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CliffsQuickReview®

Math Word

Problems

By Karen L. Anglin

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CliffsQuickReview®

Math Word

Problems

By Karen L. Anglin

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About the Author

Karen Anglin has been a mathematics instructor at

Blinn College in Brenham, Texas, since 1990. She

has given numerous award-winning presentations

to educators on the topic of math word problems.

Mrs. Anglin has a B.S. degree in Mathematics with

a secondary teaching certiﬁcate.

Publisher’s Acknowledgments

Editorial

Senior Acquisitions Editor: Greg Tubach

Project Editor: Marcia L. Johnson

Copy Editor: Tere Drenth

Technical Editor: David Herzog

Editorial Assistant: Amanda Harbin

Production

Indexer: Joan Grifﬁtts

Proofreader: Linda Quigley

Wiley Publishing, Inc. Composition Services

CliffsQuickReview®Math Word Problems

Published by:

Wiley Publishing, Inc.

111 River Street

Hoboken, NJ 07030-5774

www.wiley.com

Published by Wiley, Hoboken, NJ

Published simultaneously in Canada

ISBN: 0-7645-4492-6

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

1O/SY/QU/QU/IN

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,

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Library of Congress Cataloging-in-Publication Data

Anglin, Karen L., 1958-

Math word problems / by Karen L. Anglin.

p. cm. — (CliffsQuickReview)

Includes index.

ISBN 0-7645-4492-6 (pbk.)

1. Word problems (Mathematics)—Outlines, syllabi, etc. 2. Prob-

lem solving—Outlines, syllabi, etc. I. Title. II. Cliffs quick review.

QA63.A64 2004

510—dc22

2004005075

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Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Visit Our Web Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 1: Translating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Keywords of Basic Mathematical Operations . . . . . . . . . . . . . . . . . . . . 3

Addition keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Subtraction keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Multiplication keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Division keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Keywords That Indicate a Change in Order . . . . . . . . . . . . . . . . . . . . 11

Basic turnaround words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Addition turnaround words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Subtraction turnaround words . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Multiplication turnaround words . . . . . . . . . . . . . . . . . . . . . . . . . 14

Division turnaround words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 2: Inserting Parentheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

Translating a Comma and IS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Leading Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Adjacent Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Turnaround Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Chapter 3: Simplifying Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Distributive Property of Multiplication over Addition . . . . . . . . . . . . 30

Combining Like Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Distributing a Negative Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Chapter 4: Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

Keywords Indicating Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Solving Simple Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Deﬁning an algebraic expression . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Deﬁning an algebraic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Four steps for solving simple linear equations . . . . . . . . . . . . . . . . 40

Checking Solutions and Translations . . . . . . . . . . . . . . . . . . . . . . . . . 43

Checking solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Checking your translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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iv CliffsQuickReview Math Word Problems

Chapter 5: Geometry Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Perimeter of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Area of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Sum of the measures of the angles . . . . . . . . . . . . . . . . . . . . . . . . . 56

Rectangles and Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Perimeter of a rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Area of a rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Perimeter of a square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Area of a square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Multiple Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Other Easily Visualized Word Problems . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 6: Proportions and Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . .73

Proportion Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Proportion Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Percent Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Chapter 7: Summation Problems Using the Board Method . . . . . . . . .89

Simple Board Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Consecutive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Consecutive Even and Odd Integers . . . . . . . . . . . . . . . . . . . . . . . . . 99

Other Problems that Can Be Represented with a Board . . . . . . . . . . 105

Chapter 8: Solving Word Problems Using Tried-and-True Methods . . . .117

Polya’s Four-Step Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Expanding on Polya’s four steps . . . . . . . . . . . . . . . . . . . . . . . . . . 118

The ﬁve steps of the tried-and-true methods . . . . . . . . . . . . . . . . 119

Consecutive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Consecutive Even and Odd Integers . . . . . . . . . . . . . . . . . . . . . . . . 123

Other Problems that Cannot Be Represented with a Board . . . . . . . 126

Chapter 9: Systems of Equations Using the Board Method . . . . . . .135

Money Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Investment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Mixture Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Distance Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Chapter 10: Systems of Equations Using Tried-and-True Methods . . . .155

Money Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Investment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Mixture Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Distance Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

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Table of Contents v

Chapter 11: Common Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175

Variable Omission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Variable Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Methods for Avoiding Common Errors . . . . . . . . . . . . . . . . . . . . . . 182

A ﬁrm foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Errors in translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Errors in solving an equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Errors in answering the question . . . . . . . . . . . . . . . . . . . . . . . . . 184

CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187

CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203

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INTRODUCTION

CliffsQuickReview Math Word Problems is designed to give a clear, con-

cise, easy-to-use review of the basics of solving math word problems.

Introducing each topic, defining key terms, and carefully “walking” through

each sample problem type in a step-by-step manner gives you insight and

understanding to solving math word problems. The goal of this book is to

guide you through successful basic algebraic problem solving.

The prerequisite to benefit the most from this book is some basic pre-

algebra skills and knowledge—knowing what a variable is, being able to

use the order of operations, being able to perform arithmetic with nega-

tive numbers, being able to find the least common multiple, and being

familiar with solving a simple equation. The process of solving a simple

equation is reviewed in detail. If you need to be reminded of some of the

terminology and properties, use the Glossary at the back of this book.

One of the skills you develop by reading this book is the ability to identify

what type of word problems you are asked to solve. Another skill is the abil-

ity to match a word problem with the appropriate problem solving strategy.

For this reason, at the end of each chapter, you are given review problems

of several different types. These can be challenging and will test what you

have learned. Remember that, in life, problems are not in section order!

Chapter 4 is the first chapter in which you begin solving story problems.

The first three chapters set a firm foundation upon which you build your

skills for solving word problems.

Why You Need This Book

Can you answer yes to any of these questions?

■Do you need to review math word problems?

■Do you need a course supplement to a math class?

■Do you need a concise, comprehensive reference for math word problems?

■Do you want to have more confidence when faced with a word problem?

If so, then CliffsQuickReview Math Word Problems is for you!

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2CliffsQuickReview Math Word Problems

How to Use This Book

You can use this book in any way that fits your personal style for study and

review—you decide what works best with your needs. You can either read

the book from cover to cover or just look for the information you want

and put it back on the shelf for later. Here are a few ways you can search

for topics:

■Use the Pocket Guide to find essential information, such as keywords

to identify operations, keywords for translating in the correct order,

indications to insert parentheses, ways to solve equations, and ways

to choose the appropriate strategy for the word problem.

■Look for areas of interest in the book’s Table of Contents, or use the

index to find specific topics.

■Flip through the book looking for subject areas at the top of each page.

■Get a glimpse of what you’ll gain from a chapter by reading through

the “Chapter Check-In” at the beginning of each chapter.

■Use the “Chapter Checkout” at the end of each chapter to gauge your

grasp of the important information you need to know.

■Test your knowledge more completely in the CQR Review and look

for additional sources of information in the CQR Resource Center.

■Use the Glossary to find key terms fast. This book defines new terms

and concepts where they first appear in the chapter. If a word is bold-

faced, you can find a more complete definition in the book’s Glossary.

■Or flip through the book until you find what you’re looking for—we

organized this book to gradually build on key concepts.

Visit Our Web Site

A great resource, www.cliffsnotes.com, features timely articles and tips,

plus downloadable versions of many CliffsNotes books.

When you stop by our site, don’t hesitate to share your thoughts about

this book or any other Wiley product. Just click the Talk to Us button. We

welcome your feedback!

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CliffsQuickReview®MathWordProblemsByKarenL.Anglin544926FM.qxd4/12/0410:09PMPagei544926FM.qxd4/12/0410:09PMPageviCliffsQuickReview®MathWordProblemsByKarenL.Anglin544926FM.qxd4/12/0410:09PMPageiAbouttheAuthorKarenAnglinhasbeenamathematicsinstructoratBlinnCollegeinBrenham,Texas,since1990.Shehasgivennum...

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