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Prealgebra 2e

9.1 Use a Problem Solving Strategy

Prealgebra 2e9.1 Use a Problem Solving Strategy
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  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

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 xx 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?

A cartoon image of a girl with a sad expression writing on a piece of paper is shown. There are 5 thought bubbles. They read, “I don't know whether to add, subtract multiply, or divide!,” then “I don't understand word problems!,” then “My teachers never explained this!,” then “If I just skip all the word problems, I can probably still pass the class,” and lastly, “I just can't do this!”
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.

A cartoon image of a girl with a confident expression holding some books is shown.  There are 4 thought bubbles. They read, “While word problems were hard in the past, I think I can try them now,” then “I am better prepared now. I think I will begin to understand word problems,” then “I think I can! I think I can!,” and lastly, “It may take time, but I can begin to solve word problems.”
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 $504$504 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 $30$30 per week. What was her original weekly salary?

11.

Tim left a $9$9 tip for a $50$50 restaurant bill. What percent tip did he leave?

12.

Rashid left a $15$15 tip for a $75$75 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 $35$35 for a purse on sale for $10$10 off the original price. What was the original price of the purse?

44.

Shopping Travis bought a pair of boots on sale for $25$25 off the original price. He paid $60$60 for the boots. What was the original price of the boots?

45.

Shopping Minh spent $6.25$6.25 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 $1,166.40$1,166.40 for a new refrigerator, including $86.40$86.40 tax. What was the price of the refrigerator before tax?

48.

Shopping Kenji paid $2,279$2,279 for a new living room set, including $129$129 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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