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# CliffsQuickReview Math Word Problems

## Overview

CliffsQuickReview course guides cover the essentials of your toughest classes. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions.

*CliffsQuickReview Math Word Problems* gives you a clear, concise, easy-to-use review of the basics of solving math word problems. Introducing each topic, defining key terms, and carefully walking you through each sample problem gives you insight and understanding to solving math word problems. You begin by building a strong foundation in translating expressions, inserting parentheses, and simplifying expressions. On top of that base, you can build your skills for solving word problems:

- Discover the six basic steps for solving word problems
- Translate English-language statements into equations and then solve them
- Solve geometry problems involving single and multiple shapes
- Work on proportion and percent problems
- Solve summation problems by using the Board Method
- Use tried-and-true methods to solve problems about money, investments, mixtures, and distance

*CliffsQuickReview Math Word Problems* acts as a supplement to your textbook and to classroom lectures. Use this reference in any way that fits your personal style for study and review — you decide what works best with your needs. Here are just a few ways you can search for information:

- View the chapter on common errors and how to avoid them
- 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

With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades.

## Product Details

ISBN-13: | 9780764544927 |
---|---|

Publisher: | Houghton Mifflin Harcourt |

Publication date: | 05/14/2004 |

Series: | CliffsQuickReview Series |

Pages: | 220 |

Sales rank: | 693,717 |

Product dimensions: | 5.25(w) x 8.25(h) x (d) |

## About the Author

Karen Anglin, a mathematics instructor at Blinn College in Brenham, Texas, since 1990, regularly presents workshops to teachers on best practices for teaching math word problems. She holds an MS in Statistics and a BS in Mathematics from Texas A&M University.

## Read an Excerpt

#### CliffsQuickReview Math Word Problems

**By Karen L. Anglin**

** John Wiley & Sons **

**ISBN: 0-7645-4492-6**

#### Chapter One

**TRANSLATING EXPRESSIONS**

** Chapter Check-In**

Addition and subtraction keywords

Multiplication and division keywords

Turnaround words

The first step in solving a word problem is always to read the problem. You need to be able to **translate** words into mathematical symbols, focusing on **keywords** that indicate the mathematical procedures required to solve the problem-both the operation and the order of the expression. In much the same way that you can translate Spanish into English, you can translate English words into symbols, the language of mathematics. Many (if not all) keywords that indicate mathematical operations are familiar words.

**Keywords of Basic Mathematical Operations**

To begin, you translate English phrases into algebraic *expressions.* An algebraic expression is a collection of numbers, *variables,* operations, and **grouping symbols.** You will translate an unknown number as the variable *x* or *n*. The grouping symbols are usually a set of parentheses, but they can also be sets of brackets or braces.

In translating expressions, you want to be well acquainted with basic keywords that translate into mathematical operations: addition keywords, subtraction keywords, multiplication keywords, and division keywords, which are covered in the four following sections.

**Addition keywords**

Some common examples of addition keywords are as follows:

SUM OF______ AND ______ TOTAL OF ______ AND ______ ______ PLUS ______ ______ INCREASED BY ______ GAIN RAISE MORE INCREASE OF

The first two keywords (SUM and TOTAL) are called **leading keywords** because they lead the expression. The second two keywords (PLUS and INCREASED BY) are keywords that indicate the exact placement of the plus sign. The last four keywords can be found in word problems and may indicate addition.

When an expression begins with the leading keywords SUM or TOTAL, the leading keyword defines the corresponding AND. The plus sign then physically replaces the AND in the expression.

**Example 1:** Translate the following: the sum of five and a number

The following steps help you translate this problem:

**1. Underline the words before and after AND when it corresponds to the leading keyword SUM OF.** the sum of five and a number

**2. Circle the leading keyword and indicate the corresponding AND that it defines.**

the sum of five and a number

**3. Translate each underlined expression and replace AND with a plus sign.**

The expression translates to 5 + *x*.

**Example 2:** Translate the following: the total of a number and negative three

Use the following steps to translate this problem: the sum of five and a number

**1. The keyword TOTAL OF is a leading keyword that defines AND, so underline the words before and after AND: "a number" and "negative three."**

the total of a number and negative three

**2. Circle the leading keyword and indicate the corresponding AND that it defines.** the total of a number and negative three

**3. Translate each underlined expression and replace AND with a plus sign.**

The expression translates to *x* + -3.

**Example 3:** Translate the following: the sum of seven and negative four Translate this example in the following way:

**1. The word SUM OF is a leading keyword that defines AND , so underline the words before and after AND: "seven" and "negative four."**

the sum of seven and negative four

**2. Circle the leading keyword and indicate the corresponding AND that it defines.**

the sum of seven and negative four

**3. Translate each underlined expression and replace AND with a plus sign.**

The expression translates to 7 + - 4.

**Reminder:** The AND keyword translates to mean "plus" because the leading keyword is SUM OF. With other leading keywords (discussed in the following sections), AND can mean other things. Also notice that you do not simplify the expression and get "3" for the answer because you are just translating words into symbols and not performing the math.

Two other keywords on the addition keyword list, PLUS and INCREASED BY, can be correctly translated by the **direct translation strategy.** In the direct translation strategy, you translate each word into its corresponding algebraic symbol, one at a time, in the same order as written, as shown in Example 4.

**Example 4:** Translate the following: a number increased by twenty-four

The expression translates to *x* + 24.

Some additional keywords, such as GAIN, MORE, INCREASE OF, and RAISE, are commonly found in story problems, as in Example 5.

**Example 5:** Translate the following story problem into a mathematical expression about the weight of the linebacker: The defensive linebacker weighed two hundred twenty-two pounds at the beginning of spring training. He had a gain of seventeen pounds after working out with the team for four weeks.

The expression translates to 222 + 17.

**Note:** Not all numbers mentioned in a word problem should be included in the mathematical expression. The number "four" is just interesting fact, but it is not information you need in order to write an expression about the linebacker's weight.

You may also be wondering why the answer isn't 239 pounds. That's because the question asks you to translate the story problem into a mathematical expression, not to evaluate the expression.

**Example 6:** Translate the following word problem into a mathematical expression about the cashier's current hourly wage: A cashier at the corner grocery was earning $6.25 an hour. He received a raise of 25 cents an hour.

The expression translates to 6.25 + 0.25.

**Note:** The hourly wage is stated in dollars, and the raise is stated in cents. Any time you are adding two numbers that have **units,** make sure both numbers are measured with the same units; if they aren't, convert one of the numbers to the same units as the other. Having both numbers measured with the same units is called **homogeneous units.** In this example, you convert his raise, the 25 cents, to $0.25 because his hourly wage is measured in dollars, not cents, so the raise must also be in dollars.

**Subtraction keywords**

Subtraction keywords also include leading keywords, keywords that can be translated one word at a time, and keywords that are found in story problems. Look at the following list of subtraction keywords:

DIFFERENCE BETWEEN ______ AND ______ ______ MINUS ______ ______ DECREASED BY ______ LOSS LESS FEWER TAKE AWAY

One subtraction keyword (DIFFERENCE BETWEEN) is a two-part expression that begins with a leading keyword that defines the corresponding AND. You can use the same methods of underlining and circling the keywords shown in the preceding section to translate these expressions.

**Example 7:** Translate the following: the difference between four and six

Here is how you translate Example 7:

**1. Because the keyword DIFFERENCE BETWEEN is a leading keyword that defines the corresponding AND, underline the words before and after AND: "four" and "six."**

the difference between four and six

**2. Circle the leading keyword and indicate the corresponding AND that it defines.**

the difference between four and six

**3. Translate each underlined expression and replace AND with a minus sign.**

The expression translates to 4 - 6.

**Note:** AND is not always translated to mean addition. Here, the DIFFERENCE BETWEEN is the leading keyword that defines the AND to mean subtraction.

Other subtraction keywords, such as MINUS and DECREASED BY, use the direct translation strategy. Example 8 is a subtraction word problem that is translated one keyword at a time, in the exact order of the expression.

**Example 8:** Translate the following: twenty-four decreased by a number

The expression translates to 24 - *x*.

In a story problem, you may find the subtraction keywords LOSS, LESS, FEWER, and TAKE AWAY, as shown in Example 9.

**Example 9:** Translate the following word problem into a mathematical expression about the current value of materials at the job site: A construction company stored $1,253 worth of materials at the job site. The company suffered a loss of $300 due to storm damage.

The expression translates to 1,253 - 300.

**Multiplication keywords**

Some common examples of multiplication keywords are as follows:

MULTIPLY ______ BY ______ PRODUCT OF ______ AND ______ ______ TIMES ______ DOUBLE ______ TWICE ______ TRIPLE ______ PERCENT OF ______ FRACTION OF ______

For two of the multiplication keywords, MULTIPLY and PRODUCT OF, a leading keyword defines the corresponding BY or AND, as shown in Example 10.

**Example 10:** Translate the following: the product of seven and a number

Translate this example in the following way:

**1. Because PRODUCT OF is a leading keyword that corresponds to AND, underline the words before and after AND: "seven" and "a number."**

the product of seven and a number

**2. Circle the leading keyword and indicate the corresponding AND that it defines.**

the product of seven and a number

**3. Translate each underlined expression and replace AND with a times sign.**

The expression translates to 7 × *x*. the product of seven and a number

**Note:** Keep in mind that AND does not always indicate addition. The keyword PRODUCT OF defines the AND in this expression to mean multiplication.

A multiplication expression that is translated by the direct translation method is shown in Example 11.

**Example 11:** Translate the following: a number times fifteen

The expression translates to *x* × 15.

Some multiplication keywords, such as DOUBLE, TWICE, and TRIPLE, translate into a number and the operation of multiplication, as shown in Examples 12 and 13.

**Example 12:** Translate the following: twice a number

The expression translates to 2 × *x*.

**Example 13:** Translate the following word problem into a mathematical expression: Jennifer had $15 dollars in the bank. Over the next two weeks she doubled her money.

The expression translates to 2 × 15.

One of the keywords that indicates multiplication is OF. In word problems, however, you may see more than one use of the word "of." The only OF that indicates multiplication is the one that follows the keyword PERCENT, the percent sign, the keyword FRACTION, or a fraction. See Examples 14 and 15.

**Example 14:** Translate the following: twenty five percent of four hundred dollars

The expression translates to 0.25 × 400.

**Note:** Remember that a percent is changed to a decimal before multiplying.

**Example 15:** Translate the following: one-third of twenty-seven

The expression translates to 1/3 × 27.

**Division keywords**

Some common examples of division keywords are as follows:

QUOTIENT OF ______ AND ______ DIVIDE ______ BY ______ ______ DIVIDED BY ______ DIVIDED EQUALLY PER

The keywords PRODUCT OF and QUOTIENT OF are difficult for some people to differentiate. Here is a hint to help you remember which one indicates division and which one indicates multiplication: Quotient is a "harder" word than "product," and division is a "harder" operation than multiplication.

**Remember:** Leading keywords define the corresponding AND or BY to mean divide, usually designated with the symbol ÷.

**Example 16:** Translate the following: the quotient of seven and a number

**1. Because the keyword QUOTIENT OF is a leading keyword that defines AND, underline the words before and after AND: "seven" and "a number."**

the quotient of seven and a number

**2. Circle the leading keyword and indicate the corresponding AND that it defines.**

the quotient of seven and a number

**3. Translate each underlined expression and replace AND with a division sign.**

The expression translates to 7 ÷ *n*.

**Note:** Here, the keyword QUOTIENT OF defines AND to mean division.

**Example 17:** Translate the following: divide negative thirty-six by nine

**1. Because the word DIVIDE is a leading keyword that defines the BY, underline the words before and after BY: "negative thirtysix" and "nine."**

divide negative thirty-six by nine

**2. Circle the leading keyword and indicate the corresponding BY that it defines.**

divide negative thirty-six by nine

**3. Translate each underlined expression and replace AND with a division sign.**

The expression translates to -36 ÷ 9, -36/9, or 9 [square root of (-36)].

**Note:** The first number goes in the numerator when using a fraction bar to indicate division. The number in the numerator (the -36) goes inside the "house" when using the long division symbol.

Some division keywords can be translated one word at a time. Instead, you just follow the sentence and replace with algebraic notations as you go along.

**Example 18:** Translate the following: a number divided by 16

The expression translates to *x* ÷ 16 or *x*/16.

Often, in story problems, the keyword that indicates division is PER. When a story problem asks for the speed of a vehicle in miles per hour, set up the expression to divide the number of miles by the number of hours. You not only directly translate "miles" ÷ "hours," but also identify the number of miles and number of hours by finding them elsewhere in the problem. See Example 19.

**Example 19:** Translate the following word problem into a mathematical expression about speed: It takes three hours to travel 150 miles to grandmother's house. How do you find your average speed in miles per hour?

You find "miles" ÷ "hours" in the question. In the first part of the word problem, you find the number of miles, 150 miles, and the number of hours, three hours.

The expression translates to 150 ÷ 3.

**Keywords That Indicate a Change in Order**

Some keywords are not included in other lists in this chapter because they are a bit different from other types of keywords. This section gives you additional keywords, called **turnaround words,** and these indicate a change in order from the original English phrase.

*Continues...*

Excerpted fromCliffsQuickReview Math Word ProblemsbyKaren L. AnglinExcerpted by permission.

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

## Table of Contents

**Introduction.**

Why You Need This Book.

How to Use This Book.

Visit Our Web Site.

**Chapter 1: Translating Expressions.**

Keywords of Basic Mathematical Operations.

Addition keywords.

Subtraction keywords.

Multiplication keywords.

Division keywords.

Keywords That Indicate a Change in Order.

Basic turnaround words.

Addition turnaround words.

Subtraction turnaround words.

Multiplication turnaround words.

Division turnaround words.

**Chapter 2: Inserting Parentheses.**

Translating a Comma and IS.

Leading Keywords.

Adjacent Keywords.

Turnaround Words.

**Chapter 3: Simplifying Expressions.**

Distributive Property of Multiplication over Addition.

Combining Like Terms.

Distributing a Negative Number.

**Chapter 4: Equations.**

Keywords Indicating Equality.

Solving Simple Linear Equations.

Defining an algebraic expression.

Defining an algebraic equation.

Four steps for solving simple linear equations.

Checking Solutions and Translations.

Checking solutions.

Checking your translation.

**Chapter 5: Geometry Problems.**

Triangles.

Perimeter of a triangle.

Area of a triangle.

Sum of the measures of the angles.

Rectangles and Squares.

Perimeter of a rectangle.

Area of a rectangle.

Perimeter of a square.

Area of a square.

Multiple Shapes.

Other Easily Visualized Word Problems.

**Chapter 6: Proportions and Percents.**

Proportion Problems.

Proportion Word Problems.

Percent Problems.

Percent Word Problems.

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

Simple Board Problems.

Consecutive Integers.

Consecutive Even and Odd Integers.

Other Problems that Can Be Represented with a Board.

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

Polya’s Four-Step Process.

Expanding on Polya’s four steps.

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

Consecutive Integers.

Consecutive Even and Odd Integers.

Other Problems that Cannot Be Represented with a Board.

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

Money Problems.

Investment Problems.

Mixture Problems.

Distance Problems.

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

Money Problems.

Investment Problems.

Mixture Problems.

Distance Problems.

**Chapter 11: Common Errors.**

Variable Omission.

Variable Reversal.

Methods for Avoiding Common Errors.

A firm foundation.

Errors in translation.

Errors in solving an equation.

Errors in answering the question.

**CQR Review.**

**CQR Resource Center.**

**Glossary.**

**Index.**