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66.1 t 5(t 2) 10

67.30x 12 1,338

68.40y 19 1,381

69.7 37 r 14

70.21 25 f 19

71.10 2y 8

72.7 7x 21

73.9 5(r 3) 6 3(r 2)

74.2 3 (n 6) 4(n 2) 21

2

75. 3 z 4 8

7

76. 5 x 9 5

77. 2(9 3s) (5s 2) 25

78. 4(x 5) 3(12 x) 7

79. 9a 2.4 7a 4.6

80. 4c 1.6 7c 3.2

8.5 Using Equations to Solve Application Problems

675

WRITING

81.To solve 3x 4 5x 1, one student began by subtracting 3x from both sides. Another student solved the same equation by first subtracting 5x from both sides. Will the students get the same solution? Explain why or why not.

82.Explain the error in the following solution.

Solve:

2x 4 30

22x 4 302

x 4 15

x 4 4 15 4 x 11

REVIEW

Name the property that is used.

83.x 9 9x

84.x 99 99 x

85.(x 1) 2 x (1 2)

86.2(30y) (2 30)y

S E C T I O N 8.5

Using Equations to Solve Application Problems

Throughout this course, we have used the steps Analyze, Form, Solve, State, and Check as a strategy to solve application problems. Now that you have had an introduction to algebra, we can modify that strategy and make use of your newly learned skills.

1 Solve application problems to find one unknown.

To become a good problem solver, you need a plan to follow, such as the following fivestep strategy.You will notice that the steps are quite similar to the strategy first introduced in Chapter 1. However, this new approach uses the concept of variable, the translation skills from Section 8.1, and the equation solving methods of Sections 8.3 and 8.4.

Strategy for Problem Solving

1.Analyze the problem by reading it carefully to understand the given facts. What information is given? What are you asked to find? What vocabulary is given? Often, a diagram or table will help you visualize the facts of the problem.

2.Form an equation by picking a variable to represent the numerical value to be found. Then express all other unknown quantities as expressions involving that variable. Key words or phrases can be helpful. Finally, translate the words of the problem into an equation.

3.Solve the equation.

4.State the conclusion clearly. Be sure to include the units (such as feet, seconds, or pounds) in your answer.

5.Check the result using the original wording of the problem, not the equation that was formed in step 2 from the words.

Objectives

1Solve application problems to find one unknown.

2Solve application problems to find two unknowns.

EXAMPLE 1

676

Chapter 8 An Introduction to Algebra

Self Check 1

APARTMENT BUILDINGS Owners of a newly constructed apartment building would have to sell 34 more units before all of the 510 units were sold. How many of the apartment units have been sold to date?

Now Try Problem 19

Systems Analysis A

company’s telephone use would have to increase by 350 calls per hour before the system would reach the maximum capacity of 1,500 calls per hour. Currently, how many calls are being made each hour on the system?

Analyze

If the number of calls increases by

 

350, the system will reach capacity.

Given

The maximum capacity of the

 

system is 1,500 calls per hour.

Given

How many calls are currently

 

being made each hour?

Find

Caution! Unlike an arithmetic approach, you do not have to determine whether to add, subtract, multiply, or divide at this stage. Simply translate the words of the problem to mathematical symbols to form an equation that describes the situation. Then solve the equation.

© iStockphoto.com/Neustockimages

Form

Let n the number of calls currently being made each hour. To form an equation involving n, we look for a key word or phrase in the problem.

Key phrase: increase by 350

Translation: addition

The key phrase tells us to add 350 to the current number of calls to obtain an expression for the maximum capacity of the system. Now we translate the words of the problem into an equation.

 

The current number

increased

350

equals

the maximum capacity

 

of calls per hour

by

 

of the system.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n

 

 

350

 

1,500

 

 

Solve

 

 

 

 

 

 

 

 

n 350 1,500

 

We need to isolate n on the left side.

 

 

n 350 350 1,500 350

To isolate n, subtract 350 from both

 

4 10

 

 

1,500

 

 

 

sides to undo the addition of 350.

 

350

 

n 1,150

 

Do the subtraction.

 

 

1,150

 

 

 

 

State

Currently, 1,150 calls per hour are being made.

Check

If the number of calls currently being made each hour is 1,150, and we increase that number by 350, we should obtain the maximum capacity of the system.

1,150350

1,500 This is the maximum capacity.

The result, 1,150, checks.

Caution! Always check the result in the original wording of the problem, not by substituting it into the equation. Why? The equation may have been solved correctly, but the danger is that you may have formed it incorrectly.

EXAMPLE 2

Small Businesses Last

year, a stylist lost 17 customers who moved away. If she now has 73 customers, how many did she have originally?

Analyze

She lost 17 customers. Given

8.5 Using Equations to Solve Application Problems

677

 

2009.Marin,copyrightUsed

Shutterstock.comfromlicense

 

Self Check 2

 

 

 

 

 

 

 

 

GASOLINE STORAGE A tank

 

 

 

 

 

 

 

 

currently contains 1,325 gallons

 

 

 

 

of gasoline. If 450 gallons were

 

 

 

 

pumped from the tank earlier,

 

 

 

 

how many gallons did it originally

 

Image

under

contain?

 

 

 

 

 

She now has 73 customers. Given

How many customers did she originally have? Find

Form

We can let c the original number of customers. To form an equation involving c, we look for a key word or phrase in the problem.

Key phrase: moved away

Translation: subtraction

Now we translate the words of the problem into an equation.

 

 

This is called the verbal model.

 

 

 

The original number

minus

17

 

is

the number of

 

 

 

of customers

 

customers she has now.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c

 

17

 

 

73

 

 

 

Solve

 

 

 

 

 

 

 

 

 

c 17 73

We need to isolate c on the left side.

1

 

c 17 17 73 17

To isolate c, add 17 to both sides to undo the

 

 

 

73

 

 

subtraction of 17.

 

 

 

17

 

c 90

Do the addition.

 

 

 

90

 

 

 

 

 

 

State

She originally had 90 customers.

Check

If the hair stylist originally had 90 customers, and we decrease that number by the 17 that moved away, we should obtain the number of customers she has now.

8 10

90

17

73 This is the number of customers the hair stylist now has.

The result, 90, checks.

 

EXAMPLE 3

Traffic Fines For speeding

 

 

 

 

 

in a construction zone, a motorist had to pay a fine of

TRAFFIC FINES

$592. The violation occurred on a highway posted

DOUBLED IN

with signs like the one shown on the right.What would

CONSTRUCTION ZONE

 

 

the fine have been if such signs were not posted?

 

 

 

 

Analyze

 

 

 

 

For speeding, the motorist was fined $592.

 

Given

 

The fine was double what it would normally have been.

Given

 

What would the fine have been, had the sign not been posted?

Find

Form

We can let f the amount that the fine would normally have been. To form an equation, we look for a key word or phrase in the problem or analysis.

Key word: double Translation: multiply by 2

Now Try Problem 20

Self Check 3

SPEED READING A speed reading course claims it can teach a person to read four times faster. After taking the course, a student can now read 700 words per minute. If the company’s claims are true, what was the student’s reading rate before taking the course?

Now Try Problem 21

678

Chapter 8 An Introduction to Algebra

Now we translate the words of the problem into an equation.

Self Check 4

CLASSICAL MUSIC A woodwind quartet was hired to play at an art exhibit. If each member made $85 for the performance, what fee did the quartet charge?

Now Try Problem 22

 

Two

 

times

the normal

 

is

the new

 

 

 

 

 

 

speeding fine

 

fine.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

f

 

592

 

 

 

 

Solve

 

 

 

 

 

 

 

 

 

296

 

2f 592

We need to isolate f on the left side.

 

 

 

 

 

 

2 592

 

2f

592

To isolate f, divide both sides by 2 to undo the

 

 

 

4

 

 

 

 

 

multiplication by 2.

 

 

 

 

 

 

2

2

 

 

 

 

 

 

 

 

 

 

 

 

19

 

f 296

Do the division.

 

 

 

 

18

 

 

 

 

 

12

State

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12

The fine would normally have been $296.

 

 

 

 

0

Check

If the normal fine was $296, and we double it, we should get the new fine.

1 1

296

2

592 This is the new fine.

The result, $296, checks.

 

EXAMPLE 4

Entertainment Costs A five-piece band worked on

New Year’s Eve. If each player earned $120, what fee did the band charge?

Analyze

 

 

 

There were 5 players in the band.

Given

 

Each player made $120.

Given

 

What fee did the band charge?

Find

Form

We can let f the band’s fee. To form an equation, we look for a key word or phrase. In this case, we find it in the analysis of the problem. If each player earned the same amount ($120), the band’s fee must have been divided into 5 equal parts.

Key phrase: divided into 5 equal parts

Translation: division

Now we translate the words of the problem into an equation.

The

divided

the number of

is

each person’s

band’s fee

by

players in the band

share.

 

 

 

 

 

 

f

 

5

 

120

Solve

f

5 120

f

5 5 5 120

f 600

State

We need to isolate f on the left side.

To isolate f, multiply both sides by 5 to undo the

 

1

 

120

division by 5.

 

5

Do the multiplication.

 

600

 

 

The band’s fee was $600.

Analyze
To receive a degree in child development, students at one college must complete 135 hours of volunteer service by working 3-hour shifts at a local preschool. If a student has already volunteered 87 hours, how many more 3-hour shifts must she work to meet the service requirement for her degree?
EXAMPLE 5
0 0
The result, $600, checks.
Volunteer Service Hours

679

Check

If the band’s fee was $600, and we divide it into 5 equal parts, we should get the amount that each player earned.

120

5 600 This is the amount each band member earned.

5

1010

00

 

Students must complete 135 hours of volunteer service.

Given

 

 

 

 

 

 

 

 

Students work 3-hour shifts.

Given

 

 

A student has already completed 87 hours of service.

Given

 

 

How many more 3-hour shifts must she work?

Find

 

Form

Let x the number of shifts needed to complete the service requirement. Since each shift is 3 hours long, multiplying 3 by the number of shifts will give the number of additional hours the student needs to volunteer.

 

The number of

 

 

the number of

 

the number

 

 

hours she has

plus

3 times

shifts yet to be

is

of hours

 

 

already completed

 

 

completed

 

required.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

87

 

 

 

 

3

x

 

135

 

 

 

 

Solve

 

 

 

 

 

 

 

 

 

 

1215

 

87 3x

135

 

We need to isolate x on the left side.

 

 

 

 

1 3 5

 

87 3x 87

135 87

To isolate the variable term 3x, subtract 87

 

 

 

8 7

 

from both sides to undo the addition of 87.

 

 

 

4 8

 

 

3x

48

 

 

 

 

 

 

 

 

 

Do the subtraction.

 

 

 

 

 

 

 

 

 

 

 

3x

48

 

 

To isolate x, divide both sides by 3 to undo

 

 

16

 

 

 

 

 

 

 

 

 

 

 

 

the multiplication by 3.

 

 

 

 

 

3

 

 

 

 

3

3

 

 

 

 

 

 

 

48

 

 

x

16

 

 

Do the division.

 

 

 

 

 

 

3

 

 

 

 

 

 

 

 

 

 

18

State

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

18

The student needs to complete 16 more 3-hour shifts of volunteer service.

 

 

0

Check

The student has already completed 87 hours. If she works 16 more shifts, each 3 hours long, she will have 16 3 48 more hours. Adding the two sets of hours, we get:

8748

135 This is the total number of hours needed.

The result, 16, checks.

Self Check 5

SERVICE CLUBS To become a member of a service club, students at one college must complete 72 hours of volunteer service by working 4-hour shifts at the tutoring center. If a student has already volunteered 48 hours, how many more 4- hour shifts must she work to meet the service requirement for membership in the club?

Now Try Problem 23

EXAMPLE 6
Attorney’s Fees

680

Chapter 8 An Introduction to Algebra

Self Check 6

YARD SALES A husband and wife split the money equally that they made on a yard sale. The husband gave $75 of his share to charity, leaving him with $210. How much money did the couple make at their yard sale?

Now Try Problem 24

In return for her services, an attorney and her client split the jury’s cash award equally. After paying her assistant $1,000, the attorney ended up making $10,000 from the case. What was the amount of the award?

Analyze

The attorney and client split the award equally.

Given

The attorney’s assistant was paid $1,000.

Given

The attorney made $10,000.

Given

What was the amount of the award?

Find

Form

Let x the amount of the award. Two key phrases in the problem help us form an equation.

Key phrase:

split the award equally

Translation:

divide by 2

Key phrase:

paying her assistant $1,000

Translation:

subtract $1,000

Now we translate the words of the problem into an equation.

The award

 

the amount

 

the amount the

minus

paid to the

is

split in half

attorney makes.

 

assistant

 

 

 

 

 

 

 

 

x

 

1,000

 

10,000

 

 

 

2

 

 

 

 

 

 

Solve

x

2 1,000 10,000

x2 1,000 1,000 10,000 1,000

x

2 11,000

2 x2 2 11,000

x 22,000

We need to isolate x on the left side.

To isolate the variable term 2x , add 1,000 to both sides to undo the subtraction of 1,000.

Do the addition.

 

 

To isolate the variable x,

 

 

11,000

 

multiply both sides by 2

 

 

 

2

to undo the division by 2.

 

 

22,000

Do the multiplication.

State

The amount of the award was $22,000.

Check

If the award of $22,000 is split in half, the attorney’s share is $11,000. If $1,000 is paid to her assistant, we subtract to get:

$11,0001,000

$10,000 This is what the attorney made.

The result, $22,000, checks.

2 Solve application problems to find two unknowns.

When solving application problems, we usually let the variable stand for the quantity we are asked to find. In the next two examples, each problem contains a second unknown quantity. We will look for a key word or phrase in the problem to help us describe it using an algebraic expression.

EXAMPLE 7

8.5 Using Equations to Solve Application Problems

681

Civil Service A candidate for a position with the FBI scored 12 points higher on the written part of the civil service exam than she did on her interview. If her combined score was 92, what were her scores on the interview and on the written part of the exam?

Analyze

She scored 12 points higher on the written part than on the

 

 

interview.

Given

Her combined score was 92.

Given

What were her scores on the interview and on the written part?

Find

Form

Since we are told that her score on the written part was related to her score on the interview, we let x her score on the interview.

There is a second unknown quantity—her score on the written part of the exam. We look for a key phrase to help us decide how to represent that score using an algebraic expression.

Key phrase: 12 points higher on the

Translation: add 12 points to the

written part than on

interview score

the interview

 

So x 12 her score on the written part of the test. Now we translate the words of the problem into an equation.

The score on

plus

the score on

is

the overall

the interview

the written part

score.

 

 

 

 

 

 

 

x

 

x 12

 

92

Solve

x x 12 92 2x 12 92

2x 12 12 92 12

2x 80

22x 802

x 40

We need to isolate x on the left side.

On the left side, combine like terms: x x 2x.

To isolate the variable term, 2x, subtract 12 from both sides to undo the addition of 12.

Do the subtraction.

To isolate the variable x, divide both sides by 2 to undo the multiplication by 2.

Do the division. This is her score on the interview.

To find the second unknown, we substitute 40 for x in the expression that represents her score on the written part.

x 12 40 12

52

This is her score on the written part.

Self Check 7

CIVIL SERVICE A candidate for a position with the IRS scored 15 points higher on the written part of the civil service exam than he did on his interview. If his combined score was 155, what were his scores on the

interview and on the written part?

Now Try Problem 25

State

Her score on the interview was 40 and her score on the written part was 52.

Check

Her score of 52 on the written exam was 12 points higher than her score of 40 on the interview. Also, if we add the two scores, we get:

40

 

52

 

92

This is her combined score.

The results, 40 and 52, check.

EXAMPLE 8

682

Chapter 8 An Introduction to Algebra

Self Check 8

CRIME SCENES Police used

800 feet of yellow tape to fence off a rectangular-shaped lot for an investigation. Fifty less feet of tape was used for each width as for each length. Find the length and the width of the lot.

Playgrounds After

receiving a donation of 400 feet of chain link fencing, the staff of a preschool decided to use it to enclose a playground that is rectangular. Find the length and the width of the playground if the length is three times the width.

The perimeter is 400 ft.

Width

 

 

The length is three times as long as the width.

Analyze

Now Try Problem 26

The perimeter is 400 ft.

Given

The length is three times as long as the width.

Given

 

 

What is the length and what is the width of the rectangle?

Find

Form

We will let w the width of the playground. There is a second unknown quantity: the length of the playground. We look for a key phrase to help us decide how to represent it using an algebraic expression.

Key phrase: length three times the width Translation: multiply width by 3

So 3w the length of the playground.

The formula for the perimeter of a rectangle is P 2l 2w. In words, we can

write

 

 

 

 

 

 

 

 

 

 

 

 

 

2

the length of

plus

2

the width of

is

the

the playground

the playground

perimeter.

 

 

 

 

 

 

 

 

 

 

 

2

3w

 

2

w

 

400

Solve

2 3w 2w

400

6w 2w

400

 

8w

400

 

8w

400

 

 

 

 

8

8

 

w

50

We need to isolate w on the left side.

Do the multiplication: 2 3w 6w.

On the left side, combine like terms: 6w 2w 8w.

To isolate w, divide both sides by 8 to undo the multiplication by 8.

Do the division.

50

8 400

40

00

0

0

To find the second unknown, we substitute 50 for w in the expression that represents the length of the playground.

3w 3(50) Substitute 50 for w.

150 This is the length of the playground.

State

The width of the playground is 50 feet and the length is 150 feet.

Check

If we add two lengths and two widths, we get 2(150) 2(50) 300 100 400. Also, the length (150 ft) is three times the width (50 ft). The results check.

ANSWERS TO SELF CHECKS

1. 476 units have been sold. 2. The tank originally contained 1,775 gallons of gasoline. 3. The student used to read 175 words per minute. 4. The quartet charged $340 for the performance. 5. The student needs to complete 6 more 4-hour shifts of volunteer service. 6. The couple made $570 at the yard sale. 7. His score on the interview was 70 and his score on the written part was 85. 8. The length of the lot is 225 feet and the width of the lot is 175 feet.

S E C T I O N 8.5 STUDY SET

VOCABULARY

Fill in the blanks.

1.The five-step problem-solving strategy is:

the problem

Form an

the equation

State the

the result

2.Words such as doubled and tripled indicate the

 

operation of

 

 

.

 

 

 

 

3.

Phrases such as distributed equally and sectioned off

 

uniformly indicate the operation of

.

 

 

 

 

 

4.

Words such as trimmed, removed, and melted indicate

 

the operation of

 

 

.

 

 

5.

Words such as extended and reclaimed indicate the

 

operation of

 

 

.

 

 

 

 

 

6.

A letter (or symbol) that is used to represent a

 

number is called a

 

.

 

 

 

CONCEPTS

In each of the following problems, find the key word or phrase and tell how it translates. You do not have to solve the problem.

7.FAST FOOD The franchise fee and startup costs for a Taco Bell restaurant total $1,324,300. If an entrepreneur has $550,000 to invest, how much money will she need to borrow to open her own Taco Bell restaurant? (Source: yumfranchises.com)

Key word:

Translation:

8.GRADUATION ANNOUNCEMENTS Six of Tom’s graduation announcements were returned by the post office stamped “no longer at this address,” but 27 were delivered. How many announcements did he send?

Key word:

Translation:

9.WORKING IN GROUPS When a history teacher had the students in her class form equal-size discussion groups, there were seven complete groups, with five students in a group. How many students were in the class?

Key word:

Translation:

8.5 Using Equations to Solve Application Problems

683

10.SELF-HELP BOOKS An author book claimed that the information in his book could double a salesperson’s monthly income. If a medical supplies salesperson currently earns $5,000 a month, what monthly income can she expect to make after reading the book?

Key word:

Translation:

11.SCHOLARSHIPS See the illustration. How many scholarships were awarded this year?

Last year, s scholarships

Six more scholarships

were awarded.

were awarded this year

 

than last year.

12.OCEAN TRAVEL See the illustration. How many miles did the passenger ship travel?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Port

The freighter

 

 

The passenger ship

traveled m miles.

 

 

traveled 3 times farther

 

 

 

 

 

than the freighter.

13.SERVICE STATIONS See the illustration. How many gallons does the smaller tank hold?

Premium

 

Regular

This tank holds

This tank holds

g gallons.

100 gallons less

 

than the

 

premium tank.

14.Complete this statement about the perimeter of the rectangle shown.

2 2 240

The perimeter is 240 ft.

w

 

 

5w

684Chapter 8 An Introduction to Algebra

15.HISTORY A 1,700-year-old scroll is 425 years older than the clay jar in which it was found. How old is the jar?

Analyze

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The scroll is

 

 

 

 

years old.

 

 

 

 

 

The scroll is

 

 

 

 

 

years older than the jar.

 

 

 

 

How old is the

 

 

 

 

?

 

 

 

 

 

 

 

 

Form Let x

the

 

 

 

 

of the jar. Now we look for a key

phrase in the problem.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Key phrase: older than

 

 

 

 

 

 

 

Translation:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now we translate the words of the problem into an

equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The age

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

the age

of the

 

is

 

425 years

 

plus

of the

 

scroll

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

jar.

 

 

 

 

 

 

 

 

 

 

 

425

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solve

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

425 x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1,700

 

425 x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State The jar is

years old.

 

 

 

 

Check

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

425

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is the age of the scroll.

 

 

 

 

 

 

The result checks.

16.BANKING After a student wrote a $1,500 check to pay for a car, he had a balance of $750 in his account. How much did he have in the account before he wrote the check?

Analyze

 

 

 

 

 

 

 

 

 

 

 

 

 

A

 

 

check was written.

 

 

 

 

 

 

The new balance in the account was

 

 

 

.

 

 

 

 

 

 

 

 

 

 

 

 

How much did he have in the account

 

 

 

 

he

wrote the check?

 

 

 

 

 

 

 

 

 

 

 

Form Let x the account balance

 

 

he wrote the

check. Now we look for a key phrase in the problem.

Key phrase: wrote a check

 

 

 

 

 

 

 

 

 

 

 

Translation:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now we translate the words of the problem into an

equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The account

 

 

the amount

 

 

 

 

 

 

the

balance before

minus

of the

 

 

 

is

new

writing the check

 

 

check

 

 

 

 

 

balance.

 

 

 

 

 

 

 

 

1,500

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solve

1,500 750

x 1,500 750 x

State The account balance before writing the check was .

Check

1,500

 

 

 

This is the new balance.

 

 

 

The result checks.

 

17.AIRLINE SEATING An 88-seat passenger plane has ten times as many economy seats as first-class seats. Find the number of first-class seats and the number of economy seats.

Analyze

There are seats on the plane.

There are times as many economy as first-class

seats.

 

 

Find the number of

 

 

seats and the

number of

 

seats.

Form Since the number of economy seats is related to the number of first-class seats, we let x the number of

seats.

To represent the number of economy seats, look for a key phrase in the problem.

Key phrase: ten times as many

Translation: multiply by

So the number of economy seats.

The number of

plus

the number of

is

88.

first-class seats

economy seats

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

88

 

 

 

Solve

x 10x

88 11x 88

x

State There are first-class seats and economy seats.

Check The number of economy seats, 80, is times the number of first-class seats, 8. Also, if we add the numbers of seats, we get:

8

 

This is the total number of seats.

The results check.

18.THE STOCK MARKET An investor has seen the value of his stock double in the last 12 months. If the current value of his stock is $274,552, what was its value one year ago?

Analyze

 

The value of the stock

 

 

in 12 months.

 

The current value of the stock is

 

 

.

 

What was the

 

 

 

 

 

of the stock one year ago?

Form

 

 

 

 

 

 

 

 

 

 

 

 

 

We can let x = the

 

 

 

 

 

of the stock one year ago. We

now look for a key word in the problem.

 

 

Key phrase: double

 

 

 

 

 

 

 

 

 

 

 

 

Translation:

 

 

 

 

by 2

 

 

 

 

 

Now we translate the words of the problem into an

equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

the value

 

 

 

the current

2

times

of the stock

is

value of

 

 

 

one year ago

 

 

 

the stock.

2

 

 

 

 

 

 

 

 

 

 

 

274,552

 

 

 

 

 

 

 

 

Solve

2x

2x 274,552

x

State

The value of the stock one year ago was

 

.

Check

2

This is the current

value of the stock.

The result checks.

GUIDED PRACTICE

Form an equation and solve it to answer each question.

See Example 1.

19.FAST FOOD The franchise fee and startup costs for a Pizza Hut restaurant are $316,500. If an entrepreneur has $68,500 to invest, how much money will she need to borrow to open her own Pizza Hut restaurant?

See Example 2.

8.5 Using Equations to Solve Application Problems

685

See Example 3.

21.SPEED READING An advertisement for a speed reading program claimed that successful completion of the course could triple a person’s reading rate.

After taking the course, Alicia can now read 399 words per minute. If the company’s claims are true, what was her reading rate before taking the course?

See Example 4.

22.PHYSICAL EDUCATION A high school PE teacher had the students in her class form

three-person teams for a basketball tournament. Thirty-two teams participated in the tournament. How many students were in the PE class?

See Example 5.

23.BUSINESS After beginning a new position with 15 established accounts, a salesman made it his objective to add 5 new accounts every month. His goal was to reach 100 accounts. At this rate, how many months would it take to reach his goal?

See Example 6.

24.TAX REFUNDS After receiving their tax refund, a husband and wife split the refunded money equally. The husband then gave $50 of his money to charity, leaving him with $70. What was the amount of the tax refund check?

See Example 7.

25.SCHOLARSHIPS Because of increased giving, a college scholarship program awarded six more scholarships this year than last year. If a total of

20 scholarships were awarded over the last two years, how many were awarded last year and how many were awarded this year?

See Example 8.

26.GEOMETRY The perimeter of a rectangle is 150 inches. Find the length and the width if the length is four times the width.

APPLICATIONS

20.PARTY INVITATIONS Three of Mia’s party invitations were lost in the mail, but 59 were delivered. How many invitations did she send?

Form an equation and solve it to answer each question.

27.LOANS A student plans to pay back a $600 loan with monthly payments of $30. How many

payments has she made if she now only owes $420?

686Chapter 8 An Introduction to Algebra

28.ANTIQUES A woman purchases 8 antique spoons each year. She now owns 56 spoons. In how many years will she have 200 spoons in her collection?

29.HIP HOP Forbes magazine estimates that in 2008, Shawn “Jay-Z” Carter earned $82 million. If this was $68 million less than Curtis “50 Cent” Jackson’s earnings, how much did 50 Cent earn in 2008?

30.BUYING GOLF CLUBS A man needs $345 for a new set of golf clubs. How much more money does he need if he now has $317?

31.INTERIOR DECORATING As part of redecorating, crown molding was installed around the ceiling of a room. Sixty feet of molding was needed for the project. Find the length and the width of the room if its length is twice the width.

Molding

Paint

Wallpaper

32.SPRINKLER SYSTEMS A landscaper buried a water line around a rectangular lawn to serve as a supply line for a sprinkler system. The length of the lawn is 5 times its width. If 240 feet of pipe was used to do the job, what is the length and the width of the lawn?

Lawn

33.GRAVITY The weight of an object on Earth is 6 times greater than what it is on the moon. The situation shown below took place on Earth. If it took place on the moon, what weight would the scale register?

300

330

Pounds

360

34.INFOMERCIALS The number of orders received each week by a company selling skin care products increased fivefold after a Hollywood celebrity was added to the company’s infomercial. After adding the celebrity, the company received about 175 orders each week. How many orders were received each week before the celebrity took part?

35.THEATER The play Romeo and Juliet, by William Shakespeare, has 5 acts and a total of 24 scenes. The second act has the most scenes, 6. The third and fourth acts both have 5 scenes. The last act has the least number of scenes, 3. How many scenes are in the first act?

36.U.S. PRESIDENTS As of December 31, 1999, there had been 42 presidents of the United States. George Washington and John Adams were the only presidents in the18th century (1700-1799). During the 19th century (1800-1899), there were 23 presidents. How many presidents were there during the 20th cenury (1900-1999)?

37.HELP WANTED From the following ad from the classified section of a newspaper, determine the value of the benefit package. ($45K means $45,000.)

ACCOUNTS PAYABLE

2-3 yrs exp as supervisor. Degree a +. High vol company. Good pay, $45K & xlnt benefits; total compensation worth $52K. Fax resume.

38.POWER OUTAGES The electrical system in a building automatically shuts down when the meter shown reads 85. By how much must the current reading increase to cause the system to shut down?

50

30 70

1090

39.VIDEO GAMES After a week of playing Sega’s Sonic Adventure, a boy scored 11,053 points in one game— an improvement of 9,485 points over the very first

time he played. What was his score for his first game?

On Earth

40.AUTO REPAIR A woman paid $29 less to have her car fixed at a muffler shop than she would have paid at a gas station. At the gas station, she would have paid $219. How much did she pay to have her car fixed?

41.For a half-hour time slot on television, a producer scheduled 18 minutes more time for the program than time for the commercials. How many minutes of commercials and how many minutes of the program were there in that time slot? (Hint: How many minutes are there in a half hour?)

from Campus to Careers

Broadcasting

© iStockphoto.com/Dejan Ljami´c

42.SERVICE STATIONS At a service station, the underground tank storing regular gas holds 100 gallons less than the tank storing premium gas. If the total storage capacity of the tanks is 700 gallons, how much does the premium gas tank and how much does the regular gas tank hold?

43.CLASS TIME In a biology course, students spend a total of 250 minutes in lab and lecture each week. The lab time is 50 minutes shorter than the lecture time. How many minutes do the students spend in lecture and how many minutes do students spend in lab per week?

44.OCEAN TRAVEL At noon, a passenger ship and a freighter left a port traveling in opposite directions. By midnight, the passenger ship was

3 times farther from port than the freighter was. How far was the freighter and how far was the passenger ship from port if the distance between the ships was 84 miles?

45.ANIMAL SHELTERS The number of phone calls to an animal shelter quadrupled after the evening news aired a segment explaining the services the shelter offered. Before the publicity, the shelter received 8 calls a day. How many calls did the shelter receive each day after being featured on the news?

46.OPEN HOUSES The attendance at an elementary school open house was only half of what the principal had expected. If 120 people visited the school that evening, how many had she expected to attend?

47.BUS RIDERS A man had to wait 20 minutes for a bus today. Three days ago, he had to wait 15 minutes longer than he did today, because four buses passed by without stopping. How long did he wait three days ago?

 

8.5 Using Equations to Solve Application Problems

687

48.

HIT RECORDS The

 

 

 

oldest artist to have a

 

Congressof

 

was Louis Armstrong,

 

 

number one single

 

Librarythe

 

Dolly. He was 55

 

 

 

with the song Hello

 

 

 

years older than the

 

of

 

 

Courtesy

 

youngest artist to

 

 

 

have a number one

 

 

 

single, 12-year-old Jimmy Boyd, with I Saw Mommy

 

 

Kissing Santa Claus. How old was Louis Armstrong

 

 

when he had the number one song? (Source: The Top

 

10 of Everything, 2000.)

 

 

49.

COST OVERRUNS Lengthy delays and skyrocketing

 

costs caused a rapid-transit construction project to

 

 

go over budget by a factor of 10. The final audit

 

 

showed the project costing $540 million. What was

 

 

the initial cost estimate?

 

 

50.

LOTTO WINNERS The grocery store employees

 

 

listed below pooled their money to buy $120 worth

 

 

of lottery tickets each week, with the understanding

 

that they would split the prize equally if they

 

 

happened to win. One week they did have the

 

 

winning ticket and won $480,000. What was each

 

 

employee’s share of the winnings?

 

 

 

Sam M. Adler

Ronda Pellman

Manny Fernando

 

Lorrie Jenkins

Tom Sato

Sam Lin

 

 

Kiem Nguyen

H. R. Kinsella

Tejal Neeraj

 

 

Virginia Ortiz

Libby Sellez

Alicia Wen

 

51.RENTALS In renting an apartment with two other friends, Enrique agreed to pay the security deposit of $100 himself. The three of them agreed to contribute equally toward the monthly rent. Enrique’s first check to the apartment owner was for $425. What was the monthly rent for the apartment?

52.BOTTLED WATER DELIVERY A truck driver left the plant carrying 300 bottles of drinking water. His delivery route consisted of office buildings, each of which was to receive 3 bottles of water. The driver returned to the plant at the end of the day with

117 bottles of water on the truck. To how many office buildings did he deliver?

53.CONSTRUCTION To get a heavy-equipment operator’s certificate, 48 hours of on-the-job training are required. If a woman has completed 24 hours, and the training sessions last for 6 hours, how many more sessions must she take to get the certificate?