Math Word Problems Demystified 2/E

By Allan G. Bluman

Your solution to MATH word PROBLEMS!

Find yourself stuck on the tracks when two trains are traveling at different speeds? Help has arrived! Math Word Problems Demystified, Second Edition is your ticket to problem-solving success.

Based on mathematician George Polya's proven four-step process, this practical guide helps you master the basic procedures and develop a plan of action you can use to solve many different types of word problems. Tips for using systems of equations and quadratic equations are included. Detailed examples and concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.

It's a no-brainer! You'll learn to solve:

• Decimal, fraction, and percent problems
• Proportion and formula problems
• Number and digit problems
• Distance and mixture problems
• Finance, lever, and work problems
• Geometry, probability, and statistics problems

Simple enough for a beginner, but challenging enough for an advanced student, Math Word Problems Demystified, Second Edition helps you master this essential mathematics skill.

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Aug 22, 2011
• ISBN: 9780071763851

Book preview

chapter 1

Introduction to Problem Solving

This chapter explains the basic four-step problem-solving technique developed by George Polya. In addition, some basic problem-solving strategies such as drawing a picture, making a list, etc., are explained.

CHAPTER OBJECTIVES

In this chapter, you will learn how to

• Use the four-step problem-solving method

• Solve word problems using general problem-solving strategies

Four-Step Method

In every area of mathematics, you will encounter word problems. Some students are very good at solving word problems while others are not. When teaching word problems in pre-algebra and algebra, I often hear, I don’t know where to begin or I have never been able to solve word problems. A great deal has been written about solving word problems. A Hungarian mathematician, George Polya, did much in the area of problem solving. His book, entitled How to Solve It, has been translated into at least 17 languages, and it explains the basic steps of problem solving. These steps are explained next.

Step 1: Understand the problem First read the problem carefully several times. Underline or write down any information given in the problem. Next, decide what you are being asked to find. This will be called the goal.

Step 2: Select a strategy to solve the problem There are many ways to solve word problems. You may be able to use one of the basic operations such as addition, subtraction, multiplication, or division. You may be able to use an equation or formula. You may even be able to solve a given problem by trial and error. This step will be called strategy.

Step 3: Carry out the strategy Perform the operation, solve the equation, etc., and get the solution. If one strategy doesn’t work, try a different one. This step will be called implementation.

Step 4: Evaluate the answer This means to check your answer if possible. Another way to evaluate your answer is to see if it is reasonable. Finally, you can use estimation as a way to check your answer. This step will be called evaluation.

When you think about the four steps, they apply to many situations that you may encounter in life. For example, suppose that you play basketball. The goal is to get the basketball into the hoop. The strategy is to select a way to make a basket. You can use any one of several methods such as a jump shot, a layup, a one-handed push shot, or a slam dunk. The strategy you use will depend on the situation. After you decide on the type of shot to try, you implement the shot. Finally, you evaluate the action. Did you make the basket? Good for you! Did you miss it? What went wrong? Can you improve on the next shot?

Now let’s see how this procedure applies to a mathematical problem.

EXAMPLE

Find the next two numbers in the sequence

5 12 8 15 11 18 14 _____ _____

SOLUTION

Goal: You are asked to find the next two numbers in the sequence.

Strategy: Here you can use a strategy called find a pattern. Ask yourself, What’s being done to one number to get the next number in the sequence? In this case, to get from 5 to 12, you can add 7. But to get from 12 to 8, you need to subtract 4. So perhaps it is necessary to do two different things.

Implementation: Add 7 to 14 to get 21. Subtract 4 from 21 to get 17. Hence, the next two numbers should be 21 and 17.

Evaluation: In order to check the answers, you need to see if the add 7, subtract 4 solution works for all the numbers in the sequence, so start with 5.

Voilà! You have found the solution!

Now let’s try another one.

EXAMPLE

Find the next two numbers in the sequence

1 3 7 13 21 31 43 _____ _____

SOLUTION

Goal: You are asked to find the next two numbers in the sequence.

Strategy: Again we will use find a pattern. Ask yourself, What is being done to the first number to get the second one? Here we are adding 2. Does adding 2 to the second number 3 give us the third number 7? No. You must add 4 to the second number to get the third number 7. How do we get from the third number to the fourth number? Add 6. Let’s apply the strategy.

Implementation:

1 + 2 = 3

3 + 4 = 7

7 + 6 = 13

13 + 8 = 21

21 + 10 = 31

31 + 12 = 43

43 + 14 = 57

57 + 16 = 73

Hence, the next two numbers in the sequence are 57 and 73.

Evaluation: Since the pattern works for the first seven numbers in the sequence, we can extend it to the next two numbers, which then makes the answers correct.

EXAMPLE

Find the next two letters in the sequence

A Z C Y E X G W _____ _____

SOLUTION

Goal: You are asked to find the next two letters in the sequence.

Strategy: Again, you can use the find a pattern strategy. Notice that the sequence starts with the first letter of the alphabet, A, and then goes to the last letter, Z, then back to C, and so on. So it looks like there are two sequences.

Implementation: The first sequence is A C E G, and the second sequence is Z Y X W. Hence, the next two letters are I and V.

Evaluation: Putting the two sequences together, you get A Z C Y E X G W I V. Now you can try a few problems to see if you understand the problem-solving procedure. Be sure to use all four steps.

TRY THESE

Find the next two numbers or letters in each sequence.

1. 5 15 14 42 41 123 122 _____ _____

2. 1 6 36 216 1,296 7,776 _____ _____

3. 80 40 44 22 26 _____ _____

4. 1 4 9 16 25 36 _____ _____

5. A 6 B 13 C 20 D 27 _____ _____

SOLUTION

1. 366 and 365. Multiply the first number by 3 to get the second number; subtract 1 from the second number to get the third number. Continue.

2. 46,656 and 279,936. Multiply each number by 6 to get the next number.

3. 13 and 17. Divide the first number by 2 to get the second number, then add 4 to get the next number. Repeat the process.

4. 49 and 64. Square the numbers in the sequence: 1, 2, 3, 4, …

5. E and 34. Use the alphabet and add 7 to each number.

Well, how did you do? You have just had an introduction to systematic problem solving. The remainder of this book is divided into three parts. Chapters 2–5 explain how to solve word problems in arithmetic and pre-algebra. Chapters 6–11 explain how to solve word problems in introductory and intermediate algebra. Chapter 12 explains how to solve word problems in geometry, probability, and statistics. After successfully completing this book, you will be well along the way to becoming a competent mathematical word problem solver.

Problem-Solving Strategies

There are some general problem-solving strategies you can use to solve real-world problems and help you check your answers when you use the strategies presented later in this book. These strategies can help you with problems found on standardized tests, in other subjects, and in everyday life.

These strategies are

1. Make an organized list

2. Guess and test

3. Draw a picture

4. Find a pattern

5. Solve a simpler problem

6. Work backwards

Make an Organized List

When you use this strategy, you make an organized list of possible solutions and then systematically work out each one until the correct answer is found. Sometimes it helps to make the list in a table format.

EXAMPLE

A person has seven bills consisting of $5 bills and $10 bills. If the total amount of the money is $50, find the number of $5 bills and $10 bills he has.

SOLUTION

Goal: You are being asked to find the number of $5 bills and $10 bills the person has.

Strategy: This problem can be solved by making an organized list and finding the total amount of money you have as shown:

One $5 bill and six $10 bills make seven bills with a value of 1 × $5 + 6 × $10 = $65. This is incorrect, so try two $5 bills and five $10 bills and keep going until a sum of $50 is reached.

Implementation: Finish the list.

Hence four $5 bills and three $10 bills are needed to get $50.

Evaluation: Four $5 bills and three $10 bills make seven bills whose total value is $50.

EXAMPLE

In a barnyard there are eight animals, chickens and cows. Chickens have two legs and cows have four legs, of course. If the total number of legs is 22, how many chickens and cows are there?

SOLUTION

Goal: You are being asked to find how many chickens and how many cows are in the barnyard.

Strategy: You can make an organized list, as shown.

The number of chickens and cows must sum to 8 and that gives a total of 30 legs:

1 × 2 + 7 × 4 = 2 + 28 = 30

Implementation: Continue the table until the correct answer (22 legs) is found.

Hence, there are five chickens and three cows in the barnyard.

Evaluation: Five chickens have 5 × 2 = 10 legs, and three cows have 3 × 4 = 12 legs, 10 + 12 = 22 legs.

Guess and Test

This strategy is similar to the previous one except you do not need to make a list. You simply take an educated guess at the solution and then try it out to see if it is correct. If not, try another guess; then test it.

EXAMPLE

The sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is nine more than the original number.

SOLUTION

Goal: You are being asked to find a two-digit number.

Strategy: You can use the guess and test strategy. First guess some two-digit numbers such that the sum of the digits is 9. For example, 18, 27, 36, 45, etc., meet this part of the solution. Then see if they meet the other condition of the problem.

Implementation:

Guess: 27; reverse the digits: 72; subtract: 72 − 27 = 45

Guess: 36; reverse the digits: 63; subtract: 63 − 36 = 27

Guess: 45; reverse the digits: 54; subtract: 54 − 45 = 9. This is the correct solution; hence, the number is 45.

Evaluation: The sum of the digits 4 + 5 = 9, and the difference 54 − 45 = 9.

EXAMPLE

The letters X and W each represent a digit from 0 through 9. Find the value of each letter so that the following is true:

SOLUTION

Goal: You are being asked to find what digits X and W represent.

Strategy: Use guess and test.

Implementation: Guess a few digits for X and see what works:

Hence X = 5 and W = 1 is the correct answer.

Evaluation: Notice that all the digits in the column are the same; that is, they are all the same number. You must add three single-digit numbers and get the same number as the one’s digit of the solution. There are only two possibilities: 0 and 5. Since the answer has two digits, 0 is disregarded.

Draw a Picture

Many times a problem can be solved using a picture, figure, or diagram. Also, drawing a picture can help you to determine which other strategy can be used to solve a problem.

EXAMPLE

Ten trees are planted in a row at three-foot intervals. How far is it from the first tree to the last tree?

SOLUTION

Goal: You are being asked to find the distance from the first tree to the last tree.

Strategy: Draw a figure and count the intervals between them; then multiply the answer by 3.

Implementation: Solve the problem. See Figure 1-1.

FIGURE 1-1

Since there are nine intervals, the distance between the first and last one is 9 × 3 = 27 feet.

Evaluation: The figure shows that 27 feet is the correct answer.

EXAMPLE

A family has three children. List the number of ways according to gender that the births can occur.

SOLUTION

Goal: You are being asked to list the total number of ways three children can be born.

Strategy: Draw a diagram showing the way the children can be born.

FIGURE 1-2

Implementation: Each child could be born as a male or a female. See Figure 1-2. Hence there are eight different possibilities:

Evaluation: Since there are two ways for each child to be born, there are 2 × 2 × 2 = 8 different ways that the births can occur.

Find a Pattern

Many problems can be solved by recognizing that there is a pattern to the solution. Once the pattern is recognized, the solution can be obtained by generalizing from the pattern.

EXAMPLE

A wealthy person decided to pay an employee $1 for the first day’s work, $2 for the second day’s work, and $4 for the third day’s work, etc. How much did the employee earn for 15 days of work?

SOLUTION

Goal: You are being asked to find the amount the employee earned for a total of 15 days of work.

Strategy: You can make a table starting with the first day and continuing until you see a pattern.

Implementation:

Notice that the amount earned each day is given by 2n − ¹ where n is the number of the day. For example, on the 6th day, the person earns 2⁶ − ¹ = 2⁵ = $32. So on the 15th day, a person earns 2¹⁵ − ¹ or 2¹⁴ = $16,384. The total amount the person earns is given by doubling the amount earned that day and subtracting one. So the total amount earned at the end of the 15 days is $16,384 × 2 − 1 = $32,767.

Evaluation: You could check your answer by continuing the pattern for 15 days.

EXAMPLE

Find the answer to 12345678 × 9 + 9 using a pattern.

1 × 9 + 2 = 11

12 × 9 + 3 = 111

123 × 9 + 4 = 1111

SOLUTION

Goal: You are being asked to find the answer to 12345678 × 9 + 9 using a pattern.

Strategy: Make a table starting with 1 × 9 + 2, 12 × 9 + 3, 123 × 9 + 4, etc. Find the answers to these problems and see if you can find a pattern.

Implementation:

1 × 9 + 2 = 11

12 × 9 + 3 = 111

123 × 9 + 4 = 1111

The pattern shows that you get an answer that has the same number of 1s as the last digit that is added. So the answer to the problem would be a number which has 9 1s, that is, 111,111,111.

Evaluation: Perform the operations on a calculator and see if the answer is correct.

Solve a Simpler Problem

To use this strategy, you should simplify the problem or make up a shorter, similar problem and figure out how to solve it. Then use the same strategy to solve the given problem.

EXAMPLE

If there are 10 people at a tennis