Applications of
Ratios and Proportions
Connections
Have you ever . . .
• Needed to convert a measurement from inches to feet?
• Calculated the unit price of something sold by the case?
• Tried to plan the amount of food needed for a party?
Ratios are a useful tool in mathematics because they apply to
so many real-world situations. Ratios are all about relationships.
Converting measurements, finding percents, and measuring distances on maps can all be done using ratios.
Ratios are fractions that relate one type of thing to something else. For example:
12 inches
1 foot
25 miles
• 25 miles per gallon can be shown as a ratio: 1 gallon
• 12 inches to a foot can be shown as a ratio:
7 pounds
• Seven pounds of pasta for 55 guests can be shown as a ratio: 55 guests
Once you have a ratio, you can divide to find the unit rate, the amount per one unit of measurement. For example, if you have driven 225 miles on five gallons of gas in your new hybrid
car, you can divide to find a unit rate of 45 miles per gallon.
You can also use ratios to solve proportions, which are equations that involve ratios and an unknown. For example, if you
know your hybrid car gets 45 miles per gallon, you can calculate
the amount of gas for a 398-mile trip with a proportion:
45 miles 398 miles
x
1 gallon =
79
Essential Math Skills
Learn
It!
Solving Problems Using Ratios
A ratio is simply a fraction, but it may have different units in the numerator and
denominator.
You will often want to find a unit rate from a ratio. How many miles per gallon did you get
on that last road trip? How many stuffed mushrooms will you have for each guest? Miles per
gallon, price per pound, and inches per foot are all unit rates. When you write a unit rate as a
fraction, the denominator is one.
To find the unit rate if the denominator is not one, divide. If you have seven pounds of pasta
7
for 55 guests, that would be 55 , or about 0.13 pounds per guest.
Marie is following a pattern for building a birdhouse, but the
measurements are in centimeters and her measuring tape is in
inches. She knows that there are 2.54 centimeters in an inch. The
roof of the birdhouse should be 25 cm by 30 cm. What are its
dimensions in inches?
Write a Ratio
You can write a relationship between two values as a ratio. It is a correct ratio whichever
units are in the numerator and the denominator, but to avoid extra steps to solve the problem, put the units that you want in your final answer in the numerator.
Write a ratio relating centimeters to inches.
?1.
When you have a relationship between two things, such as centimeters and inches, you have
a ratio. You can write the ratio of 2.54 centimeters to one inch in two ways:
1 inch
2.54 centimeters
2.54 centimeters or
1 inch
Since your final answer should be in inches, the first ratio will be easiest. The ratio of centimeters to inches is a conversion factor. Since one inch and 2.54 centimeters are equal, the
value of the ratio is one. This is true of any fraction where the numerator and the denomina4
tor are equal, just as 4 = 1 . Because the ratio is equal to one, you can multiply any number
by this ratio and not change the equivalent value of that number when the units change.
80
Applications of Ratios and Proportions
Write and Solve an Equation
Since ratios are fractions, you can cancel common units to help you solve an equation. To
get rid of the old units (centimeters in this case) make sure that your conversion factor has
centimeters in the denominator. Once you cancel all the units you can, the remaining units
should match the units you want in your final answer.
Multiply the original measurement by the conversion factor to find the measurements
?2.
in inches. Round your answer to the nearest tenth.
Multiply both the length and width by the conversion factor.
25 cm
1 inch
25
1 # 2.54 cm = 2.54 inches = 9.8425 . . .
30 cm
1 inch
30
1 # 2.54 cm = 2.54 inches = 11.8110 . . .
Rounded to the nearest tenth, the dimensions are 9.8 inches by 11.8 inches.
Check Your Answer
When doing conversions, it is easy to lose track of the decimal place. Check to make sure
that your answer is sensible, and review your math for common errors.
Is a birdhouse roof that is about 10 by 10 inches reasonable?
?3.
Build Your
Math Skills
The size of the birdhouse is reasonable. If your answer were 0.98 by
1.18 inches, that would be a hummingbird house! If it were 98 by 118
inches, it would be more appropriate for a family of eagles. The measurements sound reasonable.
Try solving this
problem using the
conversion ratio
with one inch in the
denominator:
2.54 cm
1 inch
What extra steps do
you need? Why?
81
Essential Math Skills
e
ic
Pract
It!
Solve the following problems using ratios.
1 quart
1.The conversion factor for cups to a quart is 4 cups . Manolo is making
a party punch that calls for 14.5 cups of limeade, but he only has a quart
jar to use to measure the limeade. How many quarts should he use?
a.
Write the ratio or ratios in the best way for this problem. Explain why you wrote
the ratios this way.
b.
Find the solution and check your answer.
2.Aikiko is building a wood shed. The blueprint she is using says that one inch on the
drawing is equivalent to 1.5 feet. She needs to cut 12 two-by-fours to the right length. In
the blueprint, they are five inches long. How long will the two-by-fours be?
a.
Write the ratio or ratios in the best way for this problem. Explain why you wrote
the ratios this way.
b.
Find the solution and check your answer.
3.Javier is in chemistry lab. He needs 250 mL of 10% sodium hydroxide, w/v. This means
that for every liter of water, he will add 100 grams of sodium hydroxide. How many
grams of sodium hydroxide does he need?
a.
Write the ratio or ratios in the best way for this problem. Explain why you wrote
the ratios this way.
b.
Find the solution and check your answer.
82
Applications of Ratios and Proportions
4.Liz and Deshane work for a painting company. Liz can paint 25 square
feet in eight minutes, and Deshane can paint 25 square feet in six
minutes. Working together, how much can they paint in four hours?
a.
Write two ratios and an equation relating them.
b.
Answer the question and check the answer.
5.Destinee and Sarah are working together on a problem that says: “Convert
743 millimeters to inches using the following information. There are 1000 millimeters
in a meter, 100 centimeters in a meter, and 2.54 centimeters in an inch.” Sarah showed
the following work. She knows it is wrong because it is so large. Inches aren’t that much
larger than millimeters. How could Destinee correct it for her?
743 mm a
1000 mm ka 100 cm ka in k
m
m
2.54 cm = 29, 251, 968.5
a.
Identify the ratio or ratios that are incorrectly used.
b.
Solve the problem. Round to the nearest tenth.
c.
What advice would you give Sarah to avoid this error in the future?
83
Essential Math Skills
Learn
It!
Cross-Multiply and Divide
You can solve any proportion with the cross-multiply and divide strategy. If you
have one fraction equal to another fraction, multiplying denominators across
the equals sign eliminates fractions. Then, divide to isolate the variable.
Cross-Multiply
a
c ax = bc
b = x
4 2x = 3 ◊ 4
2
3 = x
Divide
bc
ax bc
a = a x= a
3◊4
2x 3 ◊ 4
x= 2
2 = 2
At the grocery store, peaches are priced at five pounds for $3.00. You have $4.50.
Assuming there is no tax, how many pounds of peaches could you buy?
Find the Relationship
Find the relationship in the question. Look for the words per or for. These words usually
indicate a relationship that can be written as a ratio. Write the ratio as a fraction, including
the units.
Identify the given relationship in the problem and write it as a fraction with units.
?1.
The given relationship is five pounds for $3.00. You can write this as a ratio:
5 pounds
3 dollars
3 pounds
You could also say that three dollars buys you five pounds and write it 5 dollars .
The ratio can be written either way, since it is equivalent to one. Three dollars is equivalent
to five pounds of peaches at the grocery store, so the ratio is equivalent to one.
Write a Proportion
Using the ratio on one side of the equation, write a proportion. There should be a fraction
on the other side of the equation. The other fraction will have the same units, in the same
places, and will use another given value (in this case, $4.50). You can use x to represent the
unknown. Once you’re sure your units line up, you can remove them.
84
Applications of Ratios and Proportions
Write a proportion to solve the problem.
?2.
Review &
Practice
Using the ratio of pounds to dollars, you can write a proportion to
solve the problem:
5 pounds
x
3 dollars = 4.5 dollars
5
x
3 = 4.5
Get extra practice
with the worksheet
for ratios and
proportions on
pages 325–326.
Cross-Multiply and Divide
You want to know the value of x, so you want to isolate x on one side of the equation. You
can solve any proportion by first cross-multiplying (multiply the denominator of each fraction across the equal sign) and then dividing by the new coefficient of x.
Solve the proportion.
?3.
Step 1: Cross-multiply.
5
x
3 = 4.5
h
g
4.5 # 5 = 3x
22.5 = 3x
Step 2: Divide.
22.5 3x
3 = 3
You can buy
22.5
3 =x
22.5
3 or 7½ pounds of peaches.
Check Your Answer
Make sure that the answer is logical and check for errors in your math.
If you could buy five pounds for $3.00, does it make sense that you can buy seven and a
?4.
half pounds for $4.50? Why?
$4.50 is one and a half times $3.00. It makes sense that you could buy one and a half times
as many peaches for $4.50 as for $3.00. Since one-and-a-half times five pounds is seven and a
half pounds, the answer makes sense.
85
Essential Math Skills
e
ic
Pract
It!
Write and solve proportions to answer the following questions.
1.Sophia wants to find the gas mileage of her car. When
she fills the tank, the odometer reads 92,481 miles.
When she fills it again, the odometer reads 92,692. It
takes 9.3 gallons to completely fill the tank. What is the
gas mileage of Sophia’s car, to the nearest hundredth?
a.
Find the relationship and write a proportion.
b.
Solve the proportion and check your answer.
c.
Identify an error you might make in solving this problem. How could you avoid
that error?
2.A new test for Lyme disease is 87% accurate in detecting if someone has the disease. If
12,400 people who have Lyme disease are tested, how many cases will not be detected?
a.
Find the relationship and write a proportion.
b.
Solve the proportion and check your answer.
c.
Identify an error you might make in solving this problem. How could you avoid
that error?
86
Applications of Ratios and Proportions
3.You are driving across the country from Phoenix to Boston, a distance of 2,648 miles.
You make the first 500 miles in seven hours, including stops. If you can keep up this
rate, how many more hours will it take you to get to Boston? Round your answer to the
nearest tenth.
a.
Find the relationship and write a proportion.
b.
Solve the proportion and check your answer.
c.
Identify an error you might make in solving this problem. How could you avoid
that error?
2.5 4.9
4. 19 = x
a.
Solve the proportion.
b.
What is a different way that you could solve this proportion? Compare the
two methods.
5.Henderson’s Groceries offers three pounds of medium prawns for $8.61, and Grocery
Time offers five pounds for $12.50. Which store has the best price for prawns?
a.
Solve the problem.
b.
What is a different way that you could solve this problem? Compare the
two methods.
87
Essential Math Skills
6.John and Hasim are studying together. The problem they are working on says: The sales
tax on a $400 stereo is $38. At the same tax rate, how much would the tax be on a $650
stereo? John does the following work. Hasim knows that the tax wouldn’t be less on a
more expensive item.
400 dollars
x
650 dollars = 38 dollars
38 (400) = 650x
38 (400)
x = 650 = $23.38
a.
Identify the mistake.
b.
Solve the proportion.
c.
What advice would you give John to avoid this error in the future?
7.Carlos and Tam both commute to work.
Carlos traveled nine miles through
bumper to bumper traffic at 15 miles per
hour. Tam lives farther away, 21 miles. His
commute as distance d in miles to time t
in minutes is represented by the graph.
Who arrives at work sooner, and by how
much time?
d
22
20
18
16
14
12
10
8
6
4
2
0
88
t
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Applications of Ratios and Proportions
Check Your Skills
Answer the following questions using your knowledge of ratios and proportions.
1.If your truck gets 15 miles to the gallon, how many miles can you go on an 18 gallon tank
of gas?
a.
1.2 miles
b.
270 miles
c.
27 miles
d.
2,700 miles
2.
You are looking at buying a used car. One car costs $8,100 and gets 22 miles per
gallon. Another costs $10,500 and gets 27 miles per gallon. With gas prices
averaging $4 per gallon, how many miles would it take for the more expensive car to
make up for the difference in cost?
a.
2,400 miles
b.
3,000 miles
c.
600 miles
d.
1,920 miles
3.Construction has begun on the lot next door to your house. On the blueprint, the
building measures 27 cm wide and 15 cm long. The scale of the blueprint is 1 cm = 3 feet.
If the lot for this building is 100 feet wide, and the building is centered on the lot, how
many feet are there between the building and the side of the lot?
4.A manufacturing company has set a goal of a maximum of two defects per 1,000
capacitors produced. The company ships out 350,828 capacitors. Use the following
figures to create an equation showing how many capacitors should be functional for
the company to meet its goal.
1,000 2 350,828 x
=
89
Essential Math Skills
5.
The large meteorite that blasted through Russia’s sky in
February 2013 had an average density of 3.6 g/cm3. Dmitri
found a rock that weighed 0.12 pounds and had a volume of 13.21
cm3. The density would need to be within 0.2 g/cm3 of the average
for the rock to have been possibly part of the meteorite. How far
is the density of the rock from 3.6 g/cm3? Round your answer to
the nearest tenth.
Math Tip
Learn common
values for
conversions.
1 mile = 1,609 meters
1 mile = 5,280 feet
1 pound = 453.6
grams
6.Grayson and Mo are heading downtown, a distance of 3.5 miles.
The bicycle odometer says they are going 24 miles per hour. How
many meters per minute are they traveling?
See page 304 for a
table of common
units of measure.
7.Andre’s Grocery has a special on mangos, four for $3.50. Pearson’s is advertising
mangos for five for $7.25. How much do you save if you are buying 10 mangos from the
cheaper store?
a.
$4.00
b.
$5.75
c.
$6.50
d.
$7.25
8.An experienced mechanic can change a head gasket in about five hours. Her apprentice
would take eight hours to complete the task. If the mechanic works for two hours
and then turns the job over to the apprentice, how long does it take the apprentice to
complete the job?
9.
Jude is on the swim team. He swam the 50-meter
breaststroke in 32.74 seconds. What was his speed in
miles per hour, rounded to the nearest hundredth?
a.
0.68 mph
b.
40.95 mph
c.
3.42 mph
d.
12.67 mph
90
Remember
the Concept
Ratios are fractions that
show a relationship.
Use a ratio to convert
units of measure.
Cross-multiply and
divide to solve
proportion problems.
Answers and Explanations
Applications of Ratios and
Proportions
Deshane: 25 ft2 per 6 min
page 79
4 hr = 240 min
2
Practice It!
pages 82–83
1 qt
1a.14.5 c #
4c
With the measurement that you have (cups) in the
denominator of the conversion factor, you can multiply the quantity you have by the conversion factor.
Your answer should be smaller than the original number, since quarts are larger than cups.
1b.35/8 quarts
1 1
5
14 2 # 4 = 3 8
1.5 ft
2a.5 in #
1 in
With the measurement that you have (inches) in
the denominator of the conversion factor, you can
multiply the quantity you have by the conversion
factor. Your answer should be larger than the original
number, since one inch in the drawing equals more
than one foot.
2b.7.5 feet
5 × 1.5 = 7.5
When there is a one in the denominator, you can
remove it.
1L
=
3a.250 mL #
1000 mL 0.25 L
With the measurement that you have (mL) in the
denominator of the conversion factor, you can multiply the quantity you have by the conversion factor.
Your answer should be smaller than the original number, since liters are larger than milliliters.
0.25 L × 100 g sodium hydroxide/1 L
The conversion factor tells you how much sodium
hydroxide to add for every liter of solution. You want
liters in the denominator of the conversion factor so
that you are left with grams of sodium hydroxide.
3b.25 g
1
250 # 1000 = 0.25
0.25 ×
100
= 25
1
2
25 ft
25 ft
( 8 min # 240 min) + ( 6 min # 240 min) = x
Solving Problems Using Ratios
4b.1,750 square feet
# 240 j + ` 25 # 240 j = x
` 25
8
6
750 + 1000 = 1750
5a.The ratio of millimeters to meters should have the
millimeters in the denominator. The equation should
change millimeters to meters by multiplying with millimeters in the denominator.
5b.29.25 inches
100 cm
m
nd
1000 mm
m
1
1 in
743 ` 1000 j^100h` 2.54 j
743 mm d
na 1 in k
2.54 cm
1
1
743 ` 10 j` 2.54 j in
74.3
= 29.251 . . .
2.54
5c.You might advise Sarah to make sure that her units
cancel to avoid this error in the future.
Cross-Multiply and Divide
Practice It!
pages 86–88
92, 692 - 92, 481 miles
x
=
1a.
9.3 gallons
1 gallon
1b.22.69 mpg
92, 692 - 92, 481
211
=
9.3
9.3
x = 22.688 . . .
1c.An error you might make is not realizing you need to
subtract the first odometer reading from the second
to find the miles driven in 9.3 gallons. To avoid this
error, think about the end result you want: miles
driven per gallon. In this problem, you know the
amount of gallons used (9.3). Ask, how many miles
were driven with 9.3 gallons?
2a.100 - 87 = 13 undetected cases
13
x
=
100 tested
12, 400 tested
4a.Liz: 25 ft2 per 8 min
i
Essential Math Skills
2b.1,612 undetected cases
13
x
=
100
12, 400
13 × 12,400 = 100x
161,200 = 100x
x = 1,612
2c.An error you might make is not realizing that a
percent can be written as a ratio. When a percent is
written as a fraction, it also is a ratio: x out of 100.
500 miles
2648 miles
=
3a.
7 hours
x
3
8.61
5
> 12.50 ?
0.35 $ 0.4
Since the answer is no, Henderson’s Groceries has the
better price. These methods are very similar, since
you are performing the same math. This is a trial-anderror method, since you start by guessing whether to
use a greater than or less than sign.
3b.30.1 hours
6a.The proportion is set up incorrectly. The first ratio
must have a similar relationship to the second ratio.
You can set it up in different ways:
500
2648
=
7
x
400 dollars
650 dollars
=
38 tax
x
500x = 7 × 2648
400 dollars
38 tax
=
650 dollars
x
500x = 18536
x = 37.072
37.1 - 7 = 30.1
3c.One mistake you might make is forgetting to subtract
the seven hours already driven from the total hours
for the trip. You could subtract 500 miles from the
total mileage before writing your proportion to avoid
this problem. Or, you could double-check what the
question is asking for before finalizing your answer.
4a.
2.5
4.9
=
19
x
2.5x = 19(4.9)
2.5x = 93.1
x = 37.24
4b.You could also solve the proportion by changing 2.5/19
to a decimal. This method may be more difficult for
many people, since 2.5 divided by 19 creates a long,
difficult decimal.
5a.Henderson’s Groceries has the better price.
Compare:
3 lb
x
5 lb
x
=
=
8.61
1 and 12.50
1
ii
5b.You might create an inequality to solve this problem
and see if it is true.
6b.$61.75
400
38
=
650
x
8
38
=
13
x
8x = 13(38)
8x = 494
x = 61.75
6c.You might tell John to examine his proportion. Do the
numerator and denominator on one side represent
a similar relationship to the numerator and denominator on the other side? In the first ratio of John’s
equation, the numerator is the first price and the
denominator is the second price. In the second ratio,
the numerator is the tax for the second price and the
denominator is the tax for the first price. The relationships are different.
7.
Tam arrives six minutes earlier.
Carlos: 9 miles #
1 hour
=x
15 miles
9
hours = x
15
x = 0.6 hours
Approximately 0.3484 lb per dollar and 0.4 lb
per dollar
0.6 hours (
The first price is lower.
60 min
) = 36 min
1 hour
Tam: On the graph, you can see that Tam drives
21 miles in 30 minutes.
Answers and Explanations
Check Your Skills
pages 89–90
1.
b. 270 miles
15 miles
x
=
1 gallon
18 gallons
5
10
=
7.25
y
5y = 72.5
y = $14.50
x
15 = 18
14.5 - 8.75 = 5.75
x = 270
8.
4.8 hours
2.
b. 3,000 miles
2 hours
2
= complete
5 hours
5
10500 - 8100 = 2400
3/5 or 0.6 left to do
$2400
=
$4 per gallon 600 gallons
8 hours
x
=
1 job
0.6 job
27 miles per gallon - 22 miles per gallon = 5 mpg
x
8 = 0.6
5 miles
x
=
1 gallon
600 gallons
x
5 = 600
3000 = x
3.
9.5 feet
x = 8 × 0.6 = 4.8 hours
9.
c. 3.42 mph
50 m
3600 sec
c
mc 1 mile m = 180, 000 . 3.42
52, 678.66
1 hr
32.74 sec
1609 m
1 cm
27 cm
=
3 ft
x
x = 3 × 27 = 81
The building is 81 feet wide.
100 - 81 = 19 feet left in the lot
19 ÷ 2 = 9.5 feet on each side of the building
2 defects
x
=
4.
1000 capacitors
350, 828 capacitors
2
x
=
1000
350, 828
5.
0.5 g/cm3
0.12 lb
453.6 g
.432
3
3
m = 54
3c
13.21 g/cm . 4.1 g/cm
13.21 cm
1 lb
4.1 - 3.6 = 0.5 g/cm3
6.
643.6 meters per minute
1 hr
24 mi 1609 m
c
mc 60 min m = 643.6 m/min
1 hr
1 mi
7.
b. $5.75
4
10
=
3.5
x
4x = 35
x = $8.75
iii