sum versus Annuities John Sands made deposits of $200 semiannually to Floor Bank, which pays 8% interest compounded semiannually. After 5 years, John makes no more deposits. What will be the balance in the account 6 years after the last deposit?

EXAMPLE

Calculate amount of annuity: Table 13.1 10 periods, 4% $200 12.0061 $2,401.22 Step 2. Calculate how much the final value of the annuity will grow by the compound interest table. Table 12.1 12 periods, 4% $2,401.22 1.6010 $3,844.35 Step 1.

For John, the stream of payments grows to $2,401.22. Then this lump sum grows for 6 years to $3,844.35. Now let’ s look at a present value example. Mel Rich decided to retire in 8 years to New Mexico. What amount should Mel invest today so he will be able to withdraw $40,000 at the end of each year for 25 years after he retires? Assume Mel can invest money at 5% interest (compounded annually).

EXAMPLE

Calculate the present value of the annuity: Table 13.2 25 periods, 5% $40,000 14.0939 $563,756 Step 2. Find the present value of $563,756 since Mel will not retire for 8 years: Table 12.3 Step 1.

8 periods, 5% (PV table)

$563,756 .6768 $381,550.06

If Mel deposits $381,550 in year 1, it will grow to $563,756 after 8 years. It’s time to try the Practice Quiz and check your understanding of this unit. 6

Off due to rounding.

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Chapter 13 Annuities and Sinking Funds

LU 13–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

What must you invest today to receive an $18,000 annuity for 5 years semiannually at a 10% annual rate? All withdrawals will be made at the end of each period. Rase High School wants to set up a scholarship fund to provide five $2,000 scholarships for the next 10 years. If money can be invested at an annual rate of 9%, how much should the scholarship committee invest today? Joe Wood decided to retire in 5 years in Arizona. What amount should Joe invest today so he can withdraw $60,000 at the end of each year for 30 years after he retires? Assume Joe can invest money at 6% compounded annually .

1. 2.

DVD 3. (Use tables in Business Math Handbook)

✓

Solutions

1.

2.

3.

LU 13–2a

Periods 5 years 2 10; Rate 10% 2 5% Factor, 7.7217 $18,000 7.7217 $138,990.60 Periods 10; Rate 9% Factor, 6.4177 $10,000 6.4177 $64,177 Calculate present value of annuity: 30 periods, 6%. $60,000 13.7648 $825,888 Step 2. Find present value of $825,888 for 5 years: 5 periods, 6%. $825,888 .7473 $617,186.10

Step 1. Step 2. Step 3. Step 1. Step 2. Step 3. Step 1.

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 329)

1. 2.

3.

What must you invest today to receive a $20,000 annuity for 5 years semiannually at a 5% annual rate? All withdrawals will be made at the end of each period. Rase High School wants to set up a scholarship fund to provide five $3,000 scholarships for the next 10 years. If money can be invested at an annual rate of 4%, how much should the scholarship committee invest today? Joe Wood decided to retire in 5 years in Arizona. What amount should Joe invest today so he can withdraw $80,000 at the end of each year for 30 years after he retires? Assume Joe can invest money at 3% compounded annually .

Learning Unit 13–3: Sinking Funds (Find Periodic Payments) A sinking fund is a financial arrangement that sets aside regular periodic payments of a particular amount of money . Compound interest accumulates on these payments to a specific sum at a predetermined future date. Corporations use sinking funds to discharge bonded indebtedness, to replace worn-out equipment, to purchase plant expansion, and so on. A sinking fund is a dif ferent type of an annuity . In a sinking fund, you determine the amount of periodic payments you need to achieve a given financial goal. In the annuity , you know the amount of each payment and must determine its future value. Let’ s work with the following formula: Sinking fund payment Future value Sinking fund table factor7

To retire a bond issue, Moore Company needs $60,000 in 18 years from today . The interest rate is 10% compounded annually . What payment must Moore make at the end of each year? Use Table 13.3 (p. 327).

EXAMPLE

7

Sinking fund table is the reciprocal of the ordinary annuity table.

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327

Learning Unit 13–3

TABLE

13.3

Period

2%

3%

4%

5%

6%

8%

10%

1

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

2

0.4951

0.4926

0.4902

0.4878

0.4854

0.4808

0.4762

3

0.3268

0.3235

0.3203

0.3172

0.3141

0.3080

0.3021

4

0.2426

0.2390

0.2355

0.2320

0.2286

0.2219

0.2155

5

0.1922

0.1884

0.1846

0.1810

0.1774

0.1705

0.1638

6

0.1585

0.1546

0.1508

0.1470

0.1434

0.1363

0.1296

7

0.1345

0.1305

0.1266

0.1228

0.1191

0.1121

0.1054

8

0.1165

0.1125

0.1085

0.1047

0.1010

0.0940

0.0874

Sinking fund table based on $1

9

0.1025

0.0984

0.0945

0.0907

0.0870

0.0801

0.0736

10

0.0913

0.0872

0.0833

0.0795

0.0759

0.0690

0.0627

11

0.0822

0.0781

0.0741

0.0704

0.0668

0.0601

0.0540

12

0.0746

0.0705

0.0666

0.0628

0.0593

0.0527

0.0468

13

0.0681

0.0640

0.0601

0.0565

0.0530

0.0465

0.0408

14

0.0626

0.0585

0.0547

0.0510

0.0476

0.0413

0.0357

15

0.0578

0.0538

0.0499

0.0463

0.0430

0.0368

0.0315

16

0.0537

0.0496

0.0458

0.0423

0.0390

0.0330

0.0278

17

0.0500

0.0460

0.0422

0.0387

0.0354

0.0296

0.0247

18

0.0467

0.0427

0.0390

0.0355

0.0324

0.0267

0.0219

19

0.0438

0.0398

0.0361

0.0327

0.0296

0.0241

0.0195

20

0.0412

0.0372

0.0336

0.0302

0.0272

0.0219

0.0175

24

0.0329

0.0290

0.0256

0.0225

0.0197

0.0150

0.0113

28

0.0270

0.0233

0.0200

0.0171

0.0146

0.0105

0.0075

32

0.0226

0.0190

0.0159

0.0133

0.0110

0.0075

0.0050

36

0.0192

0.0158

0.0129

0.0104

0.0084

0.0053

0.0033

40

0.0166

0.0133

0.0105

0.0083

0.0065

0.0039

0.0023

We begin by looking down the Period column in Table 13.3 until we come to 18. Then we go across until we reach the 10% column. The table factor is .0219. Now we multiply $60,000 by the factor as follows: $60,000 .0219 $1,314 This states that if Moore Company pays $1,314 at the end of each period for 18 years, then $60,000 will be available to pay of f the bond issue at maturity . We can check this by using Table 13.1 on p. 320 (the ordinary annuity table): $1,314 45.5992 $59,917.358 It’s time to try the following Practice Quiz.

LU 13–3

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

Today, Arrow Company issued bonds that will mature to a value of $90,000 in 10 years. Arrow’s controller is planning to set up a sinking fund. Interest rates are 12% compounded semiannually. What will Arrow Company have to set aside to meet its obligation in 10 years? Check your answer . Your answer will be of f due to the rounding of Table 13.3.

DVD

✓

Solution

10 years 2 20 periods

8

Off due to rounding.

12% 6% 2

$90,000 .0272 $2,448 Check $2,448 36.7855 $90,050.90

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Chapter 13 Annuities and Sinking Funds

LU 13–3a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 329)

Today Arrow Company issued bonds that will mature to a value of $120,000 in 20 years. Arrow’s controller is planning to set up a sinking fund. Interest rates are 6% compounded semiannually. What will Arrow Company have to set aside to meet its obligation in 10 years? Check your answer . Your answer will be of f due to rounding of Table 13.3.

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Ordinary annuities (find future value), p. 319

Invest money at end of each period. Find future value at maturity. Answers question of how much money accumulates. Future Annuity Ordinary value of payment annuity ordinary each table annuity period factor

Use Table 13.1: 2 years, $4,000 ordinary annuity at 8% annually. Value $4,000 2.0800 $8,320 (2 periods, 8%)

FV PMT c Annuities due (find future value), p. 321

FV 4,000 c

(1 .08)2 1 d $8,320 .08

(1 i)n 1 d i

Invest money at beginning of each period. Find future value at maturity. Should be higher than ordinary annuity since it is invested at beginning of each period. Use Table 13.1, but add one period and subtract one payment from answer. Future Annuity Ordinary* value payment annuity of an ° ¢ 1 Payment each table annuity period factor due

Example: Same example as above but invest money at beginning of period. $4,000 3.2464 $12,985.60 4,000.00 $ 8,985.60 (3 periods, 8%) (1 .08)2 1 b(1 .08) .08 $8,985.60

FVdue 4,000a

*Add 1 period.

FVdue PMT c

Present value of an ordinary annuity (find present value), p. 323

(1 i )n 1 d (1 i) i

Calculate number of periods and rate per period. Use Table 13.2 to find table factor for present value of $1. Multiply withdrawal for each period by table factor to get present value of an ordinary annuity. Present Present value of an value of Annuity ordinary ordinary payment annuity annuity payment table PV PMT c

Example: Receive $10,000 for 5 years. Interest is 10% compounded annually. Table 13.2: 5 periods, 10% 3.7908 $10,000 What you put in today $37,908 PV 10,000 c

1 (1 .1)5 d $37,907.88 .1

1 (1 i )n d i

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

329

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Sinking funds (find periodic payment), p. 326

Paying a particular amount of money for a set number of periodic payments to accumulate a specific sum. We know the future and must calculate the periodic payments needed. Answer can be proved by ordinary annuity table. Sinking Sinking Future fund fund table value payment factor

Example: $200,000 bond to retire 15 years from now. Interest is 6% compounded annually. By Table 13.3: $200,000 .0430 $8,600 Check by Table 13.1: $8,600 23.2759 $200,172.74

KEY TERMS

Annuities certain, p. 318 Annuity, p. 317 Annuity due, p. 318 Contingent annuities, p. 318

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 13–1a (p. 323) 1. a. $42,914.50 b. $43,773 c. $36,544.50

Future value of an annuity, p. 318 Ordinary annuity, p. 318 Payment periods, p. 318 LU 13–2a (p. 326) 1. $175,042 2. $121,663.50 3. $1,352,584.40

Present value of an annuity, p. 323 Sinking fund, p. 326 Term of the annuity, p. 318 LU 13–3a (p. 328) $1,596

Critical Thinking Discussion Questions 1. What is the dif ference between an ordinary annuity and an annuity due? If you were to save money in an annuity, which would you choose and why? 2. Explain how you would calculate ordinary annuities and annuities due by table lookup. Create an example to explain the meaning of a table factor from an ordinary annuity . 3. What is a present value of an ordinary annuity? Create an example showing how one of your relatives might plan for

retirement by using the present value of an ordinary annuity. Would you ever have to use lump-sum payments in your calculation from Chapter 12? 4. What is a sinking fund? Why could an ordinary annuity table be used to check the sinking fund payment?

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Complete the ordinary annuities for the following using tables in the Business Math Handbook: Amount of payment

Payment payable

Years

Interest rate

13–1. $10,000

Quarterly

7

4%

13–2. $7,000

Semiannually

8

7%

Value of annuity

Redo Problem 13–1 as an annuity due: 13–3.

Calculate the value of the following annuity due without a table. Check your results by Table 13.1 or the Business Math Handbook (they will be slightly off due to rounding): Amount of payment 13–4. $2,000

Payment payable

Years

Interest rate

Annually

3

6%

Complete the following using Table 13.2 or the Business Math Handbook for the present value of an ordinary annuity: Amount of annuity expected

Payment

Time

Interest rate

13–5. $900

Annually

4 years

6%

13–6. $15,000

Quarterly

4 years

8%

Present value (amount needed now to invest to receive annuity)

13–7. Check Problem 13–5 without the use of Table 13.2.

Using the sinking fund Table 13.3 or the Business Math Handbook, complete the following: Required amount

Frequency of payment

Length of time

Interest rate

13–8. $25,000

Quarterly

6 years

8%

13–9. $15,000

Annually

8 years

8%

Payment amount end of each period

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13–10. Check the answer in Problem 13–9 by Table 13.1.

WORD PROBLEMS (Use Tables in the Business Math Handbook) 13–11. John Regan, an employee at Home Depot, made deposits of $800 at the end of each year for 4 years. Interest is 4% compounded annually. What is the value of Regan’s annuity at the end of 4 years?

13–12. Pete King promised to pay his son $300 semiannually for 9 years. Assume Pete can invest his money at 8% in an ordinary annuity. How much must Pete invest today to pay his son $300 semiannually for 9 years?

13–13. “The most powerful force in the universe is compound interest,” according to an article in the Morningstar Column dated February 13, 2007. Patricia Wiseman is 30 years old and she invests $2,000 in an annuity , earning 5% compound annual return at the beginning of each period, for 18 years. What is the cash value of this annuity due at the end of 18 years?

13–14. The Toronto Star on February 15, 2007, described getting rich slowly , but surely. You have 40 years to save. If you start early, with the power of compounding, what a situation you would be in.Valerie Wise is 25 years old and invests $3,000 for only six years in an ordinary annuity at 8% interest compounded annually . What is the final value of Valerie’s investment at the end of year 6?

13–15. “Pay Dirt; It’s time for a Clean Sweep”, was the title of an article that appeared in the Minneapolis, Star Tribune on March 15, 2007. Plant your coins in the bank: during a traditional spring cleaning, coins sprout from couch cushions and junk drawers. The average American has $99 lying about. Stick $99 in an ordinary annuity account each year for 10 years at 5% interest and watch it grow. What is the cash value of this annuity at the end of year 10? Round to the nearest dollar.

13–16. Patricia and Joe Payne are divorced.The divorce settlement stipulated that Joe pay $525 a month for their daughter Suzanne until she turns 18 in 4 years. How much must Joe set aside today to meet the settlement? Interest is 6% a year.

13–17. Josef Company borrowed money that must be repaid in 20 years. The company wants to make sure the loan will be repaid at the end of year 20. So it invests $12,500 at the end of each year at 12% interest compounded annually . What was the amount of the original loan?

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13–18. Jane Frost wants to receive yearly payments of $15,000 for 10 years. How much must she deposit at her bank today at 1 % interest compounded annually?

13–19. Toby Martin invests $2,000 at the end of each year for 10 years in an ordinary annuity at 1 1% interest compounded annually. What is the final value of Toby’s investment at the end of year 10?

13–20. Alice Longtree has decided to invest $400 quarterly for 4 years in an ordinary annuity at 8%. As her financial adviser, calculate for Alice the total cash value of the annuity at the end of year 4.

13–21. At the beginning of each period for 10 years, Merl Agnes invests $500 semiannually at 6%. What is the cash value of this annuity due at the end of year 10?

13–22. Jeff Associates borrowed $30,000. The company plans to set up a sinking fund that will repay the loan at the end of 8 years. Assume a 12% interest rate compounded semiannually. What must Jeff pay into the fund each period of time? Check your answer by Table 13.1.

13–23. On Joe Martin’s graduation from college, Joe’s uncle promised him a gift of $12,000 in cash or $900 every quarter for the next 4 years after graduation. If money could be invested at 8% compounded quarterly, which offer is better for Joe?

13–24. You are earning an average of $46,500 and will retire in 10 years. If you put 20% of your gross average income in an ordinary annuity compounded at 7% annually, what will be the value of the annuity when you retire?

13–25. GU Corporation must buy a new piece of equipment in 5 years that will cost $88,000. The company is setting up a sinking fund to finance the purchase. What will the quarterly deposit be if the fund earns 8% interest?

13–26. Mike Macaro is selling a piece of land. Two offers are on the table. Morton Company offered a $40,000 down payment and $35,000 a year for the next 5 years. Flynn Company offered $25,000 down and $38,000 a year for the next 5 years. If money can be invested at 8% compounded annually, which offer is better for Mike?

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13–27. Al Vincent has decided to retire to Arizona in 10 years. What amount should Al invest today so that he will be able to withdraw $28,000 at the end of each year for 15 years after he retires? Assume he can invest the money at 8% interest compounded annually.

13–28. Victor French made deposits of $5,000 at the end of each quarter to Book Bank, which pays 8% interest compounded quarterly. After 3 years, Victor made no more deposits. What will be the balance in the account 2 years after the last deposit?

13–29. Janet Woo decided to retire to Florida in 6 years. What amount should Janet invest today so she can withdraw $50,000 at the end of each year for 20 years after she retires? Assume Janet can invest money at 6% compounded annually.

CHALLENGE PROBLEMS 13–30. David Stokke must determine how much money he needs to set aside now for future home repairs. He has determined his roof has only 10 more years of useful life. His roof would require 15 bundles of shingles.The roof he prefers costs $145 per bundle for total replacement. Assume a 1.5% inflation rate per year over the next 10 years. David is placing his money in a sinking fund at 6% compounded semiannually. (a) What is today’s cost of replacing the roof? (b) What is the cost of replacing the roof in 10 years? (c) What amount will David have to put away each year to have enough money to replace the roof?

13–31. Ajax Corporation has hired Brad O’Brien as its new president. Terms included the company’s agreeing to pay retirement benefits of $18,000 at the end of each semiannual period for 10 years. This will begin in 3,285 days. If the money can be invested at 8% compounded semiannually, what must the company deposit today to fulfill its obligation to Brad?

DVD SUMMARY PRACTICE TEST (Use Tables in the Business Math Handbook) 1. Lin Lowe plans to deposit $1,800 at the end of every 6 months for the next 15 years at 8% interest compounded semiannually. What is the value of Lin’s annuity at the end of 15 years? (p. 319)

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2. On Abby Ellen’s graduation from law school, Abby’s uncle, Bull Brady, promised her a gift of $24,000 or $2,400 every quarter for the next 4 years after graduating from law school. If the money could be invested at 6% compounded quarterly, which offer should Abby choose? (p. 325)

3. Sanka Blunck wants to receive $8,000 each year for 20 years. How much must Sanka invest today at 4% interest compounded annually? (p. 325) 4. In 9 years, Rollo Company will have to repay a $100,000 loan. Assume a 6% interest rate compounded quarterly. How much must Rollo Company pay each period to have $100,000 at the end of 9 years? (p. 325)

5. Lance Industries borrowed $130,000. The company plans to set up a sinking fund that will repay the loan at the end of 18 years. Assume a 6% interest rate compounded semiannually. What amount must Lance Industries pay into the fund each period? Check your answer by Table 13.1 (p. 326)

6. Joe Jan wants to receive $22,000 each year for the next 22 years. Assume a 6% interest rate compounded annually. How much must Joe invest today? (p. 325) 7. Twice a year for 15 years, Warren Ford invested $1,700 compounded semiannually at 6% interest. What is the value of this annuity due? (p. 321)

8. Scupper Molly invested $1,800 semiannually for 23 years at 8% interest compounded semiannually. What is the value of this annuity due? (p. 321)

9. Nelson Collins decided to retire to Canada in 10 years. What amount should Nelson deposit so that he will be able to withdraw $80,000 at the end of each year for 25 years after he retires? Assume Nelson can invest money at 7% interest compounded annually. (p. 325)

10. Bob Bryan made deposits of $10,000 at the end of each quarter to Lion Bank, which pays 8% interest compounded quarterly. After 9 years, Bob made no more deposits. What will be the account’s balance 4 years after the last deposit? (p. 319)

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Personal Finance A KIPLINGER APPROACH PORTFOLIO DOCTOR

A retiree ponders converting his traditional IRA. By Jeffrey R. Kosnett

An overlooked way to SHEAR your taxes ike most people, Tom Berry wants to pay less tax. Tom, a retired 63-year-old computer engineer from Edwardsville, Ill., has a plan that may enable him to do just that: He’s considering converting his $165,000 traditional IRA to a Roth IRA. Traditional IRA distributions are taxable; Roth distributions aren’t. Tom must begin taking distributions from his traditional IRA at age 701⁄2. At that point, Tom figures, the combination of taxable withdrawals, pension income and Social Security would push him and his wife, Paula, into the 25% federal tax bracket. Converting his IRA to a Roth now could keep him in a lower tax bracket later. “Why waste that 15% bracket?” Tom asks. Tom says he has no pressing financial needs and does not intend to draw on his traditional IRA until the law requires him to. Tom, who could live into his nineties based on his family history, invests in an assortment of mutual funds, with about two-thirds in stock funds.

L

Assuming a 7% annual return, Tom’s traditional IRA could reach nearly $300,000 by the time he’s 701⁄2. Tom would have to take out a minimum of about $11,000 that year, based on an IRS schedule designed to deplete the account over 27 years. Beyond the sweet smell of tax-free with-

Stumped by your investments? Write to us at portfoliodoc @kiplinger.com.

drawals in retirement, converting to a Roth is appealing because Tom could avoid those mandatory withdrawals. The original owner of a Roth never has to tap the account, so investments can grow indefinitely. To be eligible to convert to a Roth, your income (on a single or joint return) must be less than $100,000. Tom qualifies. (The $100,000 limit disappears in 2010.) The drawback with Tom’s plan is that he would have to pay taxes on all of the money he moves from his traditional IRA to a Roth. And if Tom switched all the money at once, he would catapult from the 15% bracket to the 33% bracket, and he’d lose one-third of his $165,000 kitty. But there’s a way to limit the pain. Tom can convert to a Roth gradually so that he’s not pushed into a higher tax bracket in any given year. It’s important to figure out how much you can convert each year without “pushing yourself into an outrageous tax

bracket,” says Curtis Chen, a financial planner in Belmont, Cal. That amount will vary with your other income and with tweaks in tax brackets. Test case. If Tom and Paula’s taxable income this year before any Roth conversion is, say, $50,000, Tom could move $11,300 to a Roth before tripping into the 25% bracket. Want to try Tom’s strategy yourself? For an idea of your own “conversion capacity,” compare your estimated taxable income for the year with the income stepping stones in the tax brackets. (To find the latest brackets, search “tax rates” at www.irs.gov.) Converting to a Roth also holds promise for your heirs: Avoiding mandatory payouts means there might be more money left for them. Even better, money in an inherited Roth IRA is tax-free while cash in a traditional IRA is taxed in the beneficiary’s top tax bracket. “If you want to leave money to someone, you’ll leave a bigger amount” with a Roth because the taxes have already been paid, says Donald Duncan, a planner with D3 Financial Counselors, in Downers Grove, Ill. He adds that the conversion strategy works best if you pay the taxes on the converted money from other sources rather than from the IRA. That enables more of your money to grow tax-free.

PHOTOGRAPH BY ANNA KNOTT

BUSINESS MATH ISSUE Kiplinger’s © 2006

Changing to a Roth is silly because you have to pay taxes upfront. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A Go to the Internet to find the latest change to the Roth 401 since this article was published.

2005 Wall Street Journal ©

338

site xt Web e t e e S ts: The t Projec /slater9e) and e n ide. r e t In urce Gu .com o e s h e h R t .m (www Interne ss Math Busine

338

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CUMULATIVE REVIEW

A Word Problem Approach—Chapters 10, 11, 12, 13 1.

Amy O’Mally graduated from high school. Her uncle promised her as a gift a check for $2,000 or $275 every quarter for 2 years. If money could be invested at 6% compounded quarterly, which offer is better for Amy? (Use the tables in the Business Math Handbook.) (p. 325)

2.

Alan Angel made deposits of $400 semiannually to Sag Bank, which pays 10% interest compounded semiannually. After 4 years, Alan made no more deposits. What will be the balance in the account 3 years after the last deposit? (Use the tables in the Business Math Handbook.) (pp. 319, 299)

3.

Roger Disney decides to retire to Florida in 12 years. What amount should Roger invest today so that he will be able to withdraw $30,000 at the end of each year for 20 years after he retires? Assume he can invest money at 8% interest compounded annually. (Use tables in the Business Math Handbook.) (p. 325)

4.

On September 15, Arthur Westering borrowed $3,000 from Vermont Bank at 1012% interest. Arthur plans to repay the loan on January 25. Assume the loan is based on exact interest. How much will Arthur totally repay? (p. 260)

5.

Sue Cooper borrowed $6,000 on an 1134%, 120-day note. Sue paid $300 toward the note on day 50. On day 90, Sue paid an additional $200. Using the U.S. Rule, Sue’s adjusted balance after her first payment is the following. (p. 261)

6.

On November 18, Northwest Company discounted an $18,000, 12%, 120-day note dated September 8. Assume a 10% discount rate. What will be the proceeds? Use ordinary interest. (p. 279)

7.

Alice Reed deposits $16,500 into Rye Bank, which pays 10% interest compounded semiannually. Using the appropriate table, what will Alice have in her account at the end of 6 years? (p. 299)

8.

Peter Regan needs $90,000 in 5 years from today to retire in Arizona. Peter’s bank pays 10% interest compounded semiannually. What will Peter have to put in the bank today to have $90,000 in 5 years? (p. 304)

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CHAPTER

14

Installment Buying, Rule of 78, and Revolving Charge Credit Cards

LEARNING UNIT OBJECTIVES LU 14–1: Cost of Installment Buying • Calculate the amount financed, finance charge, and deferred payment (p. 342). • Calculate the estimated APR by table lookup (p. 343). • Calculate the monthly payment by formula and by table lookup (p. 346).

LU 14–2: Paying Off Installment Loans before Due Date • Calculate the rebate and payoff for Rule of 78 (p. 347 ).

LU 14–3: Revolving Charge Credit Cards • Calculate the finance charges on revolving charge credit card accounts (pp. 350–352).

urnal © 2005 Wall Street Jo

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Chapter 14 Installment Buying, Rule of 78, and Revolving Charge Credit Cards

Are you interested in buying a Land Rover Range Rover Sport car? The Wall Street Journal clipping shows that the car has an APR of 7.48%. In this chapter we will explain what APR means and how you can calculate APR. This chapter also discusses the cost of buying products by installments (closed-end credit) and the revolving credit card (open-end credit).

Wall Street Journal © 2005

Learning Unit 14–1: Cost of Installment Buying Installment buying, a form of closed-end credit, can add a substantial amount to the cost of big-ticket purchases. To illustrate this, we follow the procedure of buying a pickup truck, including the amount financed, finance char ge, and deferred payment price. Then we study the effect of the Truth in Lending Act.

Amount Financed, Finance Charge, and Deferred Payment

Ford Motor Company/AP Wide World

This advertisement for the sale of a pickup truck appeared in a local paper . As you can see from this advertisement, after customers make a down payment, they can buy the truck with an installment loan. This loan is paid off with a series of equal periodic payments. These payments include both interest and principal. The payment process is called amortization. In the promissory notes of earlier chapters, the loan was paid of f in one ending payment. Now let’s look at the calculations involved in buying a pickup truck.

Checking Calculations in Pickup Advertisement Calculating Amount Financed The amount financed is what you actually borrow . To

calculate this amount, use the following formula:

Amount financed Cash price Down payment $9,045

$9,345

$300

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Learning Unit 14–1

343

Calculating Finance Charge The words finance charge in the advertisement represent

the interest charge. The interest char ge resulting in the finance char ge includes the cost of credit reports, mandatory bank fees, and so on. You can use the following formula to calculate the total interest on the loan: Total finance charge Total of all Amount (interest charge) monthly payments financed

$2,617.80

$11,662.80 $9,045 ($194.38 60 months)

Calculating Deferred Payment Price The deferred payment price represents the total

of all monthly payments plus the down payment. late the deferred payment price: Deferred payment price

$11,962.80

The following formula is used to calcu-

Total of all Down monthly payments payment $11,662.80 $300 ($194.38 60)

Truth in Lending: APR Defined and Calculated In 1969, the Federal Reserve Board established the Truth in Lending Act (Regulation Z). The law doesn’t regulate interest char ges; its purpose is to make the consumer aware of the true cost of credit. The Truth in Lending Act requires that creditors provide certain basic information about the actual cost of buying on credit. Before buyers sign a credit agreement, creditors must inform them in writing of the amount of the finance char ge and the annual percentage rate (APR). The APR represents the true or ef fective annual interest creditors char ge. This is helpful to buyers who repay loans over dif ferent periods of time (1 month, 48 months, and so on). To illustrate how the APR affects the interest rate, assume you borrow $100 for 1 year and pay a finance char ge of $9. Your interest rate would be 9% if you waited until the end of the year to pay back the loan. Now let’ s say you pay of f the loan and the finance char ge in 12 monthly payments. Each month that you make a payment, you are losing some of the value or use of that money . So the true or ef fective APR is actually greater than 9%. The APR can be calculated by formula or by tables. We will use the table method since it is more exact. Calculating APR Rate by Table 14.1 (p. 344) Note the following steps for using a table to calculate APR: CALCULATING APR BY TABLE Step 1.

Divide the finance charge by amount financed and multiply by $100 to get the table lookup factor.

Step 2.

Go to APR Table 14.1. At the left side of the table are listed the number of payments that will be made.

Step 3.

When you find the number of payments you are looking for, move to the right and look for the two numbers closest to the table lookup number. This will indicate the APR.

Now let’s determine the APR for the pickup truck advertisement given earlier in the chapter.

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TABLE

14.1

Annual percentage rate table per $100

Note: For a more detailed set of tables from 2% to 21.75%, see the reference tables in the Business Math Handbook.

As stated in Step 1 on p. 343, we begin by dividing the finance char financed and multiply by $100: Finance charge Table 14.1 ⫻ $100 ⫽ Amount financed lookup number

$2,617.80 $9,045.00

⫻ $100 ⫽

ge by the amount

We multiply by $100, since the table is based on $100 of financing.

$28.94

To look up $28.94 in Table 14.1, we go down the left side of the table until we come to 60 payments (the advertisement states 60 months). Then, moving to the right, we look for $28.94 or the two numbers closest to it. The number $28.94 is between $28.22 and $28.96.

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Learning Unit 14–1

TABLE

14.1

345

(concluded)

So we look at the column headings and see a rate between 10.25% and 10.5% . The Truth in Lending Act requires that when creditors state the APR, it must be accurate to the near1 est of 1%.1 4 Calculating the Monthly Payment by Formula and Table 14.2 (p. 346) The pickup truck advertisement showed a $194.38 monthly payment. We can check this by formula and by table lookup.

1

If we wanted an exact reading of APR when the number is not exactly in the table, we would use the process of interpolating. We do not cover this method in this course.

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TABLE

14.2

Terms in months

7.50%

8%

8.50%

9%

10.00%

10.50%

11.00%

11.50%

12.00%

Loan amortization table (monthly payment per $1,000 to pay principal and interest on installment loan)

6

$170.34

$170.58

$170.83

$171.20

$171.56

$171.81

$172.05

$172.30

$172.55

12

86.76

86.99

87.22

87.46

87.92

88.15

88.38

88.62

88.85

18

58.92

59.15

59.37

59.60

60.06

60.29

60.52

60.75

60.98

24

45.00

45.23

45.46

45.69

46.14

46.38

46.61

46.84

47.07

30

36.66

36.89

37.12

37.35

37.81

38.04

38.28

38.51

38.75

36

31.11

31.34

31.57

31.80

32.27

32.50

32.74

32.98

33.21

42

27.15

27.38

27.62

27.85

28.32

28.55

28.79

29.03

29.28

48

24.18

24.42

24.65

24.77

25.36

25.60

25.85

26.09

26.33

54

21.88

22.12

22.36

22.59

23.07

23.32

23.56

23.81

24.06

60

20.04

20.28

20.52

20.76

21.25

21.49

21.74

21.99

22.24

By Formula Finance charge Amount financed Number of payments of loan

$2,617.80 $9,045 $194.38 60

By Table 14.2 The loan amortization table (many variations of this table are available) in Table 14.2 can be used to calculate the monthly payment for the pickup truck. To calculate a monthly payment with a table, use the following steps: CALCULATING MONTHLY PAYMENT BY TABLE LOOKUP Step 1.

Divide the loan amount by $1,000 (since Table 14.2 is per $1,000): $9,045 9.045 $1,000

Step 2.

Look up the rate (10.5%) and number of months (60). At the intersection is the table factor showing the monthly payment per $1,000.

Step 3.

Multiply quotient in Step 1 by the table factor in Step 2: 9.045 $21.49 $194.38 .

Remember that this $194.38 fixed payment includes interest and the reduction of the balance of the loan. As the number of payments increases, interest payments get smaller and the reduction of the principal gets lar ger.2 Now let’s check your progress with the Practice Quiz.

LU 14–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

From the partial advertisement at the right calculate the following: 1. a. Amount financed. $288 per month b. Finance charge. Sale price $14,150 c. Deferred payment price. Down payment $ 1,450 d. APR by Table 14.1. Term/Number of payments 60 months e. Monthly payment by formula. 2.

2

Courtesy Brunswick Corporation

Jay Miller bought a New Brunswick boat for $7,500. Jay put down $1,000 and financed the balance at 10% for 60 months. What is his monthly payment? Use Table 14.2.

In Chapter 15 we give an amortization schedule for home mortgages that shows how much of each fixed payment goes to interest and how much reduces the principal. This repayment schedule also gives a running balance of the loan.

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Learning Unit 14–2

TABLE Terms in months

14.2

347

(concluded)

12.50%

13.00%

13.50%

14.00%

14.50%

15.00%

15.50%

16.00%

6

$172.80

$173.04

$173.29

$173.54

$173.79

$174.03

$174.28

$174.53

12

89.08

89.32

89.55

89.79

90.02

90.26

90.49

90.73

18

61.21

61.45

61.68

61.92

62.15

62.38

62.62

62.86

24

47.31

47.54

47.78

48.01

48.25

48.49

48.72

48.96

30

38.98

39.22

39.46

39.70

39.94

40.18

40.42

40.66

36

33.45

33.69

33.94

34.18

34.42

34.67

34.91

35.16

42

29.52

29.76

30.01

30.25

30.50

30.75

31.00

31.25

48

26.58

26.83

27.08

27.33

27.58

27.83

28.08

28.34

54

24.31

24.56

24.81

25.06

25.32

25.58

25.84

26.10

60

22.50

22.75

23.01

23.27

23.53

23.79

24.05

24.32

✓

Solutions

1.

2.

LU 14–1a

a. $14,150 $1,450 $12,700 b. $17,280 ($288 60) $12,700 $4,580 c. $17,280 ($288 60) $1,450 $18,730 $4,580 d. $100 $36.06; between 12.75% and 13% $12,700 $4,580 $12,700 e. $288 60 $6,500 6.5 $21.25 $138.13 (10%, 60 months) $1,000

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 355)

From the partial advertisement at the right calculate the following: 1. a. Amount financed. b. Finance charge. $295 per month c. Deferred payment price. d. APR by Table 14.1. e. Monthly payment by formula. Sale price: $13,999 Down payment: $1,480 2. Jay Miller bought a New Brunswick Term/Number of payments: 60 months boat for $8,000. Jay puts down $1,000 and financed the balance at 8% for 60 months. What is his monthly payment? Use Table 14.2.

Learning Unit 14–2: Paying Off Installment Loans before Due Date In Learning Unit 10–3 (p. 264), you learned about the U.S. Rule. This rule applies partial payments to the interest first, and then the remainder of the payment reduces the principal. Many states and the federal government use this rule. Some states use another method for prepaying a loan called the Rule of 78. It is a variation of the U.S. Rule. The Rule of 78 got its name because it bases the finance char ge rebate and the payof f on a 12-month loan. (Any number of months can be used.) The Rule of 78 is used less today . However, GMAC says that about 50% of its auto loans still use the Rule of 78. For loans of 61 months or longer , the Rule of 78 is not allowed (some states have even shorter requirements).

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TABLE

14.3

Rebate fraction table based on Rule of 78

Months to go

Sum of digits

Months to go

Sum of digits

1

1

31

496

2

3

32

528

3

6

33

561

4

10

34

595

5

15

35

630

6

21

36

666

7

28

37

703

8

36

38

741

9

45

39

780

10

55

40

820

11

66

41

861

12

78

42

903

13

91

43

946

14

105

44

990

15

120

45

1,035

16

136

46

1,081

17

153

47

1,128

18

171

48

1,176

19

190

49

1,225

20

210

50

1,275

21

231

51

1,326

22

253

52

1,378

23

276

53

1,431

24

300

54

1,485

25

325

55

1,540

26

351

56

1,596

27

378

57

1,653

28

406

58

1,711

29

435

59

1,770

30

465

60

1,830

33 months to go

60 months 1,830

With the Rule of 78, the finance char ge earned the first month is 12 78 . The 78 comes from summing the digits of 12 months. The finance char ge for the second month would be 11 78 , and so on. Table 14.3 simplifies these calculations. When the installment loan is made, a lar ger portion of the interest is char ged to the earlier payments. As a result, when a loan is paid of f early, the borrower is entitled to a rebate, which is calculated as follows:

CALCULATING REBATE AND PAYOFF FOR RULE OF 78 Step 1.

Find the balance of the loan outstanding.

Step 2.

Calculate the total finance charge.

Step 3.

Find the number of payments remaining.

Step 4.

Set up the rebate fraction from Table 14.3.

Step 5.

Calculate the rebate amount of the finance charge.

Step 6.

Calculate the payoff.

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Learning Unit 14–2

349

Let’s see what the rebate of the finance char ge and payoff would be if the pickup truck loan were paid of f after 27 months (instead of 60). To find the finance char ge rebate and the final payof f, we follow six specific steps listed below. Let’s begin. Find the balance of the loan outstanding: Total of monthly payments (60 $194.38) Payments to date: 27 $194.38 Balance of loan outstanding Step 2. Calculate the total finance charge: Total of all payments (60 $194.38) $11,662.80 Amount financed ($9,345 $300) 9,045.00 $ 2,617.80 Total finance charge Step 3. Find the number of payments remaining: 60 27 33 Step 4. Set up the rebate fraction from Table 14.3.3 Step 1.

Sum of digits based on number of months to go Sum of digits based on total number of months of loan

561 1,830

$11,662.80 5,248.26 $ 6,414.54

33 months to go 60 months in loan

Note: If this loan were for 12 months, the denominator would be 78. Step 5.

Calculate the rebate amount of the finance char ge:

Rebate fraction Total finance charge Rebate amount 561 1,830 Step 6.

$2,617.80

(Step 4) Calculate the payoff:

$802.51

(Step 2)

Balance of loan outstanding Rebate $6,414.54

(Step 1)

LU 14–2

Payoff

$802.51 $5,612.03

(Step 5)

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Calculate the finance char ge rebate and payof f (calculate all six steps): Months of loan 12

Loan $5,500

✓

End-ofmonth loan is repaid 7

Monthly payment $510

Finance charge rebate

Final payoff

Solutions

Step 1.

12 $510 $6,120 Step 2. 7 $510 3,570 $2,550 (balance outstanding)

12 $510

$6,120 5,500 $ 620 (total finance charge)

3

If no table is available, the following formula is available:

N(N 1) 33(33 1) 2 2 561 T(T 1) 60(60 1) 1,830 2 2 In the numerator, N stands for number of months to go, and in the denominator, T is total months of the loan.

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Step 3. 12 7 5 Step 5.

LU 14–2a

Step 4.

15 $620 $119.23 rebate 78 (Step 4) (Step 2)

15 (by Table 14.3) 78

Step 6. Step 1 Step 5 $2,550 $119.23 $2,430.77 payoff

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 355)

Calculate the finance char ge rebate and payof f (calculate steps):

Loan $6,900

Months of loan 12

End-ofmonth loan is repaid 5

Monthly payment $690

Finance charge rebate

Final payoff

Learning Unit 14–3: Revolving Charge Credit Cards

Wall Street Journal © 2005

The above Wall Street Journal heading “Credit Cards Raise Minimums Due” af fects revolving charge credit card users who pay the minimum interest on what they owe. As a revolving char ge user , it is probably not news to you that in 2006, credit card companies have been required to raise the minimum amount due on your account. You should be aware that the higher minimum amount due can give you the problem of negative amortization. This means if you only pay the minimum amount and interest costs and fees rise, the end result is your principal could go up. Let’s look at how long it will take to pay of f your credit card balance payments with the minimum amount. Study the following clipping “Pay Just the Minimum, and Get Nowhere Fast.”

The clipping assumes that the minimum rate on the balance of a credit card is 2%. Note that if the annual interest cost is 17%, it will take 17 years, 3 months to pay of f a balance o f $1,000, and the total cost will be $2,590.35. If the balance on your revolving char ge credit card is more than $1,000, you can see how fast the total cost rises. If you cannot af ford the total cost of paying only the minimum, it is time for you to reconsider how you use your revolving credit card. This is why when you have financial dif ficulties, experts often advise you first to work on getting rid of your revolving credit card debt.

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Learning Unit 14–3

351

Do you know why revolving credit cards are so popular? Businesses encourage customers to use credit cards because consumers tend to buy more when they can use a credit card for their purchases. Consumers find credit cards convenient to use and valuable in establishing credit. The problem is that when consumers do not pay their balance in full each month, they do not realize how expensive it is to pay only the minimum of their balance. To protect consumers, Congress passed the Fair Credit and Charge Card Disclosure Act of 1988.4 This act requires that for direct-mail application or solicitation, credit card companies must provide specific details involving all fees, grace period, calculation of finance charges, and so on. We begin the unit by seeing how Moe’ s Furniture Store calculates the finance char ge on Abby Jordan’s previous month’s credit card balance. Then we learn how to calculate the average daily balance on the partial bill of Joan Ring.

Calculating Finance Charge on Previous Month’s Balance Abby Jordan bought a dining room set for $8,000 on credit. She has a revolving charge account at Moe’ s Furniture Store. A revolving char ge account gives a buyer open-end credit. Abby can make as many purchases on credit as she wants until she reaches her maximum $10,000 credit limit. Often customers do not completely pay their revolving char ge accounts at the end of a billing period. When this occurs, stores add interest char ges to the customers’ bills. Moe’s furniture store calculates its interest using the unpaid balance method. It char ges 1 12% on the previous month’s balance, or 18% per year . Moe’s has no minimum monthly payment (many stores require $10 or $15, or a percent of the outstanding balance). Abby has no other char ges on her revolving char ge account. She plans to pay $500 per month until she completely pays off her dining room set. Abby realizes that when she makes a payment, Moe’ s Furniture Store first applies the money toward the interest and then reduces the outstanding balance due. (This is the U.S. Rule we discussed in Chapter 10.) For her own information, Abby worked out the first 3-month schedule of payments, shown in Table 14.4. Note how the interest payment is the rate times the outstanding balance. Today, most companies with credit card accounts calculate the finance char ge, or interest, as a percentage of the average daily balance. Interest on credit cards can be very expensive for consumers; however , interest is a source of income for credit card companies. Calculating Average Daily Balance Let’s look at the following steps for calculating the average daily balance. Remember that a cash advance is a cash loan from a credit card company .

4

An update to this act was made in 1997.

TABLE

14.4

Schedule of payments

Monthly payment number

Outstanding balance due

112% interest payment

Amount of monthly payment

Reduction in balance due

Outstanding balance due

1

$8,000.00

$120.00

$500.00

$380.00

$7,620.00

($500.00 $120.00)

($8,000.00 $380.00)

$385.70

$7,234.30

($500.00 $114.30)

($7,620.00 $385.70)

$391.49

$6,842.81

($500.00 $108.51)

($7,234.30 $391.49)

(.015 $8,000.00) 2

$7,620.00

$114.30

$500.00

(.015 $7,620.00) 3

$7,234.30

$108.51 (.015 $7,234.30)

$500.00

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CALCULATING AVERAGE DAILY BALANCE Step 1. Calculate the daily balance or amount owed at the end of each day during the billing cycle: Daily Previous Cash ⫽ ⫹ ⫹ Purchases ⫺ Payments balance balance advances Step 2. When the daily balance is the same for more than one day, multiply it by the number of days the daily balance remained the same, or the number of days of the current balance. This gives a cumulative daily balance. Step 3. Add the cumulative daily balances. Step 4. Divide the sum of the cumulative daily balances by the number of days in the billing cycle. Step 5. Finance charge ⫽ Rate per month ⫻ Average daily balance.

Following is the partial bill of Joan Ring and an explanation of how Joan’ s average daily balance and finance char ge was calculated. Note how we calculated each daily balance and then multiplied each daily balance by the number of days the balance remained the same. Take a moment to study how we arrived at 8 days. The total of the cumulative daily balances was $16,390. To get the average daily balance, we divided by the number of days in the billing cycle—30. Joan’ s finance char ge is 1 12% per month on the average daily balance. 30-day billing cycle

7 days had a balance of $450

6/20

Billing date

6/27

Payment

6/30

Charge: JCPenney

7/9

Payment

40 cr.

7/12

Cash advance

60

No. of days of current balance Step 1 30-day cycle ⫺ 22 (7 ⫹ 3 ⫹ 9 ⫹ 3) equals 8 days left with a balance of $620.

7 3 9 3 8

Previous balance

$450 $ 50 cr. 200

Current daily balance $450 400 ($450 600 ($400 560 ($600 620 ($560

⫺ ⫹ ⫺ ⫹

Extension $50) $200) $40) $60)

30 Average daily balance ⫽ Step 5

$ 3,150 1,200 5,400 1,680 4,960

Step 2

$16,390

Step 3

$16,390 ⫽ $546.33 30

Step 4

Finance charge ⫽ $546.33 ⫻ .015 ⫽ $8.19

Now try the following Practice Quiz to check your understanding of this unit.

LU 14–3

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1.

2.

DVD

Calculate the balance outstanding at the end of month 2 (use U.S. Rule) given the following: purchased $600 desk; pay back $40 per month; and char ge of 212% interest on unpaid balance. Calculate the average daily balance and finance char ge from the information that follows.

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Learning Unit 14–3

353

31-day billing cycle 8/20

Billing date

8/27

Payment

Previous balance

$210 $50 cr.

8/31

Charge: Staples

30

9/5

Payment

10 cr.

9/10

Cash advance

60

Rate 2% per month on average daily balance.

✓

Solutions

1.

2.

Monthly Interest payment $15.00 $40 (.025 $600) 2 $575 $14.38 $40 (.025 $575) Average daily balance calculated as follows:

Month 1

Balance due $600

No. of days of current balance 7 4 5 5 10

31 21 (7 4 5 5)

Reduction in balance $25.00 ($40 $15) $25.62

Current balance $210 160 ($210 $50) 190 ($160 $30) 180 ($190 $10) 240 ($180 $60)

Balance outstanding $575.00 $549.38

Extension $1,470 640 950 900 2,400

31

$6,360 $6,360 $205.16 31 Finance charge $4.10 ($205.16 .02) Average daily balance

LU 14–3a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 355)

1.

2.

Calculate the balance outstanding at the end of month 2 (use U.S. Rule) given the following: purchased $300 desk; pay back $20 per month; and char ge of 1 14% interest on unpaid balance. Calculate the average daily balance and finance char ge from the following information: 31-day billing cycle 8/21 Billing date Previous balance 8/24 Payment 8/31 Charge: Staples 9/5 Payment 9/10 Cash Advance Finance charge is 2% on average daily balance.

100 cr. 60 20 cr. 200

$400

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Chapter 14 Installment Buying, Rule of 78, and Revolving Charge Credit Cards

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Amount financed, p. 342

Amount Cash Down financed price payment

60 payments at $125.67 per month; cash price $5,295 with a $95 down payment Cash price $5,295 Down payment 95 Amount financed $5,200

Total finance charge (interest), p. 343

Total Total of Amount finance all monthly financed charge payments

(continued from above) $125.67 60 $7,540.20 per month months Amount financed 5,200.00 Finance charge

$2,340.20

Deferred payment price, p. 343

Deferred Total of Down payment all monthly payment price payments

(continued from above) $7,540.20 $95 $7,635.20

Calculating APR by Table 14.1, p. 344

Table 14.1 Finance charge $100 Amount financed lookup number

(continued from above) $2,340.20 $100 $45.004 $5,200.00 Search in Table 14.1 between 15.50% and 15.75% for 60 payments.

Monthly payment, p. 346

By formula: Finance charge Amount financed Number of payments of loan

(continued from above) $2,340.20 $5,200.00 $125.67 60 Given: 15.5% 60 months $5,200 loan $5,200 5.2 $24.05 $125.06* $1,000

By table: Loan Table (rate, months) $1,000 factor

*Off due to rounding of rate.

Paying off installment loan before due date, p. 348

1. Find balance of loan outstanding (Total of monthly payments Payments to date). 2. Calculate total finance charge. 3. Find number of payments remaining. 4. Set up rebate fraction from Table 14.3. 5. Calculate rebate amount of finance charge. 6. Calculate payoff.

Example: Loan, $8,000; 20 monthly payments of $420; end of month repaid 7. 1. $8,400 (20 $420) 2,940 (7 $420) $5,460 (balance of loan outstanding) 2. $8,400 (total payments) 8,000 (amount financed) $ 400 (total finance charge) 3. 20 7 13 91 4 and 5. $400 $173.33 210 6. $5,460.00 (Step 1) 173.33 rebate (Step 5) $5,286.67 payoff

Open-end credit, p. 350

Monthly payment applied to interest first before reducing balance outstanding.

$4,000 purchase $250 a month payment 212% interest on unpaid balance $4,000 .025 $100 interest $250 $100 $150 to lower balance $4,000 $150 $3,850 Balance outstanding after month 1.

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

355

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

Example(s) to illustrate situation

Average daily balance and finance charge, p. 352

Daily Previous Cash balance balance advances Purchases Payments

30-day billing cycle; 112% per month Example: 8/21 Balance $100 8/29 Payment $10 9/12 Charge 50 Average daily balance equals: 8 days $100 $ 800 14 days 90 1,260 8 days 140 1,120

Sum of cumulative Average daily balances daily Number of days balance in billing cycle 30-day billing cycle less the 8 and 14. Average Finance Monthly daily charge rate balance

$3,180 30 Average daily balance $106 Finance charge $106 .015 $1.59

KEY TERMS

Amortization, p. 346 Amount financed, p. 342 Annual percentage rate (APR), p. 343 Average daily balance, p. 351 Cash advance, p. 351 Daily balance, p. 352 Deferred payment price, p. 343

Down payment, p. 342 Fair Credit and Charge Card Disclosure Act of 1988, p. 351 Finance charge, p. 343 Installment loan, p. 342 Loan amortization table, p. 346 Open-end credit, p. 351

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 14–1a (p. 347) 1. a. $12,519 b. $5,181 c. $19,180 d. Bet. 14.50%–14.75% e. $295 2. $141.96

LU 14–2a (p. 350) $4,334.62 payoff; $495.38 rebate

Outstanding balance, p. 351 Rebate, p. 349 Rebate fraction, p. 349 Revolving charge account, p. 351 Rule of 78, p. 347 Truth in Lending Act, p. 343

LU 14–3a (p. 353) 1. $267.30 end of month 2 2. $410.97 $8.22

Critical Thinking Discussion Questions 1. Explain how to calculate the amount financed, finance charge, and APR by table lookup. Do you think the Truth in Lending Act should regulate interest charges? 2. Explain how to use the loan amortization table. Check with a person who owns a home and find out what part of each payment goes to pay interest versus the amount that reduces the loan principal.

3. What are the six steps used to calculate the rebate and payoff for the Rule of 78? Do you think it is right for the Rule of 78 to charge a larger portion of the finance char ges to the earlier payments? 4. What steps are used to calculate the average daily balance? Many credit card companies charge 18% annual interest. Do you think this is a justifiable rate? Defend your answer .

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Classroom Notes

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Complete the following table: Purchase price of product

Down payment

Amount financed

Number of monthly payments

Amount of monthly payments

14–1. Ford Explorer $36,900

$10,000

60

$499

14–2. Sony digital camera $700

$100

10

$80.50

Total of monthly payments

Total finance charge

Calculate (a) the amount financed, (b) the total finance charge, and (c) APR by table lookup. Purchase price of a used car

Down payment

Number of monthly payments

14–3. $5,673

$1,223

48

$5,729.76

14–4. $4,195

$95

60

$5,944.00

Amount financed

Total of monthly payments

Total finance charge

APR

Calculate the monthly payment for Problems 14–3 and 14–4 by table lookup and formula. (Answers will not be exact due to rounding of percents in table lookup.) 14–5. (14–3) (Use 13% for table lookup.)

14–6. (14–4) (Use 15.5% for table lookup.)

Calculate the finance charge rebate and payoff: Loan 14–7. $7,000 Step 1.

Months of loan

End-of-month loan is repaid

Monthly payment

36

10

$210

Finance charge rebate

Final payoff

Step 2.

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Step 3.

Step 4.

Step 5.

Step 6.

Loan 14–8. $9,000

Months of loan

End-of-month loan is repaid

Monthly payment

24

9

$440

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

Finance charge rebate

Final payoff

14–9. Calculate the average daily balance and finance charge 30-day billing cycle 9/16

Billing date

Previous balance

$2,000

9/19

Payment

9/30

Charge: Home Depot

10/3

Payment

60

10/7

Cash advance

70

$

60

1,500

Finance charge is 112 % on average daily balance

WORD PROBLEMS 14–10. The 2007 edition of Edmunds showed the manufacturer’s suggested retail price for a 2007 BMW 3 series was $35,300. Audrey McKeown is planning to put 10 percent down and finance the BMW for 60 months. Her bank has agreed to finance the car at 8 percent. She plans to budget no more than $650.00 per month for car payments. (a) What is her monthly payment to nearest cent (use loan amortization table)? (b) Will her budgeted amount be enough to purchase the BMW?

14–11. The February 26, 2007 issue of American Banker reported the average interest rate charged by commercial banking companies for new car loans rose to 7.72%. Some analysts remain concerned that rising credit costs will create a headache for lenders. Jane Stocker plans to finance a new car, in the amount of $23,600, at 7.5% for 60 months. (a) What is her monthly payment to nearest cent (use loan amortization table)? (b) How much interest will Jane pay on her loan?

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14–12. Ramon Hernandez saw the following advertisement for a used Volkswagen Bug and decided to work out the numbers to be sure the ad had no errors. Please help Ramon by calculating (a) the amount financed, (b) the finance charge, (c) APR by table lookup, (d) the monthly payment by formula, and (e) the monthly payment by table lookup (will be off slightly). a. Amount financed:

b. Finance charge: c. APR by table lookup:

d. Monthly payment by formula: e. Monthly payment by table lookup (use 14.50%):

14–13. From this partial advertisement calculate: a. Amount financed.

d. APR by Table 14.1.

b. Finance charge.

e. Check monthly payment (by formula).

c. Deferred payment price.

14–14. Paula Westing borrowed $6,200 to travel to Sweden to see her son Arthur. Her loan was to be paid in 48 monthly installments of $170. At the end of 9 months, Paula’s daughter Irene convinced her that she should pay off the loan early. What are Paula’s rebate and her payoff amount? Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

14–15. Park Plaza Dodge placed an ad in the Chicago Sun-Times on March 28, 2007 stating a new Dodge Nitro SXT was on sale for $18,999, with $2,000 down. Shirley Stewart plans to finance the car. Citizens’ Financial Bank quoted a finance charge at 8% for 48 months - Charter One bank quoted her a finance charge at 7.50% for 60 months. (a) What would be her monthly payment to Citizens’ Financial Bank to nearest cent? (b) What would be her monthly payment to Charter One Bank to nearest cent? Use loan amortization table. (c) How much more would her monthly payment be on the 48 month loan?

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14–16. Joanne Flynn bought a new boat for $14,500. She put a $2,500 down payment on it. The bank’s loan was for 48 months. Finance charges totaled $4,400.16. Assume Joanne decided to pay off the loan at the end of the 28th month. What rebate would she be entitled to and what would be the actual payoff amount?

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

14–17. First America Bank’s monthly payment charge on a 48-month $20,000 loan is $488.26. The U.S. Bank’s monthly payment fee is $497.70 for the same loan amount. What would be the APR for an auto loan for each of these banks? (Use the Business Math Handbook.)

14–18. From the following facts, Molly Roe has requested you to calculate the average daily balance. The customer believes the average daily balance should be $877.67. Respond to the customer’s concern. 28-day billing cycle 3/18

Billing date

3/24

Payment

3/29

Charge: Sears

4/5

Payment

4/9

Charge: Macy’s

Previous balance

$800 $ 60 250 20 200

14–19. Jill bought a $500 rocking chair. The terms of her revolving charge are 121% on the unpaid balance from the previous month. If she pays $100 per month, complete a schedule for the first 3 months like Table 14.4. Be sure to use the U.S. Rule. Monthly payment number

360

Outstanding balance due

112% interest payment

Amount of monthly payment

Reduction in balance due

Outstanding balance due

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CHALLENGE PROBLEMS 14–20. The base price of a Mitsubishi Endeavor XLS AWD is $29,897. With added options, the vehicle is priced at $32,142. Linda Lennon has set aside a down payment of 10% and will finance the balance at 8% for 60 months. Linda’s net pay is $3,800 a month. She has budgeted 15% of her net pay toward the monthly payment. (a) If Linda purchases the vehicle at the base price, what will be her monthly payment? (b) How much over or under budget will Linda be? (c) If Linda purchases the vehicle with the options, what will be her monthly payments? (d) How much over or under budget will Linda be?

14–21. You have a $1,100 balance on your 15% credit card. You have lost your job and been unemployed for 6 months. You have been unable to make any payments on your balance. However, you received a tax refund and want to pay off the credit card. How much will you owe on the credit card, and how much interest will have accrued? What will be the effective rate of interest after the 6 months (to nearest hundredth percent)?

DVD SUMMARY PRACTICE TEST 1.

Walter Lantz buys a Volvo SUV for $42,500. Walter made a down payment of $16,000 and paid $510 monthly for 60 months. What are the total amount financed and the total finance charge that Walter paid at the end of the 60 months? (p. 343)

2.

Joyce Mesnic bought an HP laptop computer at Staples for $699. Laura made a $100 down payment and financed the balance at 10% for 12 months. What is her monthly payment? (Use the loan amortization table.) (p. 346)

3.

Lee Remick read the following partial advertisement: price, $22,500; down payment, $1,000 cash or trade; and $399.99 per month for 60 months. Calculate (a) the total finance charge and (b) the APR by Table 14.1 (use the tables in Business Math Handbook) to the nearest hundredth percent. (p. 344)

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4.

Nancy Billows bought a $7,000 desk at Furniture.com. Based on her income, Nancy could only afford to pay back $700 per month. The charge on the unpaid balance is 3%. The U.S. Rule is used in the calculation. Calculate the balance outstanding at the end of month 2. (p. 351) Balance Monthly Reduction in Balance Month due Interest payment balance outstanding

5.

Joan Hart borrowed $9,800 to travel to France to see her son Dick. Joan’s loan was to be paid in 50 monthly installments of $250. At the end of 7 months, Joan’s daughter Abby convinced her that she should pay off the loan early. What are Joan’s rebate and payoff amount? (p. 349) Step 1.

Step 2.

Step 3.

Step 4. Step 6.

Step 5. 6.

Calculate the average daily balance and finance charge on the statement below. (p. 353) 30-day billing cycle 7/3

Balance

$400

7/18

Payment

100

7/27

Charge Wal-Mart

250

Assume 2% finance charge on average daily balance.

362

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Personal Finance A KIPLINGER APPROACH

Should I take a 0% CREDIT CARD offer? h e sh ort answer is easy. Go for a credit card with a 0% teaser rate if you can pay off the balance before the rate expires—typically in six to 12 months. If you’ll need longer to pay off your debt, look for a card with a low fixed rate. Lest you strike a Faustian bargain, however, read the fine print. Some cards waive fees only on your first balance transfer. If you want to make additional transfers, you may have to pay a fee of up to 4%—or $400 on a balance transfer of $10,000. Not everyone qualifies for the deal advertised in bold print on the credit application. A bank might bump a 3.99% rate to 5.99% if your credit is less than perfect. And with interest rates rising, it’s tougher to get a low-rate offer. “I used to be able to call my card issuers and ask for lower rates,” says Eleni Christianson of Mont-

AMANDA FRIEDMAN

T

G Eleni Christianson is using low-rate offers to pay off $18,000 in debt.

clair, Cal. “Now they tell me I have to wait for them to send me offers in the mail.” Low-rate cards can give you a breather, but you have to use the time to pay off your debt. And card issuers won’t make it easy to pay down debt quickly. Christianson and her musician husband, John, are focused on paying off $18,000 in credit-card debt—and are frustrated that Chase allows them to make only four payments per month. “They let you use your cards limitlessly, but they limit how often you can pay,” says Eleni, a project coordinator for a contractor. Be careful if you accept a low-rate transfer offer on an existing card that already has a balance. Your payments will be used to pay down the lower-rate balance first, and interest will continue to accrue on the initial balance at the higher rate. — THOMAS M. ANDERSON

I’m living paycheck to paycheck. Can I INVEST? ivi n g up one latte a day will give you plenty of money to get started. You can put it in a high-yield savings account until you have enough to invest in a mutual fund with a minimum deposit of $1,000 or more. Or, if you don’t want to wait, a number of fund companies—Ariel Mutual Funds, Investment Company of America, TIAA-CREF

G

and T. Rowe Price—will let you invest as little as $50 a month if you set up automatic deductions from your bank account or paycheck. Your best strategy is to put your money in a fund with a diversified mix of stocks and investing styles. For instance, T. Rowe Price Spectrum Growth (symbol PRSGX; 800-638-5660) invests in ten other T. Rowe Price funds. Spectrum

Growth has an annualized return of 9% over the past ten years and 6% over the past five years. Investors pay only the expenses of the underlying funds; investing $600 a year will cost you $5. Another one-stop alternative is a target-retirement fund, which shifts its asset mix to become more conservative as you get closer to retirement. T. Rowe Price Retirement 2040

(TRRDX), launched in 2002, has returned an annualized 14% over the past three years. Think of it: If you invest $50 a month in an account that earns 8% annually, you will have $74,518 in 30 years. Not a bad start for living paycheck to paycheck— and you can always bump up your investments as that paycheck gets bigger. — THOMAS M. ANDERSON

BUSINESS MATH ISSUE Kiplinger’s © 2006

Paying off debt with low-rate offers is not a good financial decision. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

363

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A Calculate the following for the Ion-2: (a) monthly payment, (b) amount financed, (c) finance charge, and (d) APR.

Boston Daily Globe by Metrowest. Copyright ©2003 by Globe Newspaper Co (MA). Reproduced with permission of GLOBE NEWSPAPER CO (MA) in the format Textbook via Copyright Clearance Center.

Number of payments

364

ANNUAL PERCENTAGE RATE 2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

61

5.25

6.54

7.94

9.34

10.69

12.04

13.46

14.92

16.24

62

5.34

6.62

8.09

9.38

10.81

12.26

13.71

15.09

16.57

63

5.44

6.70

8.21

9.59

11.06

12.43

13.90

15.37

16.89

64

5.51

6.81

8.34

9.72

11.22

12.67

14.16

15.59

17.12

65

5.58

6.89

8.46

9.92

11.39

12.83

16.36

15.89

17.34

66

5.71

7.02

8.58

10.05

11.60

13.09

14.54

16.11

17.68

67

5.76

7.15

8.74

10.19

11.73

13.30

14.85

16.33

17.94

68

5.87

7.25

8.89

10.40

11.92

13.50

15.07

16.69

18.20

69

5.89

7.46

9.00

10.57

12.14

13.70

15.27

16.89

18.45

70

6.05

7.55

9.13

10.71

12.29

13.88

15.52

17.16

18.73

71

6.14

7.68

9.25

10.87

12.52

14.05

15.73

17.45

19.01

72

6.18

7.78

9.34

11.02

12.70

14.30

15.97

17.68

19.29

Internet Projects: See text Web site (www.mhhe.com/slater9e) and The Business Math Internet Resource Guide.

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CHAPTER

15

The Cost of Home Ownership

LEARNING UNIT OBJECTIVES LU 15–1: Types of Mortgages and the Monthly Mortgage Payment • List the types of mortgages available (pp. 367–368). • Utilize an amortization chart to compute monthly mortgage payments (p. 368). urnal © 2005 Wall Street Jo

• Calculate the total cost of interest over the life of a mortgage (p. 369).

LU 15–2: Amortization Schedule—Breaking Down the Monthly Payment • Calculate and identify the interest and principal portion of each monthly payment (p. 370). • Prepare an amortization schedule (p. 371).

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Chapter 15 The Cost of Home Ownership

FIGURE

15.1

Types of mortgages available

Loan types

Advantages

Disadvantages

30-year fixed rate mortgage

A predictable monthly payment.

If interest rates fall, you are locked in to higher rate unless you refinance. (Application and appraisal fees along with other closing costs will result.)

15-year fixed rate mortgage

Interest rate lower than 30-year fixed (usually 41 to 12 of a percent). Your equity builds up faster while interest costs are cut by more than one-half.

A larger down payment is needed. Monthly payment will be higher.

Graduated-payment mortgage (GPM)

Easier to qualify for than 30- or 15-year fixed rate. Monthly payments start low and increase over time.

May have higher APR than fixed or variable rates.

Biweekly mortgage*

Shortens term loan; saves substantial amount of interest; 26 biweekly payments per year. Builds equity twice as fast.

Not good for those not seeking an early loan payoff. Extra payment per year

Adjustable rate mortgage (ARM)

Lower rate than fixed. If rates fall, could be adjusted down without refinancing. Caps available that limit how high rate could go for each adjustment period over term of loan.

Monthly payment could rise if interest rates rise. Riskier than fixed rate mortgage in which monthly payment is stable.

Home equity loan

Cheap and reliable accessible lines of credit backed by equity in your home. Tax-deductible. Rates can be locked in. Reverse mortgages may be available to those 62 or older.

Could lose home if not paid. No annual or interest caps.

Interest-only mortgages

Borrowers pay interest but no principal in the early years (5 to 15) of the loan.

Early years build up no equity.

*A different type of mortgage loan, called a mortgage accelerator loan, has come to the United States. It uses home equity borrowing and the borrower’s paycheck to shorten the time until a mortgage is paid off, saving tens of thousands in interest expense. The biweekly mortgage loan shortens a mortgage by paying an extra mortgage payment once a year, the mortgage accelerator loan program is based on an approach common in Australia and the United Kingdom, where borrowers deposit their paychecks into an account that, every month, applies every unspent dime against the mortgage loan balance.

Figure 15.1 indicates that lenders now of fer a new type of mortgage loan to home buyers— the interest-only mortgage. It would seem that to many borrowers, delaying building equity in their home will be an important disadvantage. When buying a home, purchasers can be concerned about how much a home appreciates in value and which mortgage would be best for them. The Wall Street Journal clipping “Sticker Shock” shows that home-price appreciation for a 3-year change increases 95% in South Africa versus 29% in the United States. The Wall Street Journal clipping “How the Payments Stack Up” shows how the monthly payment for a U.S. mortgage varies for three different loan types.

Wall Street Journal © 2005

Wall Street Journal © 2005

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Learning Unit 15–1

367

Purchasing a home usually involves paying a lar ge amount of interest. Note how your author was able to save $70,121.40. Over the life of a 30-year fixed rate mortgage (Figure 15.1) of $100,000, the interest would have cost $207,235. Monthly payments would have been $849.99. This would not include taxes, insurance, and so on. Your author chose a biweekly mortgage (Figure 15.1). This meant that every two weeks (26 times a year) the bank would receive $425. By paying every two weeks instead of once a month, the mortgage would be paid of f in 23 years instead of 30—a $70,121.40 savings on interest. Why? When a payment is made every two weeks, the principal is reduced more quickly , which substantially reduces the interest cost.

Learning Unit 15–1: Types of Mortgages and the Monthly Mortgage Payment In the past several years, interest rates have been low , which has caused an increase in home sales. Today, more people are buying homes than are renting homes. The question facing prospective buyers concerns which type of mortgage will be best for them. Figure 15.1 lists the types of mortgages available to home buyers. Depending on how interest rates are moving when you purchase a home, you may find one type of mortgage to be the most advantageous for you. Have you heard that elderly people who are house-rich and cash-poor can use their home to get cash or monthly income? The Federal Housing Administration makes it possible for older homeowners to take out a reverse mortgage on their homes. Under reverse mortgages, senior homeowners borrow against the equity in their property , often getting fixed monthly checks. The debt is repaid only when the homeowners or their estate sells the home. Now let’s learn how to calculate a monthly mortgage payment and the total cost of loan interest over the life of a mortgage. We will use the following example in our discussion. Gary bought a home for $200,000. He made a 20% down payment. The 9% mortgage is for 30 years (30 12 360 payments). What are Gary’s monthly payment and total cost of interest?

EXAMPLE

Computing the Monthly Payment for Principal and Interest You can calculate the principal and interest of Gary’ s monthly payment using the amortization table shown in Table 15.1 (p. 368) and the following steps. (Remember that this is the same type of amortization table used in Chapter 14 for installment loans.) COMPUTING MONTHLY PAYMENT BY USING AN AMORTIZATION TABLE Step 1.

Divide the amount of the mortgage by $1,000.

Step 2.

Look up the rate and term in the amortization table. At the intersection is the table factor.

Step 3.

Multiply Step 1 by Step 2.

For Gary, we calculate the following: $160,000 (amount of mortgage) 160 $8.05 (table rate) $1,288 $1,000 So $160,000 is the amount of the mortgage ($200,000 less 20%). The $8.05 is the table factor of 9% for 30 years per $1,000. Since Gary is mortgaging 160 units of $1,000, the factor of $8.05 is multiplied by 160. Remember that the $1,288 payment does not include taxes, insurance, and so on.

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Chapter 15 The Cost of Home Ownership

TABLE

15.1

Amortization table (mortgage principal and interest per $1,000) INTEREST

Term in years

5%

512%

6%

621%

7%

721%

8%

812%

9%

912%

10%

1012%

11%

10

10.61

10.86

11.11

11.36

11.62

11.88

12.14

12.40

12.67

12.94

13.22

13.50

13.78

12

9.25

9.51

9.76

10.02

10.29

10.56

10.83

11.11

11.39

11.67

11.96

12.25

12.54

15

7.91

8.18

8.44

8.72

8.99

9.28

9.56

9.85

10.15

10.45

10.75

11.06

11.37

17

7.29

7.56

7.84

8.12

8.40

8.69

8.99

9.29

9.59

9.90

10.22

10.54

10.86

20

6.60

6.88

7.17

7.46

7.76

8.06

8.37

8.68

9.00

9.33

9.66

9.99

10.33

22

6.20

6.51

6.82

7.13

7.44

7.75

8.07

8.39

8.72

9.05

9.39

9.73

10.08

25

5.85

6.15

6.45

6.76

7.07

7.39

7.72

8.06

8.40

8.74

9.09

9.45

9.81

30

5.37

5.68

6.00

6.33

6.66

7.00

7.34

7.69

8.05

8.41

8.78

9.15

9.53

35

5.05

5.38

5.71

6.05

6.39

6.75

7.11

7.47

7.84

8.22

8.60

8.99

9.37

TABLE

15.2

Effect of interest rates on monthly payments 9%

Monthly payment

Total cost of interest

11%

Difference

$1,288

$1,524.80

$236.80 per month

(160 $8.05)

(160 $9.53)

$303,680

$388,928

$85,248

($1,288 360) $160,000

($1,524.80 360) $160,000

($236.80 360)

What Is the Total Cost of Interest? We can use the following formula to calculate Gary’ s total interest cost over the life of the mortgage: Total cost Total of all Amount of of interest monthly payments mortgage $303,680

$463,680 ($1,288 360)

$160,000

Effects of Interest Rates on Monthly Payment and Total Interest Cost Table 15.2 shows the ef fect that an increase in interest rates would have on Gary’ s monthly payment and his total cost of interest. Note that if Gary’ s interest rate rises to 1 1%, the 2% increase will result in Gary paying an additional $85,248 in total interest. For most people, purchasing a home is a major lifetime decision. Many factors must be considered before this decision is made. One of these factors is how to pay for the home. The purpose of this unit is to tell you that being informed about the types of available mortgages can save you thousands of dollars. In addition to the mortgage payment, buying a home can include the following costs: •

•

Closing costs: When property passes from seller to buyer , closing costs may include fees for credit reports, recording costs, lawyer ’s fees, points, title search, and so on. A point is a one-time char ge that is a percent of the mortgage. Two points means 2% of the mortgage. On the following page you can see a sample of closing costs for a $125,000 mortgage in the Wall Street Journal clipping “Add-ons.” Escrow amount: Usually, the lending institution, for its protection, requires that each month 1/12 of the insurance cost and 1/12 of the real estate taxes be kept in a special account called the escrow account. The monthly balance in this account will change depending on the cost of the insurance and taxes. Interest is paid on escrow accounts.

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Learning Unit 15–1

15.1

TABLE

369

(concluded) INTEREST

Term in years 1121%

1134%

12%

1212%

1243%

13%

1321%

1334%

14%

1421%

1443%

15%

1512%

10

14.06

14.21

14.35

14.64

14.79

14.94

15.23

15.38

15.53

15.83

15.99

16.14

16.45

12

12.84

12.99

13.14

13.44

13.60

13.75

14.06

14.22

14.38

14.69

14.85

15.01

15.34

15

11.69

11.85

12.01

12.33

12.49

12.66

12.99

13.15

13.32

13.66

13.83

14.00

14.34

17

11.19

11.35

11.52

11.85

12.02

12.19

12.53

12.71

12.88

13.23

13.41

13.58

13.94

20

10.67

10.84

11.02

11.37

11.54

11.72

12.08

12.26

12.44

12.80

12.99

13.17

13.54

22

10.43

10.61

10.78

11.14

11.33

11.51

11.87

12.06

12.24

12.62

12.81

12.99

13.37

25

10.17

10.35

10.54

10.91

11.10

11.28

11.66

11.85

12.04

12.43

12.62

12.81

13.20

30

9.91

10.10

10.29

10.68

10.87

11.07

11.46

11.66

11.85

12.25

12.45

12.65

13.05

35

9.77

9.96

10.16

10.56

10.76

10.96

11.36

11.56

11.76

12.17

12.37

12.57

12.98

Reprinted by permission of The Wall Street Journal, © 2003 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

•

Repairs and maintenance: This includes paint, wallpaper , landscaping, plumbing, electrical expenses, and so on.

As you can see, the cost of owning a home can be expensive. But remember that all interest costs of your monthly payment and your real estate taxes are deductible. For many , owning a home can have advantages over renting. Before you study Learning Unit 15–2, let’ s check your understanding of Learning Unit 15–1.

LU 15–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

Given: Price of home, $225,000; 20% down payment; 9% interest rate; 25-year mortgage. Solve for: 1. Monthly payment and total cost of interest over 25 years. 2. If rate fell to 8%, what would be the total decrease in interest cost over the life of the mortgage?

✓ Solutions 1.

$225,000 $45,000 $180,000

$180,000 180 $8.40 $1,512 $1,000 $273,600 $453,600 $180,000 ($1,512 300) 25 years 12 payments per year

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8%

$1,389.60 monthly payment (180 $7.72) Total interest cost $236,880 ($1,389.60 300) $180,000 Savings $36,720 ($273,600 $236,880)

2.

LU 15–1a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 373)

Given: Price of home, $180,000; 30% down payment; 7% interest rate; 30-year mortgage. Solve for: 1. Monthly payment and total cost of interest over 30 years. 2. If rate fell to 5%, what would be the total decrease in interest cost over the life of the mortgage?

Learning Unit 15–2: Amortization Schedule—Breaking Down the Monthly Payment In Learning Unit 15–1, we saw that over the life of Gary’ s $160,000 loan, he would pay $303,680 in interest. Now let’ s use the following steps to determine what portion of Gary’ s first monthly payment reduces the principal and what portion is interest. CALCULATING INTEREST, PRINCIPAL, AND NEW BALANCE OF MONTHLY PAYMENT Step 1.

Calculate the interest for a month (use current principal): Interest Principal Rate Time.

Step 2.

Calculate the amount used to reduce the principal: Principal reduction Monthly payment Interest (Step 1).

Step 3.

Calculate the new principal: Current principal Reduction of principal (Step 2) New principal.

Step 1. Interest (I) Principal (P) Rate (R) Time (T) $1,200

$160,000

.09

1 12

Step 2. The reduction of the $160,000 principal each month is equal to the payment less interest. So we can calculate Gary’s new principal balance at the end of month 1 as follows: Monthly payment at 9% (from Table 15.1) Interest for first month

$1,288 (160 $8.05) 1,200

Principal reduction

$

88

Step 3. As the years go by , the interest portion of the payment decreases and the principal portion increases. Principal balance Principal reduction

$160,000 88

Balance of principal

$159,912

Let’s do month 2: Step 1. Interest Principal Rate Time $159,912 .09 $1,199.34

1 12

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Learning Unit 15–2

TABLE

15.3

371

Partial amortization schedule MONTHLY PAYMENT, $1,288

Payment number

Principal (current)

Interest

Principal reduction

Balance of principal

1

$160,000.00

$1,200.00

$88.00

$159,912.00

($1,288 ⫺ $1,200)

($160,000 ⫺ $88)

2

3

1 a $160,000 ⫻ .09 ⫻ b 12 $159,912.00 1 a $159,912 ⫻ .09 ⫻ b 12 $159,823.34

4

$159,734.02

5 6 7

$1,199.34

$88.66

($1,288 ⫺ $1,199.34)

($159,912 ⫺ $88.66)

$159,823.34

$1,198.68

$89.32

$159,734.02

$1,198.01

$89.99

$159,644.03

$159,644.03

$1,197.33

$90.67

$159,553.36

$159,553.36

$1,196.65

$91.35

$159,462.01

$159,462.01

$1,195.97*

$92.04

$159,369.97

*Off 1 cent due to rounding.

$1,288.00 monthly payment ⫺ 1,199.34 interest for month 2

Step 2.

Step 3.

$ 88.66 principal reduction $159,912.00 principal balance ⫺

88.66 principal reduction

$159,823.34 balance of principal Note that in month 2, interest costs drop 66 cents ($1,200.00 ⫺ $1,199.34). So in 2 months, Gary has reduced his mortgage balance by $176.66 ($88.00 ⫹ $88.66). After 2 months, Gary has paid a total interest of $2,399.34 ($1,200.00 ⫹ $1,199.34).

Example of an Amortization Schedule The partial amortization schedule given in Table 15.3 shows the breakdown of Gary’ s monthly payment. Note the amount that goes toward reducing the principal and toward payment of actual interest. Also note how the outstanding balance of the loan is reduced. After 7 months, Gary still owes $159,369.97. Often when you take out a mortgage loan, you will receive an amortization schedule from the company that holds your mortgage. In the future, you may want to consider when it would be a good time to refinance your mortgage. The two Wall Street Journal clippings “The Costs of Refinancing” and “Running the Numbers” should be helpful in making this decision. Reprinted by permission of The Wall Street Journal, © 2002 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

Reprinted by permission of The Wall Street Journal, © 2003 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

It’s time to test your knowledge of Learning Unit 15–2 with a Practice Quiz.

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LU 15–2

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

Prepare an amortization schedule for first three periods for the following: mortgage, $100,000; 11%; 30 years.

✓

Solutions

DVD PORTION TO— $100,000 mortgage; monthly payment, $953 (100 $9.53)

Payment number 1

Principal (current)

Principal reduction

Interest

$100,000

Balance of principal

$916.67

$36.33 $99,963.67 1 b ($953.00 $916.67) ($100,000 $36.33) 12 $99,963.67 $916.33 $36.67 $99,927.00 1 a $99,963.67 .11 b ($953.00 $916.33) ($99,963.67 $36.67) 12 $99,927 $916.00 $37.00 $99,890.00 a $100,000 .11

2

3

a $99,927 .11

LU 15–2a

1 b ($953.00 $916.00) ($99,927.00 $37.00) 12

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 373)

Prepare an amortization schedule for the first two periods for the following: mortgage, $70,000; 7%; 30 years.

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES Topic

Key point, procedure, formula

Example(s) to illustrate situation

Computing monthly mortgage payment, p. 367

Based on per $1,000 Table 15.1: Amount of mortgage Table rate $1,000

Use Table 15.1: 12% on $60,000 mortgage for 30 years. $60,000 60 $10.29 $1,000 $617.40

Calculating total interest cost, p. 368

Total of all Amount of monthly payments mortgage

Using example above: 30 years 360 (payments) $617.40 $222,264 60,000 $162,264 (mortgage interest over life of mortgage)

Amortization schedule, p. 371

IPRT

Using same example:

a I for month P R

1 b 12

Principal Monthly Interest reduction payment New Current Reduction of principal principal principal

Payment number 1

Portion to— Principal Interest reduction $600 $17.40

a$60,000 .12

Balance of principal $59,982.60

$617.40 $60,000.00 ba b $600.00 $17.40 2 $17.57 $59,965.03 1 $617.40 $59,982.60 ba a$59,982.60 .12 ba b 12 $599.83 $17.57 1 b 12 $599.83

a

(continues)

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Chapter Organizer and Study Guide with Check Figures for Extra Practice Quizzes

373

CHAPTER ORGANIZER AND STUDY GUIDE WITH CHECK FIGURES FOR EXTRA PRACTICE QUIZZES (concluded) Topic

Key point, procedure, formula

KEY TERMS

Adjustable rate mortgage, (ARM), p. 366 Amortization schedule, p. 371 Amortization table, p. 367 Biweekly mortgage, p. 367 Closing costs, p. 368

CHECK FIGURES FOR EXTRA PRACTICE QUIZZES WITH PAGE REFERENCES

LU 15–1a (p. 370) 1. $839.16 $176,097.60 2. $117,583.20 $58,514.40

Example(s) to illustrate situation

Escrow account, p. 368 Fixed rate mortgage, p. 367 Graduated-payment mortgages (GPM), p. 366 Home equity loan, p. 366 Interest-only mortgage, p. 366

Mortgage accelerator p. 366 Monthly payment, p. 367 Mortgages, p. 367 Points, p. 368 Reverse mortgage, p. 367

LU 15–2a (p. 372) $408.33 $57.87 $69,942.13 $408.00 $58.20 $69,833.93

Critical Thinking Discussion Questions 1. Explain the advantages and disadvantages of the following loan types: 30-year fixed rate, 15-year fixed rate, graduatedpayment mortgage, biweekly mortgage, adjustable rate mortgage, and home equity loan. Why might a bank require a home buyer to establish an escrow account?

2. How is an amortization schedule calculated? Is there a best time to refinance a mortgage? 3. What is a point? Is paying points worth the cost? 4. Would you ever consider a jumbo mortgage?

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Classroom Notes

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END-OF-CHAPTER PROBLEMS Name

Date

DRILL PROBLEMS Complete the following amortization chart by using Table 15.1. Selling price of home

Down payment

Principal (loan)

Rate of interest

Years

15–1. $140,000

$10,000

7%

25

15–2. $90,000

$5,000

512%

30

15–3. $340,000

$70,000

6%

35

Payment per $1,000

Monthly mortgage payment

15–4. What is the total cost of interest in Problem 15–2?

15–5. If the interest rate rises to 7% in Problem 15–2, what is the total cost of interest?

Complete the following: Selling price

Down payment

Amount mortgage

Rate

Years

15–6. $125,000

$5,000

7%

30

15–7. $199,000

$40,000

1212%

35

Monthly payment

First Payment Broken Down Into— Interest

Principal

Balance at end of month

15–8. Bob Jones bought a new log cabin for $70,000 at 1 1% interest for 30 years. Prepare an amortization schedule for first 3 periods. Portion to— Payment Balance of loan number

Interest

Principal

outstanding

375

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WORD PROBLEMS 15–9. The March 9, 2007 edition of the Amarillo Globe-News reported mortgage rates at a low for the year . The local rates for a 30 year mortgage at the Citibank are 6.00% and the First United Bank at 6.50%. Tom Burke plans to purchase a $350,000 home with 20 percent down. (a) What would Tom’s monthly payment be at the Citibank? (b) What would Tom’s monthly payment be at the First United Bank?

15–10. Oprah Winfrey has closed on a 42-acre estate near Santa Barbara, California, for $50,000,000. If Oprah puts 20% down and finances at 7% for 30 years, what would her monthly payment be?

15–11. Joe Levi bought a home in Arlington, Texas, for $140,000. He put down 20% and obtained a mortgage for 30 years at 512%. What is Joe’s monthly payment? What is the total interest cost of the loan?

15–12. If in Problem 15–11 the rate of interest is 7 12%, what is the difference in interest cost?

15–13. Mike Jones bought a new split-level home for $150,000 with 20% down. He decided to use Victory Bank for his mortgage. They were offering 1334% for 25-year mortgages. Provide Mike with an amortization schedule for the first three periods. Portion to— Payment Balance of loan number

Interest

Principal

outstanding

15–14. Harriet Marcus is concerned about the financing of a home. She saw a small cottage that sells for $50,000. If she puts 20% down, what will her monthly payment be at (a) 25 years, 1112%; (b) 25 years, 12 21%; (c) 25 years, 13 21%; (d) 25 years, 15%? What is the total cost of interest over the cost of the loan for each assumption? (e) What is the savings in interest cost between 1112% and 15%? (f) If Harriet uses 30 years instead of 25 for both 1 112% and 15%, what is the dif ference in interest?

376

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15–15. The Los Angeles Times on March 10, 2007 listed a home in BelAir at $12,850,000. Willard Paine is interested in purchasing this home. (a) if he put 20% down and finances at 7% for 30 years, what would be his monthly payment? (b) If he put 30% down at 7% for 30 years, what would be his monthly payment?

15–16. On January 9, 2007 the New York Times reported mortgage applications were up as home buyers see a break in rates. Daniel and Jan were part of that sur ge. Last month, the couple agreed to pay $560,000 for a four -bedroom colonial home in Waltham, Mass., with $60,000 down payment. They have a 30 year mortgage at a fixed rate of 6.00%. (a) How much is their monthly payment? (b) After the first payment, what would be the balance of the principal?

CHALLENGE PROBLEMS 15–17. A Boston historical home built in 1903 has an appraised price of $625,000 and $3,900 yearly property taxes. A Chicago historical home built in 1889 has a price of $425,000 and $5,020 yearly property taxes. Ronald Albert is offered employment in both Boston and Chicago at $125,000 a year . Ron loves historical homes, and his job acceptance will be based to a large extent on which home he is able to af ford. His local banker will finance either home at 7 12% interest with 20% down for 30 years. Ron does not want to spend more than 35% of his gross salary on monthly payments. (a) What would be the monthly payments for the home in Boston? (b) What would be the monthly payments for the home in Chicago? (c) How much more are the monthly payments for the Boston home? (d) Can Ron afford either home?

377

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15–18. Sharon Fox decided to buy a home in Marblehead, Massachusetts, for $275,000. Her bank requires a 30% down payment. Sue Willis, an attorney, has notified Sharon that besides the 30% down payment there will be the following additional costs: Recording of the deed $ 30.00 A credit and appraisal report 155.00 Preparation of appropriate documents 48.00 A transfer tax of 1.8% of the purchase price and a loan origination fee of 2.5% of the mortgage amount Assume a 30-year mortgage at a rate of 10%. a.

What is the initial amount of cash Sharon will need?

b. What is her monthly payment? c.

What is the total cost of interest over the life of the mortgage?

DVD SUMMARY PRACTICE TEST 1.

Pat Lavoie bought a home for $180,000 with a down payment of $10,000. Her rate of interest is 6% for 30 years. Calculate her (a) monthly payment; (b) first payment, broken down into interest and principal; and (c) balance of mortgage at the end of the month. (pp. 367, 368)

2.

Jen Logan bought a home in Iowa for $110,000. She put down 20% and obtained a mortgage for 30 years at 512 % . What are Jen’s monthly payment and total interest cost of the loan? (p. 368)

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3.

Christina Sanders is concerned about the financing of a home. She saw a small Cape Cod–style house that sells for $90,000. If she puts 10% down, what will her monthly payment be at (a) 30 years, 5%; (b) 30 years, 512% (c) 30 years, 6%; and (d) 30 years, 612% What is the total cost of interest over the cost of the loan for each assumption? (p. 368)

4.

Loretta Scholten bought a home for $210,000 with a down payment of $30,000. Her rate of interest is 6% for 35 years. Calculate Loretta’s payment per $1,000 and her monthly mortgage payment. (p. 367)

5.

Using Problem 4, calculate the total cost of interest for Loretta Scholten. (p. 368)

379

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Personal Finance A KIPLINGER APPROACH a short-term rent-back to you for another month or two after closing. Offer a security deposit—say, a couple of thousand dollars—and a daily rent that covers their new mortgage costs. Both parties should also check with their homeowners insurance company to make sure their homes and possessions are covered during a temporary rent-back.

Buying first

hi s st r at e gy makes the most sense if you’re in a hot market where you’re likely to encounter a bidding war over prime real estate. But in a slowing market, if you buy a new home first, you could get stuck making payments on two houses if your current home doesn’t sell fast enough. Plus, you will have to come up with cash for a down payment on the new house. Many people use their home-equity line of credit as a “bridge” loan to supply the down payment on a new home. But you need a robust income to qualify for payments on the old mortgage, the bridge loan and the mortgage on your new home. The lender will probably turn you down for the new mortgage if the payments on all three (plus any other outstanding debt) total more than 36% of your gross income. The burden of three mortgages could also force you to jump at a low-ball offer on your current house—and you can expect such offers once buyers tour the empty house and realize you need to sell. Renting out your current home is another option, and it may be a better financial move provided the rent checks more than cover the mortgage and the expense of upkeep. Tax deductions could even put you ahead of the game. If you rent out your current house, most mortgage lenders will consider up to 75% of the rent payments as income, as long as you have a signed lease.

T

GOOD TIMING —and

maybe a short-term rental— can help smooth the transition from one home to another.

Should you buy or sell first? f s e l l e r s are having trouble finding a buyer, they might consider your offer to buy their home contingent on selling your current home. You can make this contingency more palatable by including a “kick-out” clause, which allows the sellers to continue to market their home while you search for a buyer for yours. But most sellers don’t like to see a home-sale continency in a purchase offer. It ties them up with an offer that is by no means a sure thing.

I

Selling first

e a l - e s tat e agents prefer that you sell your current home first and put off serious shopping until after you’ve accepted

R

an offer. That’s safer for them: There’s more risk that an offer encumbered by a home-sale contingency will fall through—and take the agent’s paycheck with it. Trouble is, if you’ve already agreed to vacate your home in 60 days, you could feel tremendous pressure to settle for a house that falls short of your ideal. And if the sellers of the house you want know you need to find a home quickly, it can be that much harder to negotiate a good price. If you sell first, you can take some of the heat off by negotiating better terms for your home sale. Aim for a longer period until closing—say, 90 days instead of 60—or ask the buyers if they would consider

BUSINESS MATH ISSUE From Kiplinger’s Buying and Selling a Home: Make the Right Choice in any Market April, 2006, Kaplan Business 8e (ISBN 1419535781) p. 5.

380

A bridge loan is the only way to buy and sell a home. 1. List the key points of the article and information to support your position. 2. Write a group defense of your position using math calculations to support your view.

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Slater’s Business Math Scrapbook with Internet Application Putting Your Skills to Work PROJECT A

Explain the relationship between interest rates and points.

urnal © 2006 Wall Street Jo 2006 Wall Street Journal ©

b site text We he e e S : s t T t Projec /slater9e) and e. Interne ce Guid m r o u .c o e s h e h R t .m e (www Intern ss Math Busine

381

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CHAPTER

16

How to Read, Analyze, and Interpret Financial Reports

LEARNING UNIT OBJECTIVES LU 16–1: Balance Sheet—Report as of a Particular Date • Explain the purpose and the key items on the balance sheet (pp. 383–386). • Explain and complete vertical and horizontal analysis (pp. 386–388).

LU 16–2: Income Statement—Report for a Specific Period of Time • Explain the purpose and the key items on the income statement (pp. 389–392). • Explain and complete vertical and horizontal analysis (p. 392–394).

LU 16–3: Trend and Ratio Analysis • Explain and complete a trend analysis (p. 394). • List, explain, and calculate key financial ratios (p. 394).

Wall Street Jo urnal © 2006

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Learning Unit 16–1

383

Barron’s © 2005

Wall Street Journal © 2005

The Wall Street Journal article that follows the heading “Living with Sarbanes-Oxley” states that after a wave of business scandals (Enron and world.com), the Sarbanes-Oxley Act was passed to ensure public companies are accurately reporting their financial statements. As you will see in this chapter , an understatement of expenses overstates the reported earnings or net income of a company . This overstatement presents a false picture of the company’ s financial position. This chapter explains how to analyze two key financial reports: the balance sheet (shows a company’ s financial condition at a particular date) and the income statement (shows a company’s profitability over a time period). 1 Business owners must understand their financial statements to avoid financial dif ficulties. This includes knowing how to read, analyze, and interpret financial reports.

Learning Unit 16–1: Balance Sheet—Report As of a Particular Date The Wall Street Journal clipping “Moving the Goalposts” is a subheading of the clipping “Living with Sarbanes-Oxley.” Under the sub-head it is stated that SarbanesOxley requires internal controls so that public companies record assets, liabilities, and other item accurately on financial statements.

Wall Street Journal © 2005 1

The third key financial report is the statement of cash flows. We do not discuss this statement. For more information on the st ment of cash flows, check your accounting text.

ate-

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Chapter 16 How to Read, Analyze, and Interpret Financial Reports

The balance sheet gives a financial picture of what a company is worth as of a particular date, usually at the end of a month or year . This report lists (1) how much the company owns (assets), (2) how much the company owes (liabilities), and (3) how much the owner (owner ’s equity) is worth. Note that assets and liabilities are divided into two groups: current ( short term, usually less than one year); and long term, usually more than one year . The basic formula for a balance sheet is as follows: Assets Liabilities Owner’s equity Like all formulas, the items on both sides of the equal sign must balance. By reversing the above formula, we have the following common balance sheet layout: Assets Liabilities Owner’s equity

David Young Wolff/PhotoEdit

To introduce you to the balance sheet, let’ s assume that you collect baseball cards and decide to open a baseball card shop. As the owner of The Card Shop, your investment, or owner’s equity, is called capital. Since your business is small, your balance sheet is short. After the first year of operation, The Card Shop balance sheet looks like this below . The heading gives the name of the company , title of the report, and date of the report. Note how the totals of both sides of the balance sheet are the same. This is true of all balance sheets. THE CARD SHOP Balance Sheet December 31, 2009

Liabilities

Assets Capital does not mean cash. It is the owner’s investment in the company.

Cash

$ 3,000

Merchandise inventory (baseball cards)

Report as of a particular date

Accounts payable Owner’s Equity

4,000

Equipment

3,000

Total assets

$10,000

$ 2,500

E. Slott, capital Total liabilities and owner’s equity

We can take figures from the balance sheet of mula to determine how much the business is worth:

7,500 $10,000

The Card Shop and use our first for-

Assets Liabilities Owner’s equity (capital) $10,000 $2,500

$7,500

Since you are the single owner of The Card Shop, your business is a sole proprietorship. If a business has two or more owners, it is a partnership. A corporation has many owners or stockholders, and the equity of these owners is called stockholders’ equity. Now let’s study the balance sheet elements of a corporation.

Elements of the Balance Sheet The format and contents of all corporation balance sheets are similar . Figure 16.1 (p. 385) shows the balance sheet of Mool Company . As you can see, the formula Assets Liabilities Stockholders’ equity (we have a corporation in this example) is also the framework of this balance sheet. To help you understand the three main balance sheet groups (assets, liabilities, and stockholders’ equity) and their elements, we have labeled them in Figure 16.1. An explanation of these groups and their elements follows. Do not try to memorize the elements. Just try to understand their meaning. Think of Figure 16.1 as a reference aid. You will find that the more you work with balance sheets, the easier it is for you to understand them. 1.

Assets: Things of value owned by a company (economic resources of the company) that can be measured and expressed in monetary terms. a. Current assets: Assets that companies consume or convert to cash within 1 year or a normal operating cycle.

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Learning Unit 16–1

FIGURE

16.1

Balance sheet MOOL COMPANY Balance Sheet December 31, 2009

Assets broken down into current assets and plant and equipment

Put in heading who, what, when

$ 7,000

Accounts receivable

d.

Merchandise inventory

e. f.

9,000 2.

30,000

Prepaid expenses

15,000

Total current assets

$ 61,000

g. Plant and equipment: h.

Building (net)

i.

Land

j.

a. Current liabilities:

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

Cash

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

1.

a. Current assets:

c.

Liabilities broken down into current and long-term

Liabilities

Assets

b.

385

a. Common stock

b.

Accounts payable

c.

Salaries payable

d.

$80,000 12,000

Total current liabilities

$ 92,000

e. Long-term liabilities: f. g.

Mortgage note payable

58,000

Total liabilities

$150,000

$60,000 Stockholders’ Equity

84,000

Total plant and equipment

144,000 3.

k. Total assets

$205,000 Total of current assets and plant and equipment

2.

$20,000

b. Retained earnings c.

35,000

Total stockholders’ equity

d. Total liabilities and stockholders’ equity

Total is double-ruled

55,000 $205,000

Total of all liabilities and stockholders’ equity

b. Cash: Total cash in checking accounts, savings accounts, and on hand. c. Accounts receivable: Money owed to a company by customers from sales on account (buy now, pay later). d. Merchandise inventory: Cost of goods in stock for resale to customers. e. Prepaid expenses: The purchases of a company are assets until they expire (insurance or rent) or are consumed (supplies). f. Total current assets: Total of all assets that the company will consume or convert to cash within 1 year. g. Plant and equipment: Assets that will last longer than 1 year. These assets are used in the operation of the company. h. Building (net): The cost of the building minus the depreciation that has accumulated. Usually, balance sheets show this as “Building less accumulated depreciation.” In Chapter 17 we discuss accumulated depreciation in greater detail. i. Land: An asset that does not depreciate, but it can increase or decrease in value. j. Total plant and equipment: Total of building and land, including machinery and equipment. k. Total assets: Total of current assets and plant and equipment. Liabilities: Debts or obligations of the company. a. Current liabilities: Debts or obligations of the company that are due within 1 year. b. Accounts payable: A current liability that shows the amount the company owes to creditors for services or items purchased. c. Salaries payable: Obligations that the company must pay within 1 year for salaries earned but unpaid. d. Total current liabilities: Total obligations that the company must pay within 1 year. e. Long-term liabilities: Debts or obligations that the company does not have to pay within 1 year. f. Mortgage note payable: Debt owed on a building that is a long-term liability; often the building is the collateral. g. Total liabilities: Total of current and long-term liabilities.

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Chapter 16 How to Read, Analyze, and Interpret Financial Reports

3.

Stockholders’ equity (owner’s equity): The rights or interest of the stockholders to assets of a corporation. If the company is not a corporation, the term owner’s equity is used. The word capital follows the owner’s name under the title Owner’s Equity. a. Common stock: Amount of the initial and additional investment of corporation owners by the purchase of stock. b. Retained earnings: The amount of corporation earnings that the company retains, not necessarily in cash form. c. Total stockholders’ equity: Total of stock plus retained earnings. d. Total liabilities and stockholders’ equity: Total current liabilities, long-term liabilities, stock, and retained earnings. This total represents all the claims on assets— prior and present claims of creditors, owners’ residual claims, and any other claims.

Now that you are familiar with the common balance sheet items, you are ready to analyze a balance sheet.

Vertical Analysis and the Balance Sheet Often financial statement readers want to analyze reports that contain data for two or more successive accounting periods. To make this possible, companies present a statement

FIGURE

16.2

ROGER COMPANY Comparative Balance Sheet December 31, 2008 and 2009

Comparative balance sheet: Vertical analysis

We divide each item by the total of assets.

2009

2008

Amount

Percent

Amount

Percent

$22,000

25.88

$18,000

22.22

8,000

9.41

9,000

11.11

Merchandise inventory

9,000

10.59

7,000

8.64

Prepaid rent

4,000

4.71

5,000

6.17

$43,000

50.59

$39,000

48.15*

$18,000

21.18

$18,000

22.22

24,000

29.63

Assets Current assets: Cash Portion ($8,000)

Base Rate ($85,000) (?) ($29,000)

Accounts receivable

Total current assets Plant and equipment: Building (net) Land

24,000

Total plant and equipment Total assets

$42,000

28.24 49.41*

$42,000

51.85 100.00

$85,000

100.00

$81,000

$14,000

16.47

$8,000

9.88

18,000

21.18

17,000

20.99

$32,000

37.65

$25,000

12,000

14.12

Liabilities Current liabilities: Accounts payable Salaries payable We divide each item by the total of liabilities and stockholders’ equity.

Total current liabilities

Mortgage note payable Total liabilities Portion ($20,000) Base Rate (?) ($85,000) ($29,000)

30.86*

Long-term liabilities: $44,000

20,000

24.69

51.76*

$45,000

55.56*

$20,000

23.53

$20,000

24.69

21,000

24.71

16,000

19.75

$41,000

48.24

$36,000

44.44

$85,000

100.00

$81,000

100.00

Stockholders’ Equity Common stock Retained earnings Total stockholders’ equity Total liabilities and stockholders’ equity

Note: All percents are rounded to the nearest hundredth percent. *Due to rounding.

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Learning Unit 16–1

showing the data from these periods side by side. As you might expect, this statement is called a comparative statement. Comparative reports help illustrate changes in data. Financial statement readers should compare the percents in the reports to industry percents and the percents of competitors. Figure 16.2 (p. 386) shows the comparative balance sheet of Roger Company . Note that the statement analyzes each asset as a percent of total assets for a single period. The statement then analyzes each liability and equity as a percent of total liabilities and stockholders’ equity. We call this type of analysis vertical analysis. The following steps use the portion formula to prepare a vertical analysis of a balance sheet. PREPARING A VERTICAL ANALYSIS OF A BALANCE SHEET Step 1.

Divide each asset (the portion) as a percent of total assets (the base). Round as indicated.

Step 2.

Round each liability and stockholders’ equity (the portions) as a percent of total liabilities and stockholders’ equity (the base). Round as indicated.

We can also analyze balance sheets for two or more periods by using horizontal analysis. Horizontal analysis compares each item in one year by amount, percent, or both with the same item of the previous year . Note the Abby Ellen Company horizontal analysis

FIGURE

16.3

ABBY ELLEN COMPANY Comparative Balance Sheet December 31, 2008 and 2009

Comparative balance sheet: Horizontal analysis

Difference between 2008 and 2009 Portion –($1,000) Base Rate ($6,000) (?)

Increase (decrease) 2008

Amount

$ 6,000

$ 4,000

$2,000

Percent

Assets Current assets: Cash

50.00*

Accounts receivable

5,000

6,000

(1,000)

16.67

Merchandise inventory

9,000

4,000

5,000

125.00

(2,000)

28.57

Prepaid rent Total current assets

2008

2009

5,000

7,000

$25,000

$21,000

$12,000

$12,000

–0–

–0–

18,000

18,000

–0–

–0–

$30,000

$30,000

–0–

–0–

$55,000

$51,000

$4,000

7.84

$ 3,200

$ 1,800

$1,400

77.78

$4,000

19.05

Plant and equipment: Building (net) Land Total plant and equipment Total assets Liabilities Current liabilities: Accounts payable Salaries payable Total current liabilities

2,900

3,200

$ 6,100

$ 5,000

(300)

9.38

$1,100

22.00

Long-term liabilities: Mortgage note payable

17,000

15,000

2,000

13.33

$23,100

$20,000

$3,100

15.50

Abby Ellen, capital

$31,900

$31,000

$ 900

2.90

Total liabilities and owner’s equity

$55,000

$51,000

$4,000

7.84

Total liabilities Owner’s Equity

*The percents are not summed vertically in horizontal analysis.

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shown in Figure 16.3 (p. 387). To make a horizontal analysis, we use the portion formula and the steps that follow . PREPARING A HORIZONTAL ANALYSIS OF A COMPARATIVE BALANCE SHEET Step 1.

Calculate the increase or decrease (portion) in each item from the base year.

Step 2.

Divide the increase or decrease in Step 1 by the old or base year.

Step 3.

Round as indicated.

You can see the dif ference between vertical analysis and horizontal analysis by looking at the example of vertical analysis in Figure 16.2 (p. 386). The percent calculations in Figure 16.2 are for each item of a particular year as a percent of that year ’s total assets or total liabilities and stockholders’ equity. Horizontal analysis needs comparative columns because we take the dif ference between periods. In Figure 16.3, for example, the accounts receivable decreased $1,000 from 2008 to 2009. Thus, by dividing $1,000 (amount of change) by $6,000 (base year), we see that Abby’s receivables decreased 16.67%. Let’s now try the following Practice Quiz.

LU 16–1

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

1.

Complete this partial comparative balance sheet by vertical analysis. Round percents to the nearest hundredth. 2009

DVD

Amount

Percent

2008 Amount

Percent

Assets Current assets: a. Cash b. Accounts receivable

✓ 1.

$ 40,000

18,000

17,000

c. Merchandise inventory

15,000

12,000

d. Prepaid expenses

17,000

14,000

•

•

•

•

•

•

•

•

•

$160,000

$150,000

Total current assets

2.

$ 42,000

What is the amount of change in merchandise inventory and the percent increase?

Solutions

a.

Cash

b. Accounts receivable c.

Merchandise inventory

d. Prepaid expenses $15,000 12,000

2. Amount

$ 3,000

2009 $42,000 $160,000 $18,000 $160,000 $15,000 $160,000 $17,000 $160,000 Percent

26.25% 11.25% 9.38% 10.63%

$3,000 25% $12,000

2008 $40,000 $150,000 $17,000 $150,000 $12,000 $150,000 $14,000 $150,000

26.67% 11.33% 8.00% 9.33%

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LU 16–1a

389

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 398)

1.

Complete this partial comparative balance sheet by vertical analysis. Round percents to the nearest hundredth. 2009 Amount

Percent

2008 Amount

Percent

Assets Current assets: a. Cash

$ 38,000

$ 35,000

b. Accounts receivable

$ 19,000

$ 18,000

c. Merchandise inventory

$ 16,000

$ 11,000

d. Prepaid expenses

$ 20,000

$ 16,000

•

•

•

•

•

•

• Total current assets

•

•

$180,000

$140,000

2. What is the amount of change in merchandise inventory and the percent increase?

Learning Unit 16–2: Income Statement—Report for a Specific Period of Time One of the most important departments in a company is its accounting department. The job of the accounting department is to determine the financial results of the company’s operations. Is the company making money or losing money? Are the numbers presented by the accounting department correct? What happens if the accounting department has made accounting errors? The Wall Street Journal clipping “Kodak Restates Losses on Accounting Errors” answers these questions. As you can see, the Kodak accounting department made errors; as a result, the errors will af fect the company’ s losses.

The McGraw-Hill Companies, Jill Braaten photographer

Wall Street Journal © 2005

In this learning unit we look at the income statement—a financial report that tells how well a company is performing (its profitability or net profit) during a specific period of time (month, year, etc.). In general, the income statement reveals the inward flow of revenues (sales) against the outward or potential outward flow of costs and expenses. The form of income statements varies depending on the company’ s type of business. However, the basic formula of the income is the same: Revenues ⫺ Operating expenses ⫽ Net income In a merchandising business like The Card Shop, we can enlar ge on this formula:

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Chapter 16 How to Read, Analyze, and Interpret Financial Reports

FIGURE

16.4

Income statement MOOL COMPANY Income Statement For Month Ended December 31, 2009

Report for a specific period

Put in heading who, what, when

Revenues: ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

1.

a.

Gross sales

b.

Less: Sales returns and allowances

c. d.

$22,080

Actual sales after discounts and returns

$ 1,082

Sales discounts

432

1,514

Net sales

$20,566

Cost of merchandise (goods) sold: ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

2.

a.

Merchandise inventory December 1, 2009

b.

Purchases

$1,248

c.

Less: Purchase returns and allowances

d.

Less: Purchase discounts

e.

Cost of net purchases

f.

Cost of merchandise (goods available for sale)

g.

Less: Merchandise inventory, December 31, 2009

$10,512 $336 204

540 9,972 1,600

h. Cost of merchandise (goods) sold

3. {

Inventory not yet sold

$11,220

9,620

Gross profit from sales

$10,946

Operating expenses: ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

4.

5. {

a.

Salary

$ 2,200

b.

Insurance

c.

Utilities

400

d.

Plumbing

120

e.

Rent

410

f.

Depreciation

200

g.

Net sales Cost of merchandise (goods) sold

1,300

Total operating expenses

4,630

Net income

$ 6,316

Gross profit Operating expenses

Note: Numbers are subtotaled from left to right.

After any returns, allowances, or discounts Revenues (sales) Cost of merchandise or goods Baseball cards Gross profit from sales Operating expenses Net income (profit)

THE CARD SHOP Income Statement For Month Ended December 31, 2009 Revenues (sales) Cost of merchandise (goods) sold Gross profit from sales Operating expenses Net income

$8,000 3,000 $5,000 750 $4,250

Now let’s look at The Card Shop’ s income statement to see how much profit The Card Shop made during its first year of operation. For simplicity, we assume The Card Shop sold all the cards it bought during the year . For its first year of business, The Card Shop made a profit of $4,250. We can now go more deeply into the income statement elements as we study the income statement of a corporation.

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Elements of the Corporation Income Statement Figure 16.4 (p. 390) gives the format and content of the Mool Company income statement— a corporation. The five main items of an income statement are revenues, cost of merchandise (goods) sold, gross profit on sales, operating expenses, and net income. We will follow the same pattern we used in explaining the balance sheet and define the main items and the letter-coded subitems. 1.

Revenues: Total earned sales (cash or credit) less any sales returns and allowances or sales discounts. a. Gross sales: Total earned sales before sales returns and allowances or sales discounts. Note in the Wall Street Journal clipping “Heinz Profit Falls 19% Despite Sales Increase” that sales increased for Heinz but its profits were down.

Toby Talbot/AP Wide World

Wall Street Journal © 2005

b. Sales returns and allowances: Reductions in price or reductions in revenue due to goods returned because of product defects, errors, and so on. When the buyer keeps the damaged goods, an allowance results. c. Sales (not trade) discounts: Reductions in the selling price of goods due to early customer payment. For example, a store may give a 2% discount to a customer who pays a bill within 10 days. d. Net sales: Gross sales less sales returns and allowances less sales discounts. 2.

Cost of merchandise (goods) sold: All the costs of getting the merchandise that the company sold. The cost of all unsold merchandise (goods) will be subtracted from this item (ending inventory). a. Merchandise inventory, December 1, 2009: Cost of inventory in the store that was for sale to customers at the beginning of the month. b. Purchases: Cost of additional merchandise brought into the store for resale to customers. c. Purchase returns and allowances: Cost of merchandise returned to the store due to damage, defects, errors, and so on. Damaged goods kept by the buyer result in a cost reduction called an allowance. d. Purchase discounts: Savings received by the buyer for paying for merchandise before a certain date. These discounts can result in a substantial savings to a company. e. Cost of net purchases: Cost of purchases less purchase returns and allowances less purchase discounts. f. Cost of merchandise (goods available for sale): Sum of beginning inventory plus cost of net purchases.

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Merchandise inventory, December 31, 2009: Cost of inventory remaining in the store to be sold. h. Cost of merchandise (goods) sold: Beginning inventory plus net purchases less ending inventory. Gross profit from sales: Net sales less cost of merchandise (goods) sold. Operating expenses: Additional costs of operating the business beyond the actual cost of inventory sold. a.–f. Expenses: Individual expenses broken down. g. Total operating expenses: Total of all the individual expenses. Net income: Gross profit less operating expenses. g.

3. 4.

5.

In the next section you will learn some formulas that companies use to calculate various items on the income statement.

Calculating Net Sales, Cost of Merchandise (Goods) Sold, Gross Profit, and Net Income of an Income Statement It is time to look closely at Figure 16.4 (p. 390) and see how each section is built. Use the previous vocabulary as a reference. We will study Figure 16.4 step by step. Step 1.

Calculate the net sales—what Mool earned: Net sales Gross sales

$20,566 $22,080 Step 2.

Sales returns Sales discounts and allowances

$1,082

$432

Calculate the cost of merchandise (goods) sold: Cost of Net purchases Beginning Ending merchandise (purchases less inventory inventory (goods) sold returns and discounts)

$9,620 Step 3.

$1,248

$9,972

$1,600

Calculate the gross profit from sales—profit before operating expenses: Gross profit Cost of merchandise Net sales from sales (goods) sold

$10,946 $20,566 Step 4.

$9,620

Calculate the net income—profit after operating expenses: Net income Gross profit Operating expenses

$6,316

$10,946

$4,630

Analyzing Comparative Income Statements We can apply the same procedures of vertical and horizontal analysis to the income statement that we used in analyzing the balance sheet. Let’ s first look at the vertical analysis for Royal Company , Figure 16.5 (p. 393). Then we will look at the horizontal analysis of Flint Company’s 2008 and 2009 income statements shown in Figure 16.6 (p. 393). Note in the margin how numbers are calculated. The following Practice Quiz will test your understanding of this unit.

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Learning Unit 16–2

16.5

FIGURE

ROYAL COMPANY Comparative Income Statement For Years Ended December 31, 2008 and 2009

Vertical analysis

Net sales Individual amount

393

Cost of merchandise sold Gross profit from sales

2009

Percent of net

2008

Percent of net

$45,000

100.00

$29,000

100.00*

19,000

42.22

12,000

41.38

$26,000

57.78

$17,000

58.62

Operating expenses:

Portion ($12,000)

Depreciation

Base Rate ($29,000) (?) ($29000)

$ 1,000

2.22

500

1.72

Selling and advertising

4,200

9.33

1,600

5.52

Research

2,900

6.44

2,000

6.90

500

1.11

200

.69

19.11†

$ 4,300

14.83

Miscellaneous

Net sales

$

Total operating expenses

$ 8,600

Income before interest and taxes

$17,400

38.67

$12,700

43.79

6,000

13.33

3,000

10.34

$ 9,700

33.45

Interest expense Income before taxes

$11,400

Provision for taxes Net income

25.33†

5,500

12.22

3,000

$ 5,900

13.11

$ 6,700

10.34 23.10†

*Net sales 100% † Off due to rounding.

16.6

FIGURE

FLINT COMPANY Comparative Income Statement For Years Ended December 31, 2008 and 2009

Horizontal analysis

INCREASE (DECREASE)

Difference from 2008 and 2009

Sales Sales returns and allowances Net sales

Portion ($4,000)

Cost of merchandise (goods) sold Gross profit from sales

Base Rate ($12,000) (?) ($29000)

Depreciation Selling and administrative Research

Amount

Percent

$90,000

$80,000

$10,000

2,000

2,000

–0–

$88,000

$78,000

$10,000

12.82

45,000

40,000

5,000

12.50

$43,000

$38,000

$ 5,000

13.16

$ 6,000

$ 5,000

$ 1,000

20.00

16,000

12,000

4,000

600

1,000

(400)

33.33 40.00

1,200

500

700

140.00

Total operating expenses

$23,800

$18,500

$ 5,300

28.65

Income before interest and taxes

$19,200

$19,500

$

(300)

1.54

$

(300)

1.94

(200)

5.00

(100)

.87

Miscellaneous

Interest expense Income before taxes Provision for taxes Net income

4,000

4,000

$15,200

$15,500

3,800

4,000

$11,400

$11,500

–0–

$

PRACTICE QUIZ

Complete this Practice Quiz to see how you are doing

DVD

2008

Operating expenses:

Old year—2008

LU 16–2

2009

From the following information, calculate: a. Net sales. c. Gross profit from sales. b. Cost of merchandise (goods) sold. d. Net income. Given Gross sales, $35,000; sales returns and allowances, $3,000; beginning inventory , $6,000; net purchases, $7,000; ending inventory , $5,500; operating expenses, $7,900.

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✓

Solutions

a. $35,000 $3,000 $32,000 (Gross sales Sales returns and allowances) b. $6,000 $7,000 $5,500 $7,500 (Beginning inventory Net purchases Ending inventory) c. $32,000 $7,500 $24,500 (Net sales Cost of merchandise sold) d. $24,500 $7,900 $16,600 (Gross profit from sales Operating expenses)

LU 16–2a

EXTRA PRACTICE QUIZ

Need more practice? Try this Extra Practice Quiz (check figures in Chapter Organizer, p. 398)

From the following information, calculate: a. Net sales c. Gross profit from sales b. Cost of merchandise (goods) sold d. Net income Given: Gross sales, $36,000; sales returns and allowances, $2,800; beginning inventory , $5,900; net purchases, $6,800; ending inventory , $5,200; operating expenses, $8,100.

Learning Unit 16–3: Trend and Ratio Analysis Now that you understand the purpose of balance sheets and income statements, you are ready to study how experts look for various trends as they analyze the financial reports of companies. This learning unit discusses trend analysis and ratio analysis. The study of these trends is valuable to businesses, financial institutions, and consumers.

Trend Analysis Many tools are available to analyze financial reports. When data cover several years, we can analyze changes that occur by expressing each number as a percent of the base year . The base year is a past period of time that we use to compare sales, profits, and so on, with other years. We call this trend analysis. Using the following example of Rose Company , we complete a trend analysis with the following steps: COMPLETING A TREND ANALYSIS Step 1.

Select the base year (100%).

Step 2.

Express each amount as a percent of the base year amount (rounded to the nearest whole percent).

GIVEN (BASE YEAR 2007) 2010

2009

2008

2007

$621,000

$460,000

$340,000

$420,000

Gross profit

182,000

141,000

112,000

124,000

Net income

48,000

41,000

22,000

38,000

Sales

TREND ANALYSIS

Sales

2010

2009

2008

2005 2007

148%

110%

81%

100%

Gross profit

147

114

90

100

Net income

126

108

58

100

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How to Calculate Trend Analysis Sales for 2008 Each item $340,000 80.95%