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Solving Applications of Proportions

5.5

5.5 OBJECTIVE

1. Solve an application that involves a proportion

Now that you have learned how to find an unknown value in a proportion, let’s see howthis can be used in the solution of applications.

443

Step 1 Read the problem carefully to determine the given information.Step 2 Write the proportion necessary to solve the problem. Use a letter to

represent the unknown quantity. Be sure to include the units in writingthe proportion.

Step 3 Solve, answer the question of the original problem, and check theproportion as before.

Step by Step: Solving Applications of Proportions

Using a Proportion to Find an Unknown Value

In a shipment of 400 parts, 14 are found to be defective. How many defective parts shouldbe expected in a shipment of 1000?

Assume that the ratio of defective parts to the total number remains the same.

Multiply:

400x � 14,000

Divide by the coefficient, 400.

x � 35

35 defective parts should be expected in the shipment.Checking the original proportion, we get

14 � 1000 � 400 � 35

14,000 � 14,000

14 defective

400 total�

x defective

1000 total

Example 1

We have decided to let x be the unknown number of defective parts.

C H E C K Y O U R S E L F 1

An investment of $3000 earned $330 for 1 year. How much will an investment of$10,000 earn at the same rate for 1 year?

Let’s look at an application involving fractions in the proportion.

444 CHAPTER 5 RATIOS AND PROPORTIONS

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Using Proportions to Find an Unknown Value

The scale on a map is given as inch (in.) = 3 miles (mi). The distance between two towns

is 4 in. on the map. How far apart are the towns in miles?

For this solution we use the fact that the ratio of inches (on the map) to miles remains thesame.

1 � x � 4 � 3 � 4

x � 48 (mi)

4�1

4� � x � 4 � 3 � 4

1

4� x � 3 � 4

1

4 in.

3 mi�

4 in.

x mi

1

4

Example 2

C H E C K Y O U R S E L F 2

Jack drives 125 mi in hours (h). At the same rate, how far will he be able to travel

in 4 h? (Hint: Write as an improper fraction.)2

12

2

12

We may also find decimals in the solution of an application.

Using Proportions to Find an Unknown Value

Jill works 4.2 h and receives $21. How much will she get if she works 10 h?The ratio of hours worked to the amount of pay remains the same.

Let a be the unknown amount of pay.4.2 h

$21�

10 h

$a

Example 3

NOTE We could divide both

sides by :

then invert and multiply

� 48

x �121

� 41

x �1214

x �3 � 4

14

14

� x

14

� 3 � 4

14

14

SOLVING APPLICATIONS OF PROPORTIONS SECTION 5.5 445©

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In Example 4 we must convert the units stated in the problem.

C H E C K Y O U R S E L F 3

A piece of cable 8.5 centimeters (cm) long weighs 68 grams (g). What will a 10-cmlength of the same cable weigh?

Using Proportions to Find an Unknown Value

A machine can produce 15 tin cans in 2 minutes (min). At this rate how many cans can itmake in an 8-h period?

In writing a proportion for this problem, we must write the times involved in terms ofthe same units.

2x � 15 � 480

or 2x � 7200

x � 3600 cans

15 cans

2 min�

x cans

480 min

Example 4

Because 1 h is 60 min, convert 8 hto 480 min.

C H E C K Y O U R S E L F 4

Instructions on a can of film developer call for 2 ounces (oz) of concentrate to1 quart (qt) of water. How much of the concentrate is needed to mix with 1 gallon(gal) of water? (4 qt � 1 gal.)

An important use of proportions is in solving problems involving similar geometricfigures. These are figures that have the same shape and whose corresponding sides areproportional. For instance, in the similar triangles shown below,

a proportion involving corresponding sides is

3

4�

6

8

6

8

3

4

4.2a � 210

Divide both sides by 4.2.

a � $50

4.2a

4.2�

210

4.2

446 CHAPTER 5 RATIOS AND PROPORTIONS

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C H E C K Y O U R S E L F 5

If a woman who is feet tall casts a shadow that is 3 feet long, how tall is a

building that casts a shadow that is 90 feet long?

5

12

C H E C K Y O U R S E L F A N S W E R S

1. $1100 2.

Divide both sides by

x � 200 mi

3. 80 g 4. 8 oz 5. 165 feet tall

5

2

5

2 x � 500

125 mi

5

2 h

�x mi

4 h

Solving an Application Using Similar Triangles

If a 6-foot-tall man casts a shadow that is 10 feet long, how tall is a tree that casts a shadowthat is 140 feet long?

Let’s look at a picture of the two triangles involved.

From the similar triangles, we have the proportion

Using the proportion rule, we have 6 � 140 � 10 � h

10 � h � 840

h � 84

The tree is 84 feet tall.

10 � h

10�

840

10

6

10�

h

140

h ft.

10 ft.6 ft

Example 5

NOTE Connect the top of thetree to the end of the shadowto create a triangle. Connectingthe top of the man to the endof his shadow creates a similartriangle.

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ExercisesSolve the following applications.

1. Book purchases. If 12 books are purchased for $40, how much will you pay for18 books at the same rate?

2. Construction. If an 8-foot (ft) two-by-four costs 96¢, what should a 12-ft two-by-four cost?

3. Consumer affairs. A box of 18 tea bags is marked 90¢. At that price, what should abox of 48 tea bags cost?

4. Consumer affairs. Cans of orange juice are marked 2 for 93¢. What would the priceof a case of 24 cans be?

5. Workload. A worker can complete the assembly of 15 tape players in 6 hours (h). Atthis rate, how many can the worker complete in a 40-h workweek?

6. Consumer affairs. If 3 pounds (lb) of apples cost 90¢, what will 10 lb cost?

7. Elections. The ratio of yes to no votes in an election was 3 to 2. How many no voteswere cast if there were 2880 yes votes?

8. College enrollment. The ratio of men to women at a college is 7 to 5. How manywomen students are there if there are 3500 men?

9. Photography. A photograph 5 inches (in.) wide by 6 in. high is to be enlarged sothat the new width is 15 in. What will be the height of the enlargement?

10. Shift work. Meg’s job is assembling lawn chairs. She can put together 55 chairs in4 h. At this rate, how many chairs can she assemble in an 8-h shift?

11. Distance. Christy can travel 110 miles (mi) in her new car on 5 gallons (gal) of gas.How far can she travel on a full tank, which has 12 usable gal?

12. Property taxes. The Changs purchased an $80,000 home, and the property taxeswere $1400. If they make improvements and the house is now valued at $120,000,what will the new property tax be?

5.5

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

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13. Distance. A car travels 165 mi in 3 h. How far will it travel in 8 h if it continues atthe same speed?

14. Consumer affairs. A battery pack is on sale at 2 for $3. At this rate, how much will7 packs cost?

15. Manufacturing. The ratio of teeth on a smaller gear to those on a larger gear is 3 to7. If the smaller gear has 15 teeth, how many teeth does the larger gear have?

16. Consumer affairs. A store has T-shirts on sale at 2 for $5.50. At this rate, what willfive shirts cost?

Using the given map, find the distances between the cities named in exercises 17 to 20.

17. Find the distance from Harrisburg to Philadelphia.

18. Find the distance from Punxsutawney (home of the groundhog) to State College(home of the Nittany Lions).

19. Find the distance from Gettysburg to Meadville.

20. Find the distance from Scranton to Waynesburg.

21. Manufacturing. An inspection reveals 30 defective parts in a shipment of 500. Howmany defective parts should be expected in a shipment of 1200?

22. Investments. You invest $4000 in a stock that pays a $180 dividend in 1 year. At thesame rate, how much will you need to invest to earn $270?

PENNSYLVANIA

Railroad

MARYLANDWEST VIRGINIA

OHIO

NEW YORK

NEW JERSEY

DEVA

L a k e E r i e

Allegheny Reservior

reviR

annaheuqsuS

Ohio

Riv e

r

revi

Rer

awa

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D

PymatuningReservior

FALLINGWATER

JamestownOlean Elmira

Ithaca

Binghamton

Camden

Vineland

MillvilleAberdeen

Baltimore

Frederick

HagerstownCumberland

Charles Town

Morgantown

Fairmont Martinsburg

ClarksburgAtlantic City

Youngstown

Wilmington

PittsburghAlquippa

Erie

Carbondale

Scranton

Williamsport

MansfieldBradford

Meadville

Oil City

Du Bois

Wilkes-Barre

Altoona

Punxsutawney

WashingtonMcKeesport

UniontownGettysburg

York Chester

Lancaster

Reading

Allentown

Chambersburg

Lebanon

Hazleton

Waynesburg

BethlehemState College

New Castle

Philadelphia

Towanda

Lewisburg

Pottstown

Norristown

Kennett Square

Carlisle

Mars Indiana

Butler

Latrobe JohnstownHershey

Warren

Franklin

Sharon

Kittanning

Milton

1

90

22

84

83

81

70

79

80

78

15

6

220

81

81

62

6

17

219

80

15

70

522

Trenton

Harrisburg

© MAGELLAN GeographixSMSanta Barbara, CA (800) 929-4MAP

N

0 40 mi

ANSWERS

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

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23. Football. A football back ran 212 yards (yd) in the first two games of the season. Ifhe continues at the same pace, how many yards should he gain in the 11-gameseason?

24. Cooking. A 6-lb roast will serve 14 people. What size roast is needed to serve21 people?

25. Lawn care. A 2-lb box of grass seed is supposed to cover 2500 square feet (ft2) oflawn. How much seed will you need for 8750 ft2 of lawn?

26. Fencing. A 6-ft fence post casts a 9-ft shadow. How tall is a nearby pole that casts a15-ft shadow?

27. Lighting. A 9-ft light pole casts a 15-ft shadow. Find the height of a nearby tree thatis casting a 40-ft shadow.

28. Construction. On the blueprint of the Wilsons’ new home, the scale is 5 in. equals7 ft. What will be the actual length of a bedroom if it measures 10 in. long on theblueprint?

29. Distance. The scale on a map is in. � 50 mi. If the distance between two towns on

the map is 6 in., how far apart are they in miles?

1

2

hh ft f

2030405040302010

10 20 30 40 50 40 30 20

10

10

ANSWERS

23.

24.

25.

26.

27.

28.

29.

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30. Science. A metal bar expands in. for each 12°F rise in temperature. How much will

it expand if the temperature rises 48°F?

31. Car maintenance. Your car burns 2 quarts (qt) of oil on a trip of 5000 mi. How many

quarts should you expect to use when driving 7200 mi?

32. Lighting. A 6-ft person casts a 7 -ft shadow. If the shadow of a nearby pole is 30 ft

long, how tall is the pole?

33. Manufacturing. A piece of tubing 10.5 centimeters (cm) long weighs 35 grams (g).What is the weight of a piece of the same tubing that is 15 cm long?

34. Salary. Jane works 7.75 h and receives $38.75 pay. What will she receive at the samerate if she works 12 h?

35. Sales tax. The sales tax on an item costing $80 is $5.20. What will the tax be for anitem costing $150?

36. Conversion. If 8 kilometers (km) is approximately 4.8 mi, how many kilometers willequal 12 mi?

37. Timing. You find that your watch gains 2 minutes (min) in 6 h. How much will itgain in 3 days?

38. Painting. If 2 qt of paint will cover 225 ft2, how many square feet will 2 gal cover?(1 gal � 4 qt)

39. Construction. Directions on a box of 4 cups of wallpaper paste are to mix thecontents with 5 qt of water. To mix a smaller batch using 1 cup of paste, how muchwater (in ounces) should be added? (1 qt � 32 oz)

40. Film processing. A film processing machine can develop three rolls of film every8 min. At this rate, how many rolls can be developed in a 4-h period?

1

2

1

2

1

4

ANSWERS

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

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41. Carpooling. Approximately 7 out of every 10 people in the U.S. workforce drive towork alone. During morning rush hour there are 115,000 cars on the streets of amedium-sized city. How many of these cars have one person in them?

42. Carpooling. Approximately 15 out of every 100 people in the U.S. workforcecarpool to work. There are an estimated 320,000 people in the workforce of a givencity. How many of these people are in car pools?

Use a proportion to find the unknown side, labeled x, in each of the following pairs ofsimilar figures.

43. 44.

45. 46.

47. A recipe for 12 servings lists the following ingredients:

12 cups ziti 7 cups spaghetti sauce 4 cups ricotta cheese

cup parsley 1 teaspoon garlic powder teaspoon pepper

4 cups mozzarella 2 tablespoons parmesancheese cheese

Determine the amount of ingredients necessary to serve 5 people.

Answers

1. $60 3. ; 18x � 4320; x � 240¢, or $2.40

5. 100 players 7. 1920 no votes 9. 18 in. 11.

5x � 1320; x � 264 mi 13. 440 mi 15. 35 teeth 17. 110 mi

19. 215 mi 21. 72 defective parts 23.212 yd

2 games�

x yd

11 games; x � 1166 yd

5 gal

110 mi�

12 gal

x mi;

18 tea bags

90 ¢�

48 tea bags

x¢

1

2

1

2

x

12

4

3

4

x

8

12

6

x

5

2

4

6

x

2

ANSWERS

41.

42.

43.

44.

45.

46.

47.

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25. 7 lb 27. 24 ft 29. 600 mi 31.

5000x � 18,000; x � 3.6 qt 33. 50 g 35. $9.75

37. (Write 3 days as 72 h)

39. 40 oz 41. 80,500 cars with one person 43. 3 45. 647.

2 min

6 h�

x min

72 h; x � 24 min

�Write 2

1

2 as

5

2�

5

2 qt

5000 mi�

x qt

7200 mi;

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453

Using Your Calculator to Solve Proportions

When the numbers involved in a proportion are large, your calculator can be very useful.Consider Example 1.

Example 1

Solving a Proportion for an Unknown Value

Solve the proportion for n.

Start the solution as before.

1505n � 43 � 105

Find n, using the calculator.

43 105 1505

Display 3

���

1505n

1505�

43 � 105

1505

n

43�

105

1505

Divide both terms by the coefficient,1505.

n has the value 3. To check, replace n with 3 and use yourcalculator to multiply.

C H E C K Y O U R S E L F 1

Solve the proportion for n.

n

27�

35

189

In practical applications, you may have to round the result after using your calculator inthe solution of a proportion. Example 2 shows such a situation.

Example 2

Using Proportions to Find an Unknown Value

Micki drives 278 miles (mi) on 13.6 gallons (gal) of gas. If the gas tank of her car holds21 gal, how far can she travel on a full tank of gas?

We can write the proportion

Multiply.

13.6 x � 278 � 21

278 mi

13.6 gal�

x mi

21 gal

454 CHAPTER 5 RATIOS AND PROPORTIONS

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Now divide both terms by 13.6.

Now to find x, we must multiply 278 by 21 and then divide by 13.6. On the calculator,

278 21 13.6

Let’s round the result to the nearest mile; Micki can drive about 429 mi on a full tank of gas.

429.26471���

13.6x

13.6�

278 � 21

13.6

C H E C K Y O U R S E L F 2

Life insurance costs $4.37 for each $1000 of insurance. How much does a $25,000policy cost?

Your calculator can be very handy for comparing prices at the grocery store.As we said in Section 5.2, to find the unit price, just divide the cost of the item by the

number of units.

Example 3

Using Proportions to Find an Unknown Value

A dishwashing liquid comes in three sizes:

(a) 12 ounces (oz) for 77¢

(b) 22 oz for $1.33

(c) 32 oz for $1.85

Which is the best buy?For each size, let’s find the unit price in cents per ounce.

(a) For the first size (77¢ for 12 oz), using your calculator, divide.

77 12

(b) To find the unit price for the second size, divide again:

133 22

(c) For the third size,

185 32

$1.85

32 oz� 5.8¢ per ounce

5.78125��

$1.33

22 oz� 6.0¢ per ounce

6.0454545��

77¢

12 oz� 6.4¢ per ounce

6.4166667��

NOTE Notice that we consider$1.33 as 133¢ to find the ratio“cents per ounce.”

We have chosen to round to thenearest tenth of a cent.

Treat $1.33 as 133¢.

Again round to the nearest tenthof a cent.

1

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SOLVING APPLICATIONS OF PROPORTIONS SECTION 5.5 455©

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Applying Proportions to Find an Unknown Value

Vegetable oil is sold in the following quantities:

(a) 16 oz for $1.27

(b) 1 pint (pt) 8 oz for $1.79

(c) 1 quart (qt) 6 oz for $2.89

Which is the best buy?

(a)

(b) Because 1 pt is 16 oz, 1 pt 8 oz is (16 � 8) oz, or 24 oz. So we write 1 pt 8 oz as 24 oz.

(c) Because 1 qt is 32 oz, 1 qt 6 oz is (32 � 6) oz, or 38 oz. Write 1 qt 6 oz as 38 oz.

In this case, by comparing the unit prices we see that the 1-pt 8-oz size is the best buy.

$2.89

38 oz� 7.6¢ per ounce

$1.79

24 oz� 7.5¢ per ounce

$1.27

16 oz� 7.9¢ per ounce

C H E C K Y O U R S E L F 3

A floor cleaner comes in three sizes:

32 oz for $2.8948 oz for $3.4370 oz for $4.96

Which is the best buy?

Example 4

C H E C K Y O U R S E L F 4

Ketchup is sold in the following quantities:

12 oz for $0.681 pt 5 oz for $1.051 qt 7 oz for $1.89

Which is the best buy?

C H E C K Y O U R S E L F A N S W E R S

1. 5 2. $109.25 3. 70-oz size at 7.09¢ per oz 4. 1 qt 7 oz for 4.8¢ per oz

Comparing the three unit prices, we see that the 32-oz size of dishwashing liquid, at 5.8¢per ounce, is the best buy.

Note: All the ratios used must be in terms of the same units. If quantities involve differ-ent units, they must be converted.

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Calculator Exercises

Solve for the unknown.

1. 2.

3. (to nearest tenth) 4. (to nearest hundredth)

5. (to nearest tenth) 6. (to nearest hundredth)

Solve the following applications.

7. Salary. Bill earns $248.40 for working 34.5 hours (h). How much will he receive ifhe works at the same pay rate for 31.75 h?

8. Construction. Construction-grade lumber costs $384.50 per 1000 board-feet. Whatwill be the cost of 686 board-feet?

9. Speed of sound. A speed of 88 feet per second (ft/s) is equal to a speed of 60 milesper hour (mi/h). If the speed of sound is 750 mi/h, what is the speed of sound in feetper second?

10. Manufacturing. A shipment of 75 parts is inspected, and 6 are found to be faulty.At the same rate, how many defective parts should be found in a shipment of 139?Round your result to the nearest whole number.

11. Taxes. The property tax on a $67,250 home is $2315. At the same rate, what will bethe tax on a home valued at $87,625? Round your result to the nearest dollar.

12. Production. A machine produces 158 items in 12 minutes (min). At the same rate,how many items will it produce in 8 h?

12.2

0.042�

x

0.08

2.7

3.8�

5.9

n

13.9

8.4�

n

9.2

x

4.7�

11.8

16.9

770

1988�

n

71

630

1365�

15

a

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

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13. Gas mileage. Sally travels 510 mi, using 11.6 gallons (gal) of gas. How manygallons of gas will she need for a trip of 1800 mi? Give your answer to the nearesttenth of a gallon.

14. Tom and Jerry operate a food concession stand at a local amusement park. Theysell the most food when the attendance at the park is at maximum capacity. Whenthis happens, they sell an average of 450 pork roll sandwiches and 550 cheesesteak sandwiches. The company that owns the park is going to expand; they willincrease the capacity of the park from 6000 to 9000 people next season. Tom andJerry plan to expand their concession stand so they can sell moresandwiches.

(a) Using the same ratio of attendance to sandwiches, how many additionalsandwiches of each kind would Tom and Jerry expect to sell?

(b) The following costs are associated with the anticipated expansion:

Currently, pork roll sandwiches sell for $1.50, and steak sandwiches sell for$2.75. Based on these prices and the information in part (a), how would you planthe expansion, and what would you charge for a sandwich?

Find the best buy in each of the following exercises.

15. Dishwashing liquid: 16. Canned corn:

(a) 12 oz for 79¢ (a) 10 oz for 21¢

(b) 22 oz for $1.29 (b) 17 oz for 39¢

17. Syrup: 18. Shampoo:

(a) 12 oz for 99¢ (a) 4 oz for $1.16

(b) 24 oz for $1.59 (b) 7 oz for $1.52

(c) 36 oz for $2.19 (c) 15 oz for $3.39

19. Salad oil (1 qt is 32 oz): 20. Tomato juice (1 pt is 16 oz):

(a) 18 oz for 89¢ (a) 8 oz for 37¢

(b) 1 qt for $1.39 (b) 1 pt 10 oz for $1.19

(c) 1 qt 16 oz for $2.19 (c) 1 qt 14 oz for $1.99

Item Cost per Unit

Construction $70/square footSupplies $0.65/pork roll sandwich

$1.25/steak sandwichEmployee costs $8/hour

ANSWERS

13.

14.

15.

16.

17.

18.

19.

20.

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21. Peanut butter (1 lb is 16 oz): 22. Laundry detergent:

(a) 12 oz for $1.25 (a) 1 lb 2 oz for $1.99

(b) 18 oz for $1.72 (b) 1 lb 12 oz for $2.89

(c) 1 lb 12 oz for $2.59 (c) 2 lb 8 oz for $4.19

(d) 2 lb 8 oz for $3.76 (d) 5 lb for $7.99

Answers1. 32.5 3. 3.3 5. 8.3 7. $228.60 9. 1100 ft/s 11. $301613. 40.9 gal 15. b 17. c 19. b 21. c

ANSWERS

21.

22.

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