6-5 Solving Proportions
Solve each proportion.
1. SOLUTION: 2. SOLUTION: 3. SOLUTION: 4. SOLUTION: eSolutions Manual - Powered by Cognero
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6-5 Solving Proportions
4. SOLUTION: 5. SOLUTION: 6. SOLUTION: 7. TELEVISION Aspect ratio is the ratio of width to height of a television screen. A widescreen television screen
has an aspect ratio of 16 inches wide to 9 inches high. If a television screen is 48 inches wide, how many inches high
is it?
SOLUTION: Let h represent the height of the screen that is 48 inches wide.
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The height of the television screen is 27 inches.
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6-5 Solving Proportions
7. TELEVISION Aspect ratio is the ratio of width to height of a television screen. A widescreen television screen
has an aspect ratio of 16 inches wide to 9 inches high. If a television screen is 48 inches wide, how many inches high
is it?
SOLUTION: Let h represent the height of the screen that is 48 inches wide.
The height of the television screen is 27 inches.
8. TIME Joaquin has a total of 12.5 hours of football practice after school five days a week. Each practice is the
same amount of time. How many hours of practice will he have for 15 school days? 22 school days?
SOLUTION: Let x represent the number of hours per day Joaquin has practice.
Joaquin practices 2.5 hours per day. Use the equation 2.5d = h to find the number of hours h he practices in d days.
Joaquin has 37.5 hours of practice for 15 school days.
Joaquin has 55 hours of practice for 22 school days.
Solve each proportion.
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Joaquin has 37.5 hours of practice for 15 school days.
6-5 Solving Proportions
Joaquin has 55 hours of practice for 22 school days.
Solve each proportion.
9. SOLUTION: 10. SOLUTION: 11. SOLUTION: 12. SOLUTION: eSolutions Manual - Powered by Cognero
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6-5 Solving Proportions
12. SOLUTION: 13. SOLUTION: 14. SOLUTION: 15. SOLUTION: eSolutions Manual - Powered by Cognero
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6-5 Solving Proportions
15. SOLUTION: 16. SOLUTION: 17. SOLUTION: 18. SOLUTION: eSolutions Manual - Powered by Cognero
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6-5 Solving Proportions
18. SOLUTION: 19. SOLUTION: 20. SOLUTION: 21. PAINTING The classrooms at Lincoln Middle School are painted every summer. If 7 gallons of paint are needed to
paint 4 classrooms, how many gallons of paint are needed to paint 16 classrooms?
SOLUTION: Let g represent the gallons of paint needed to paint 16 classrooms.
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To paint 16 classrooms, 28 gallons of paint are needed.
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22. PIZZA A principal is ordering pizzas for a school pizza party. He knows that 9 pizzas will feed 25 students. If there
6-5 Solving Proportions
21. PAINTING The classrooms at Lincoln Middle School are painted every summer. If 7 gallons of paint are needed to
paint 4 classrooms, how many gallons of paint are needed to paint 16 classrooms?
SOLUTION: Let g represent the gallons of paint needed to paint 16 classrooms.
To paint 16 classrooms, 28 gallons of paint are needed.
22. PIZZA A principal is ordering pizzas for a school pizza party. He knows that 9 pizzas will feed 25 students. If there
are 300 students in the school, how many pizzas will he need to order?
SOLUTION: Let p represent the number of pizzas needed for 300 students.
The principal will need to order 108 pizzas.
23. BOATS A boat traveled 150 feet in 9.7 seconds. How far would the boat travel in 1 minute? 1 minute 30 seconds?
SOLUTION: Let x represent the number of feet the boat travels per second.
The boat travels about 15.464 feet per second. Use the equation d = 15.464t to find the distance d the boat travels in
t seconds.
Because 1 minute = 60 seconds, substitute 60 for t and solve.
The boat travels about 927.8 feet in 1 minute.
Because 1 minute 30 seconds = 60 + 30 or 90 seconds, substitute 90 for t and solve.
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The boat travels about 1391.8 feet in 1 minute 30 seconds.
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6-5 Solving Proportions
The principal will need to order 108 pizzas.
23. BOATS A boat traveled 150 feet in 9.7 seconds. How far would the boat travel in 1 minute? 1 minute 30 seconds?
SOLUTION: Let x represent the number of feet the boat travels per second.
The boat travels about 15.464 feet per second. Use the equation d = 15.464t to find the distance d the boat travels in
t seconds.
Because 1 minute = 60 seconds, substitute 60 for t and solve.
The boat travels about 927.8 feet in 1 minute.
Because 1 minute 30 seconds = 60 + 30 or 90 seconds, substitute 90 for t and solve.
The boat travels about 1391.8 feet in 1 minute 30 seconds.
24. RAINFALL The record for the most amount of rain in the shortest amount of time in the United States was 12
inches in 42 minutes. How much rain fell in 15 minutes? 28 minutes?
SOLUTION: Let x represent the number of inches of rain per minute.
The rain fell at a rate of about 0.286 inches per minute. Use the equation a = 0.286t to find the number of inches of
rain a that fell in t minutes.
In 15 minutes, about 4.29 inches of rain fell.
In 28 minutes, about 8 inches of rain fell.
25. PUNCH The table shows the amount of each ingredient in 52 ounces of punch.
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28 minutes, about 8 inches of rain fell.
6-5 In
Solving
Proportions
25. PUNCH The table shows the amount of each ingredient in 52 ounces of punch.
a. If you have 130 ounces of punch, how much lime juice does the punch contain?
b. If there are 54 ounces of sparkling lemon water in the punch, how many ounces of cranberry concentrate does
the punch contain?
c. If the punch contains 44 ounces of water, how many ounces of punch do you have?
SOLUTION: a. There are 4 ounces of lime juice in 52 ounces of punch. Let l represent the number of ounces of lime juice in 130
ounces of punch.
There are 10 ounces of lime juice in 130 ounces of punch.
b. For every 24 ounces of sparkling lemon water, there are 12 ounces of cranberry concentrate. Let c represent the
number of ounces of cranberry concentrate in punch that contains 54 ounces of sparkling lemon water.
There are 27 ounces of cranberry concentrate in punch that contains 54 ounces of sparkling lemon water.
c. There are 12 ounces of water in 52 ounces of punch. Let p represent the number of ounces of punch that
contains 44 ounces of water.
44 ounces of water would be contained in approximately 190.67 ounces of punch.
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Write a proportion that could be used to solve for each variable. Then solve.
26. 6 goals in 14 games
9 goals in g games
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6-5 Solving Proportions
44 ounces of water would be contained in approximately 190.67 ounces of punch.
Write a proportion that could be used to solve for each variable. Then solve.
26. 6 goals in 14 games
9 goals in g games
SOLUTION: 9 goals in 21 games
27. s inches in 0.54 hour
4.55 inches in 1.89 hours
SOLUTION: 1.3 inches in 0.54 hour
28. 14 gallons for d dollars
8 gallons for $24.72
SOLUTION: 14 gallons for $43.26
29. 20 boxes on 4 shelves
b boxes on 20 shelves
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6-5 Solving Proportions
14 gallons for $43.26
29. 20 boxes on 4 shelves
b boxes on 20 shelves
SOLUTION: 100 boxes on 20 shelves
30. ART A 14-inch-wide by 20-inch-long print of the Eiffel Tower is also available as a postcard 6 inches long. What is
the width of the postcard?
SOLUTION: Let w represent the width of the postcard.
The width of the postcard is 4.2 inches.
Solve each proportion.
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6-5 Solving Proportions
32. SOLUTION: 33. SOLUTION: 34. SOLUTION: 35. eSolutions Manual - Powered by Cognero
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6-5 Solving Proportions
35. SOLUTION: 36. SOLUTION: 37. MULTIPLE REPRESENTATIONS
In this problem, you will explore proportions. A craft store is offering the specials shown for different materials.
a. Algebraic Write an equation relating the cost c to the number of yards n for each material.
b. Graphical Graph the cost of each material per yard on a coordinate plane. Which item costs the least per yard?
the most? How is this shown on the graph?
c. Numerical How much would it cost to buy 18 inches of ribbon? How much would it cost to buy 10 meters of
fleece?
SOLUTION: a. Let y represent the cost per yard of each material. To determine the total cost of a certain material, find the
product of the cost per yard and the number of yards bought.
Material
Cost
Unit Rate
Cost c of the
number of
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yards n
Ribbon
$2.67 for 3 yd
c = 0.89n
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fleece?
SOLUTION: a.
Let y represent the cost per yard of each material. To determine the total cost of a certain material, find the
6-5 product
Solving
Proportions
of the cost per yard and the number of yards bought.
Material
Cost
Unit Rate
Ribbon
$2.67 for 3 yd
Cost c of the
number of
yards n
c = 0.89n
Fleece
$9.50 for 2 yd
c = 4.75n
Quilting
Fabric
$11.96 for 4 yd
c = 2.99n
Satin
Fabric
$10.35 for 3 yd
c = 3.45n
b.
Ribbon costs the least per yard. Fleece costs the most per yard. The most expensive fabric is the line that is the
steepest. The cheapest fabric is the line that is the least steep.
c. Convert
inches
yards, then substitute for n in c = 0.89n and solve.
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Use 1 in. ≈ 2.54 cm and 100 cm ≈ 1.094 yd.
6-5 Ribbon
Solving
Proportions
costs the least per yard. Fleece costs the most per yard. The most expensive fabric is the line that is the
steepest. The cheapest fabric is the line that is the least steep.
c. Convert 18 inches to yards, then substitute for n in c = 0.89n and solve.
Use 1 in. ≈ 2.54 cm and 100 cm ≈ 1.094 yd.
It will cost $0.45 to buy 18 inches of ribbon.
Convert 10 meters to yards, then substitute for n in c = 4.75n and solve.
Use 1 m ≈ 1.094 yd.
It will cost $51.97 to buy 10 meters of fleece.
38. OPEN ENDED Give two examples that are proportional to
.
SOLUTION: and
are both proportional to .
39. REASONING Suppose a:b = 2:4 and b:c = 4:9. If a = 10, find the value of c.
SOLUTION: Substitute 10 for a in
and solve.
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Substitute
for b in
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6-5 Solving and
Proportions
are both proportional to .
39. REASONING Suppose a:b = 2:4 and b:c = 4:9. If a = 10, find the value of c.
SOLUTION: Substitute 10 for a in
and solve.
Substitute 20 for b in
and solve.
40. CHALLENGE Solve each proportion.
a.
b.
c.
d.
SOLUTION: a.
b.
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6-5 Solving Proportions
40. CHALLENGE Solve each proportion.
a.
b.
c.
d.
SOLUTION: a.
b.
c.
d.
41. ERROR ANALYSIS Trey and Morgan are solving the proportion
. Is either of them correct? Explain your
reasoning.
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6-5 Solving Proportions
41. ERROR ANALYSIS Trey and Morgan are solving the proportion
. Is either of them correct? Explain your
reasoning.
SOLUTION: Sample answer:
Trey is correct. Morgan cross multiplied incorrectly.
42. CHALLENGE Rectangle ABCD has a fixed area. As the length l and the width w change, what do you know
about their product? Is the length proportional to the width? Justify your reasoning.
SOLUTION: Sample answer: Because the area is fixed, the product of the length and width will remain constant, lw = k. The
length is not proportional to the width. For example if the area is 36 square units:
w
3
6
9
l
12
6
4
Area = lw
36
36
36
Ratio (w : l)
3 : 12 = 1 : 4
6:6=1:1
9:4
From the table, you can see the proportions are not equal.
43. WRITING IN MATH Describe a situation in which it may be easier to solve a proportion using the constant of
proportionality. Explain your reasoning. Does it matter which method you use to solve a proportion? Why or why
not?
SOLUTION: Sample answer: It is easier to use the constant of proportionality if you are solving the proportion for several
unknowns. It is easy to find a unit rate and then multiply the amounts needed to find the equivalent amounts. No,
because no matter which method you use to solve a proportion, you will arrive at the same answer.
44. Which equation could be used to find the total cost c if Fernando wanted to buy 8 pencils from the school store?
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A c = 0.15 • 8
B c = 0.45 • 8
C c = 24 • 0.45
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SOLUTION: 6-5
Sample answer: It is easier to use the constant of proportionality if you are solving the proportion for several
unknowns.
It is easy to find a unit rate and then multiply the amounts needed to find the equivalent amounts. No,
Solving Proportions
because no matter which method you use to solve a proportion, you will arrive at the same answer.
44. Which equation could be used to find the total cost c if Fernando wanted to buy 8 pencils from the school store?
A c = 0.15 • 8
B c = 0.45 • 8
C c = 24 • 0.45
D c = 0.15 • 24
SOLUTION: Three pencils cost $0.45. Find the cost of one pencil using a proportion.
Each pencil costs $0.15. The total cost is equal to the product of the cost of each pencil and the number of pencils
bought.
c = 0.15 • 8
Choice A is the correct answer.
45. A line to purchase concert tickets is moving at a rate of 5 feet every 20 minutes. At this rate, how long will a person
have been in line if they have moved 30 feet?
F 30 min
G1h
H 1 h 30 min
J 2h
SOLUTION: Let t represent the number of minutes spent standing in line in order to move 30 feet.
A person will spend 120 minutes or 2 hours in line to move 30 feet. Choice J is the correct answer.
46. During her keyboarding test, Luisa typed 145 words in three minutes. Which of the following could not be used to
determine the number of minutes it would take her to type 550 words?
A
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B
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6-5 Solving Proportions
A person will spend 120 minutes or 2 hours in line to move 30 feet. Choice J is the correct answer.
46. During her keyboarding test, Luisa typed 145 words in three minutes. Which of the following could not be used to
determine the number of minutes it would take her to type 550 words?
A
B
C
D
SOLUTION: A
B
C
D
This proportion can be
used to determine the
number of minutes it would
take her to type 550 words.
This proportion can be
used to determine the
number of minutes it would
take her to type 550 words.
This proportion can be
used to determine the
number of minutes it would
take her to type 550 words.
This proportion can not be
used to determine the
number of minutes it would
take her to type 550 words.
Choice D is the correct answer.
47. EXTENDED RESPONSE There are 258 eighth graders at Henderson Middle School. The graph shows how
many of the eighth graders participate in each sport.
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6-5 Solving Proportions
Choice D is the correct answer.
47. EXTENDED RESPONSE There are 258 eighth graders at Henderson Middle School. The graph shows how
many of the eighth graders participate in each sport.
a. If each student participates in only one sport, write a proportion that could be used to predict how many students
at Henderson Middle School participate in sports if there are 645 total students.
b. Solve the proportion in part a.
c. Write and solve a proportion that could be used to predict the number of eighth graders that participate in track if
there are 220 total eighth graders.
SOLUTION: a. Out of 258 eighth graders, the number of students participating in a sport is 25 + 38 + 21 + 7 + 9 or 100. Let s
represent the number of students predicted to participate in sports if there are 645 total students.
b.
If there are 645 total students, 250 students are predicted to participate in sports.
c. Out of 258 eighth graders, the number of students participating in track is 25. Let s represent the number of
student predicted to participate in track if there are 220 total students.
If there are 220 eighth graders, about 21 eighth graders are predicted to participate in track.
48. Determine whether the rental charge is proportional to the time. Explain your reasoning.
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6-5 Solving Proportions
If there are 220 eighth graders, about 21 eighth graders are predicted to participate in track.
48. Determine whether the rental charge is proportional to the time. Explain your reasoning.
SOLUTION: Time
(hours)
Rental
Charge
1
2
3
4
$13
$23
$33
$43
Rate
No, the rental charge is not proportional to the time because the rates are not equal.
Use dimensional analysis to complete each conversion. Round to the nearest hundredth.
49. 5 in. ≈ _ cm
(Hint: 1 in. ≈ 2.54 cm)
SOLUTION: 50. 10 km ≈ _ mi
(Hint: 1 km ≈ 0.621 mi)
SOLUTION: 51. 26.3 cm ≈ _ in.
(Hint: 1 cm ≈ 0.394 in.)
SOLUTION: 52. PLANES How long will it take an Air Force fighter jet to fly 5200 miles at 650 miles per hour?
SOLUTION: Let h = the number of hours it will take. Then the product of the speed and h gives the number of miles flown.
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It will take 8 hours for the jet to fly 5200 miles.
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52. PLANES How long will it take an Air Force fighter jet to fly 5200 miles at 650 miles per hour?
SOLUTION: Let h = the number of hours it will take. Then the product of the speed and h gives the number of miles flown.
It will take 8 hours for the jet to fly 5200 miles.
Find each product.
53. SOLUTION: 54. SOLUTION: 55. SOLUTION: 56. SOLUTION: eSolutions Manual - Powered by Cognero
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6-5 Solving Proportions
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