Prealgebra 2e

# 9.1Use a Problem Solving Strategy

Prealgebra 2e9.1 Use a Problem Solving Strategy

### Learning Objectives

By the end of this section, you will be able to:

• Approach word problems with a positive attitude
• Use a problem solving strategy for word problems
• Solve number problems

### Be Prepared 9.1

Before you get started, take this readiness quiz.

Translate $“6“6$ less than twice $x”x”$ into an algebraic expression.
If you missed this problem, review Example 2.25.

### Be Prepared 9.2

Solve: $23x=24.23x=24.$
If you missed this problem, review Example 8.16.

### Be Prepared 9.3

Solve: $3x+8=14.3x+8=14.$
If you missed this problem, review Example 8.20.

### Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in Figure 9.2?

Figure 9.2 Negative thoughts about word problems can be barriers to success.

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts like the student in Figure 9.3. Read the positive thoughts and say them out loud.

Figure 9.3 When it comes to word problems, a positive attitude is a big step toward success.

If we take control and believe we can be successful, we will be able to master word problems.

Think of something that you can do now but couldn't do three years ago. Whether it's driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!

### Use a Problem-solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you've increased your math vocabulary as you learned about more algebraic procedures, and you've had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we'll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We'll demonstrate the strategy as we solve the following problem.

### Example 9.1

Pete bought a shirt on sale for $18,18,$ which is one-half the original price. What was the original price of the shirt?

### Try It 9.1

Joaquin bought a bookcase on sale for $120,120,$ which was two-thirds the original price. What was the original price of the bookcase?

### Try It 9.2

Two-fifths of the people in the senior center dining room are men. If there are $1616$ men, what is the total number of people in the dining room?

We list the steps we took to solve the previous example.

### Problem-Solving Strategy

1. Step 1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet.
2. Step 2. Identify what you are looking for.
3. Step 3. Name what you are looking for. Choose a variable to represent that quantity.
4. Step 4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem. Make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.

Let's use this approach with another example.

### Example 9.2

Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought $1111$ apples to the picnic. How many bananas did he bring?

### Try It 9.3

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was $33$ more than the number of notebooks. He bought $55$ textbooks. How many notebooks did he buy?

### Try It 9.4

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is seven more than the number of crossword puzzles. He completed $1414$ Sudoku puzzles. How many crossword puzzles did he complete?

In Solve Sales Tax, Commission, and Discount Applications, we learned how to translate and solve basic percent equations and used them to solve sales tax and commission applications. In the next example, we will apply our Problem Solving Strategy to more applications of percent.

### Example 9.3

Nga's car insurance premium increased by $60,60,$ which was $8%8%$ of the original cost. What was the original cost of the premium?

### Try It 9.5

Pilar's rent increased by $4%.4%.$ The increase was $38.38.$ What was the original amount of Pilar's rent?

### Try It 9.6

Steve saves $12%12%$ of his paycheck each month. If he saved $504504$ last month, how much was his paycheck?

### Solve Number Problems

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem Solving Strategy. Remember to look for clue words such as difference, of, and and.

### Example 9.4

The difference of a number and six is $13.13.$ Find the number.

### Try It 9.7

The difference of a number and eight is $17.17.$ Find the number.

### Try It 9.8

The difference of a number and eleven is $−7.−7.$ Find the number.

### Example 9.5

The sum of twice a number and seven is $15.15.$ Find the number.

### Try It 9.9

The sum of four times a number and two is $14.14.$ Find the number.

### Try It 9.10

The sum of three times a number and seven is $25.25.$ Find the number.

Some number word problems ask you to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

### Example 9.6

One number is five more than another. The sum of the numbers is twenty-one. Find the numbers.

### Try It 9.11

One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

### Try It 9.12

The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

### Example 9.7

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

### Try It 9.13

The sum of two numbers is negative twenty-three. One number is $77$ less than the other. Find the numbers.

### Try It 9.14

The sum of two numbers is negative eighteen. One number is $4040$ more than the other. Find the numbers.

### Example 9.8

One number is ten more than twice another. Their sum is one. Find the numbers.

### Try It 9.15

One number is eight more than twice another. Their sum is negative four. Find the numbers.

### Try It 9.16

One number is three more than three times another. Their sum is negative five. Find the numbers.

Consecutive integers are integers that immediately follow each other. Some examples of consecutive integers are:

$...1,2,3,4,......1,2,3,4,...$
$...−10,−9,−8,−7,......−10,−9,−8,−7,...$
$...150,151,152,153,......150,151,152,153,...$

Notice that each number is one more than the number preceding it. So if we define the first integer as $n,n,$ the next consecutive integer is $n+1.n+1.$ The one after that is one more than $n+1,n+1,$ so it is $n+1+1,n+1+1,$ or $n+2.n+2.$

$n1st integern+12nd consecutive integern+23rd consecutive integern1st integern+12nd consecutive integern+23rd consecutive integer$

### Example 9.9

The sum of two consecutive integers is $47.47.$ Find the numbers.

### Try It 9.17

The sum of two consecutive integers is $95.95.$ Find the numbers.

### Try It 9.18

The sum of two consecutive integers is $−31.−31.$ Find the numbers.

### Example 9.10

Find three consecutive integers whose sum is $42.42.$

### Try It 9.19

Find three consecutive integers whose sum is $96.96.$

### Try It 9.20

Find three consecutive integers whose sum is $−36.−36.$

### Section 9.1 Exercises

#### Practice Makes Perfect

Use a Problem-solving Strategy for Word Problems

In the following exercises, use the problem-solving strategy for word problems to solve. Answer in complete sentences.

1.

Two-thirds of the children in the fourth-grade class are girls. If there are $2020$ girls, what is the total number of children in the class?

2.

Three-fifths of the members of the school choir are women. If there are $2424$ women, what is the total number of choir members?

3.

Zachary has $2525$ country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

4.

One-fourth of the candies in a bag of are red. If there are $2323$ red candies, how many candies are in the bag?

5.

There are $1616$ girls in a school club. The number of girls is $44$ more than twice the number of boys. Find the number of boys in the club.

6.

There are $1818$ Cub Scouts in Troop $645.645.$ The number of scouts is $33$ more than five times the number of adult leaders. Find the number of adult leaders.

7.

Lee is emptying dishes and glasses from the dishwasher. The number of dishes is $88$ less than the number of glasses. If there are $99$ dishes, what is the number of glasses?

8.

The number of puppies in the pet store window is twelve less than the number of dogs in the store. If there are $66$ puppies in the window, what is the number of dogs in the store?

9.

After $33$ months on a diet, Lisa had lost $12%12%$ of her original weight. She lost $2121$ pounds. What was Lisa's original weight?

10.

Tricia got a $6%6%$ raise on her weekly salary. The raise was $3030$ per week. What was her original weekly salary?

11.

Tim left a $99$ tip for a $5050$ restaurant bill. What percent tip did he leave?

12.

Rashid left a $1515$ tip for a $7575$ restaurant bill. What percent tip did he leave?

13.

Yuki bought a dress on sale for $72.72.$ The sale price was $60%60%$ of the original price. What was the original price of the dress?

14.

Kim bought a pair of shoes on sale for $40.50.40.50.$ The sale price was $45%45%$ of the original price. What was the original price of the shoes?

Solve Number Problems

In the following exercises, solve each number word problem.

15.

The sum of a number and eight is $12.12.$ Find the number.

16.

The sum of a number and nine is $17.17.$ Find the number.

17.

The difference of a number and twelve is $3.3.$ Find the number.

18.

The difference of a number and eight is $4.4.$ Find the number.

19.

The sum of three times a number and eight is $23.23.$ Find the number.

20.

The sum of twice a number and six is $14.14.$ Find the number.

21.

The difference of twice a number and seven is $17.17.$ Find the number.

22.

The difference of four times a number and seven is $21.21.$ Find the number.

23.

Three times the sum of a number and nine is $12.12.$ Find the number.

24.

Six times the sum of a number and eight is $30.30.$ Find the number.

25.

One number is six more than the other. Their sum is forty-two. Find the numbers.

26.

One number is five more than the other. Their sum is thirty-three. Find the numbers.

27.

The sum of two numbers is twenty. One number is four less than the other. Find the numbers.

28.

The sum of two numbers is twenty-seven. One number is seven less than the other. Find the numbers.

29.

A number is one more than twice another number. Their sum is negative five. Find the numbers.

30.

One number is six more than five times another. Their sum is six. Find the numbers.

31.

The sum of two numbers is fourteen. One number is two less than three times the other. Find the numbers.

32.

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

33.

One number is fourteen less than another. If their sum is increased by seven, the result is $85.85.$ Find the numbers.

34.

One number is eleven less than another. If their sum is increased by eight, the result is $71.71.$ Find the numbers.

35.

The sum of two consecutive integers is $77.77.$ Find the integers.

36.

The sum of two consecutive integers is $89.89.$ Find the integers.

37.

The sum of two consecutive integers is $−23.−23.$ Find the integers.

38.

The sum of two consecutive integers is $−37.−37.$ Find the integers.

39.

The sum of three consecutive integers is $78.78.$ Find the integers.

40.

The sum of three consecutive integers is $60.60.$ Find the integers.

41.

Find three consecutive integers whose sum is $−36.−36.$

42.

Find three consecutive integers whose sum is $−3.−3.$

#### Everyday Math

43.

Shopping Patty paid $3535$ for a purse on sale for $1010$ off the original price. What was the original price of the purse?

44.

Shopping Travis bought a pair of boots on sale for $2525$ off the original price. He paid $6060$ for the boots. What was the original price of the boots?

45.

Shopping Minh spent $6.256.25$ on $55$ sticker books to give his nephews. Find the cost of each sticker book.

46.

Shopping Alicia bought a package of $88$ peaches for $3.20.3.20.$ Find the cost of each peach.

47.

Shopping Tom paid $1,166.401,166.40$ for a new refrigerator, including $86.4086.40$ tax. What was the price of the refrigerator before tax?

48.

Shopping Kenji paid $2,2792,279$ for a new living room set, including $129129$ tax. What was the price of the living room set before tax?

#### Writing Exercises

49.

Write a few sentences about your thoughts and opinions of word problems. Are these thoughts positive, negative, or neutral? If they are negative, how might you change your way of thinking in order to do better?

50.

When you start to solve a word problem, how do you decide what to let the variable represent?

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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