Section 2.7: Decoding Word Problems Involving Rates

This content comes directly from E-Campus Ontario’s textbook Pre-Calculus Section 1.7 Use a Problem Solving Strategy.

Access this resource for free at https://ecampusontario.pressbooks.pub/sccmathtechmath1/chapter/use-a-problem-solving-strategy/

Learning Objectives

In this section, you will:

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

Let’s Get Started by Approaching Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?

How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does 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 below?

Negative thoughts can be barriers to success.

 

A student is shown with thought bubbles saying “I don’t know whether to add, subtract, multiply, or divide!,” “I don’t understand word problems!,” “My teachers never explained this!,” “If I just skip all the word problems, I can probably still pass the class,” and “I just can’t do this!”

Figure 1

 

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. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 2 and say them out loud. Thinking positive thoughts is a first step towards success.

 

A student is shown with thought bubbles saying “While word problems were hard in the past, I think I can try them now,” “I am better prepared now. I think I will begin to understand word problems,” “I think I can! I think I can!,” and “It may take time, but I can begin to solve word problems.”

Figure 2

 

Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!

Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.

 

How To

Use a Problem-Solving Strategy to Solve Word Problems.

  1. Read the problem. Make sure all the words and ideas are understood.
  2. Identify what we are looking for.
  3. Name what we are looking for. Choose a variable to represent that quantity.
  4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

 

EXAMPLE 1

Use a Problem-Solving Strategy to Solve Word Problems

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

 

Show/Hide Solution

 

image  Try It #1

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

image  Try It #2

Two-fifths of the songs in Mariel’s playlist are country. If there are 16 country songs, what is the total number of songs in the playlist?

EXAMPLE 2

Use a Problem-Solving Strategy to Solve Word Problems

Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?

 

Show/Hide Solution
Step 1. Read the problem.
Step 2. Identify what we are looking for. How many boys were in the study group?
Step 3. Name. Choose a variable to represent the number of boys. Let [latex]n=[/latex] the number of boys.
Step 4. Translate. Restate the problem in one sentence with all the important information. .
Translate into an equation. .
Step 5. Solve the equation. .
Subtract 3 from each side. .
Simplify. .
Divide each side by 2. .
Simplify. .
Step 6. Check. First, is our answer reasonable? Yes, having 4 boys in a study group seems OK. The problem says the number of girls was 3 more than twice the number of boys. If there are four boys, does that make eleven girls? Twice 4 boys is 8. Three more than 8 is 11.
Step 7. Answer the question. There were 4 boys in the study group.

 

image  Try It #3

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy?

image  Try It #4

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?

Solving Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

 

EXAMPLE 3

Solving a Number Problem

The difference of a number and six is 13. Find the number.

 

Show/Hide Solution

 

image  Try It #5

The difference of a number and eight is 17. Find the number.

image  Try It #6

The difference of a number and eleven is [latex]-7[/latex]. Find the number.

EXAMPLE 4

Use a Problem-Solving Strategy to Solve Word Problems

Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?

 

Show/Hide Solution
Step 1. Read the problem.
Step 2. Identify what we are looking for. How many boys were in the study group?
Step 3. Name. Choose a variable to represent the number of boys. Let [latex]n=[/latex] the number of boys.
Step 4. Translate. Restate the problem in one sentence with all the important information. .
Translate into an equation. .
Step 5. Solve the equation. .
Subtract 3 from each side. .
Simplify. .
Divide each side by 2. .
Simplify. .
Step 6. Check. First, is our answer reasonable? Yes, having 4 boys in a study group seems OK. The problem says the number of girls was 3 more than twice the number of boys. If there are four boys, does that make eleven girls? Twice 4 boys is 8. Three more than 8 is 11.
Step 7. Answer the question. There were 4 boys in the study group.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

 

image  Try It #7

The sum of four times a number and two is 14. Find the number.

image  Try It #8

The sum of three times a number and seven is 25. Find the number.

Problems with More Than One Variable

Some number word problems ask us 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. In order to avoid using more than 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 5

Writing all Variables in Terms of One Variable

One number is five more than another. The sum of the numbers is 21. Find the numbers.

 

Show/Hide Solution
Step 1. Read the problem.
Step 2. Identify what we are looking for. We are looking for two numbers.
Step 3. Name. We have two numbers to name and need a name for each.
Choose a variable to represent the first number. Let [latex]n={1}^{\mathrm{st}}[/latex] number.
What do we know about the second number? One number is five more than another.
[latex]n+5={2}^{\mathrm{nd}}[/latex] number
Step 4. Translate. Restate the problem as one sentence with all the important information. The sum of the 1st number and the 2nd number is 21.
Translate into an equation. .
Substitute the variable expressions. .
Step 5. Solve the equation. .
Combine like terms. .
Subtract 5 from both sides and simplify. .
Divide by 2 and simplify. .
Find the second number, too. .
.
.
Step 6. Check.
Do these numbers check in the problem?
Is one number 5 more than the other? [latex]\phantom{\rule{1.6em}{0ex}}13\stackrel{?}{=}8+5[/latex]
Is thirteen 5 more than 8? Yes. [latex]\phantom{\rule{1.6em}{0ex}}13=13[/latex]✓
Is the sum of the two numbers 21? [latex]8+13\stackrel{?}{=}21[/latex]
[latex]\phantom{\rule{1.6em}{0ex}}21=21[/latex]✓
Step 7. Answer the question. The numbers are 8 and 13.

 

image  Try It #9

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

image  Try It #10

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

EXAMPLE 6

Writing all Variables in Terms of One Variable

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

 

Show/Hide Solution

 

image  Try It #11

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

image  Try It #12

The sum of two numbers is [latex]-18[/latex]. One number is 40 more than the other. Find the numbers.

EXAMPLE 7

Writing all Variables in Terms of One Variable

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

 

Show/Hide Solution

 

image  Try It #13

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

image  Try It #14

One number is three more than three times another. Their sum is [latex]-5[/latex]. Find the numbers.

Consecutive Integers

Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other.

 Examples of consecutive integers are:

[latex]\begin{array}{c}\hfill1,2,3,4\hfill \\ \\ -10,-9,-8,-7\hfill \\ \\ \hfill 150,151,152,153\hfill \end{array}[/latex]

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

[latex]\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{integer}\hfill \\ \hfill n+1\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive integer}\hfill \\ \hfill n+2\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive integer . . . etc.}\hfill \end{array}[/latex]

 

EXAMPLE 8

Solving Problems with Consecutive Integers

The sum of two consecutive integers is 47. Find the numbers.

 

Show/Hide Solution

 

image  Try It #15

The sum of two consecutive integers is [latex]95[/latex]. Find the numbers.

image  Try It #16

The sum of two consecutive integers is [latex]-31[/latex]. Find the numbers.

EXAMPLE 9

Solving Problems with Consecutive Integers

Find three consecutive integers whose sum is [latex]-42[/latex].

 

Show/Hide Solution

 

image  Try It #17

Find three consecutive integers whose sum is [latex]-96[/latex].

image  Try It #18

Find three consecutive integers whose sum is [latex]-36[/latex].

Consecutive Even/Odd Integers

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

[latex]\begin{array}{c}\hfill 18,20,22\hfill \\ \\ \hfill 64,66,68\hfill \\ \\ \hfill -12,-10,-8\hfill \end{array}[/latex].

Notice each integer is 2 more than the number preceding it. If we call the first one [latex]n[/latex], then the next one is [latex]n+2[/latex]. The next one would be [latex]n+2+2[/latex] or [latex]n+4[/latex].

[latex]\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{even integer}\hfill \\ \hfill n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer}\hfill \\ \hfill n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer . . . etc.}\hfill \end{array}[/latex]

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81

[latex]\begin{array}{c}\hfill 77,79,81\hfill \\ \\ \hfill n,n+2,n+4\hfill \end{array}[/latex]

[latex]\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{odd integer}\hfill \\ \hfill n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer}\hfill \\ \hfill n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer . . . etc.}\hfill \end{array}[/latex]

Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

 

EXAMPLE 10

Solving Problems with Consecutive Integers

Find three consecutive even integers whose sum is 84.

 

Show/Hide Solution

 

image  Try It #19

Find three consecutive even integers whose sum is [latex]102[/latex].

image  Try It #20

Find three consecutive even integers whose sum is [latex]-24[/latex].

EXAMPLE 11

Solving Problems with Consecutive Integers

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

 

Show/Hide Solution

 

image  Try It #21

According to the National Automobile Dealers Association, the average cost of a car in 2014 was 28,500. This was 1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?

image  Try It #22

The Canadian Real Estate Association (CREA) data shows that the median price of new home in the Canada in December 2018 was $470,000. This was $14,000 more than 19 times the price in December 1967. What was the median price of a new home in December 1967?

Section 2.7 Exercises

[Answers to odd problem numbers are provided at the end of the problem set.  Just scroll down!]

Verbal

1.  List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

 

2. List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

 

Algebraic

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

3. Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

 

4. Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

 

5. Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

 

6. One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?

 

7. There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

 

8. There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

 

9. Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

 

10. Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

 

11. Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

 

12. Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

 

13. Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

 

14. Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

 

15. The sum of a number and eight is 12. Find the number.

 

16. The sum of a number and nine is 17. Find the number.

 

17. The difference of a number and 12 is three. Find the number.

 

18. The difference of a number and eight is four. Find the number.

 

19. The sum of three times a number and eight is 23. Find the number.

 

20. The sum of twice a number and six is 14. Find the number.

 

21.The difference of twice a number and seven is 17. Find the number.

 

22. The difference of four times a number and seven is 21. Find the number.

 

23. Three times the sum of a number and nine is 12. Find the number.

 

24. Six times the sum of a number and eight is 30. Find the number.

 

25. One number is six more than the other. Their sum is 42. Find the numbers.

 

26. One number is five more than the other. Their sum is 33. Find the numbers.

 

27. The sum of two numbers is 20. One number is four less than the other. Find the numbers.

 

28. The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

 

29. The sum of two numbers is [latex]-45[/latex]. One number is nine more than the other. Find the numbers.

 

30. The sum of two numbers is [latex]-61[/latex]. One number is 35 more than the other. Find the numbers.

 

31. The sum of two numbers is [latex]-316[/latex]. One number is 94 less than the other. Find the numbers.

 

32. The sum of two numbers is [latex]-284[/latex]. One number is 62 less than the other. Find the numbers.

 

33. One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

 

34. One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

 

35. One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.

 

36. One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.

 

37. One number is one more than twice another. Their sum is [latex]-5[/latex]. Find the numbers.

 

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

 

39. The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

 

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

 

41. The sum of two consecutive integers is 77. Find the integers.

 

42. The sum of two consecutive integers is 89. Find the integers.

 

43. The sum of two consecutive integers is [latex]-23[/latex]. Find the integers.

 

44. The sum of two consecutive integers is [latex]-37[/latex]. Find the integers.

 

45. The sum of three consecutive integers is 78. Find the integers.

 

46. The sum of three consecutive integers is 60. Find the integers.

 

47. Find three consecutive integers whose sum is [latex]-36[/latex].

 

48. Find three consecutive integers whose sum is [latex]-3[/latex].

 

49. Find three consecutive even integers whose sum is 258.

 

50. Find three consecutive even integers whose sum is 222.

 

51. Find three consecutive odd integers whose sum is 171.

 

52. Find three consecutive odd integers whose sum is 291.

 

53. Find three consecutive even integers whose sum is [latex]-36[/latex].

 

54. Find three consecutive even integers whose sum is [latex]-84[/latex].

 

55. Find three consecutive odd integers whose sum is [latex]-213[/latex].

 

56. Find three consecutive odd integers whose sum is [latex]-267[/latex].

 

Real-World Applications

57. Sale Price. Patty paid $35 for a purse on sale for $10 off the original price. What was the original price of the purse?

 

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

 

59. Buying in Bulk. Minh spent $6.25 on five sticker books to give his nephews. Find the cost of each sticker book.

 

60. Buying in Bulk. Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.

 

61. Price before Sales Tax. Tom paid $1,166.40 for a new refrigerator, including $86.40 tax. What was the price of the refrigerator?

 

62. Price before Sales Tax. Kenji paid $2,279 for a new living room set, including $129 tax. What was the price of the living room set?

 

Extentions

63. What has been your past experience solving word problems?

 

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

 

65. What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.

 

66. What are consecutive even integers? Name three consecutive even integers between [latex]-50[/latex] and [latex]-40[/latex].

 

Answers to Section 2.7 Odd Problems

1. Answers will vary

 

3. 30

 

5. 125

 

7. 6

 

9. 58

 

11. $750

 

13. $13,500

 

15. 4

 

17. 15

 

19. 5

 

21. 12

 

23. [latex]-5[/latex]

 

25. 18, 24

 

27. 8, 12

 

29. [latex]-18,-27[/latex]

 

31. [latex]-111,-205[/latex]

 

33. 32, 46

 

35. 23, 28

 

37. [latex]-2,-3[/latex]

 

39. 4, 10

 

41. 38, 39

 

43. [latex]-11,-12[/latex]

 

45. 25, 26, 27

 

47. [latex]-11,-12,-13[/latex]

 

49. 84, 86, 88

 

51. 55, 57, 59

 

53. [latex]-10,-12,-14[/latex]

 

55. [latex]-69,-71,-73[/latex]

 

57. $45

 

59. $1.25

 

61. $1080

 

63. Answers will vary

 

65. Consecutive odd integers are odd numbers that immediately follow each other. An example of three consecutive odd integers between 50 and 60 would be 51, 53, and 55.

 

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