Section 4.3
PRE-ACTIVITY
PREPARATION
Applications of Proportions
The previous two sections introduced you to the fact that ratios are used to represent the information in many
different circumstances. Interest and dividends, taxes, wages and salaries, currency conversion, commissions,
fuel efficiency, mixtures and solutions, unit prices for groceries, and standard conversion within and between
the US and metric systems of measurement—all derive from an understanding of rates and unit rates.
Furthermore, proportions can be used to solve application problems involving such ratios. Setting up and
solving proportions can help you answer the mathematical questions that arise in everyday tasks such as:
•
scaling a recipe up or down
•
predicting the distance your car will travel on a full tank of
gas
•
determining the distance between two cities on a map
•
comparison shopping for the most economical purchase
•
mixing a batch of cleaning solution from concentrate
•
estimating the height of the old oak tree in your back yard
In this section, you will discover the versatility of proportions and learn to recognize the sorts of applications
for which you can appropriately use this important mathematical tool. In the process, you will develop the
essential skill of how to read a problem carefully to pull out the information that is critical for answering its
question.
LEARNING OBJECTIVES
• Explore a range of applications for proportions.
• Recognize when a proportion can be used to solve an application problem.
• Know how to set up proportions for various types of application problems.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
proportion
conversion
rate
currency conversion rate
ratio
round to the nearest cent
solve an equation
standard conversion rate
unit rate
unit price
unknown
variable
407
Chapter 4 — Ratios and Proportions
408
Using a Proportion to Solve an Application Problem
When presented with a practical application problem, it is necessary to first recognize whether or not you can
identify a ratio from the information given in the problem. That ratio will be your basis for comparison in
the proportion you write.
Some ratios are set, serving as essential components in art, cooking, business, design, medicine, music,
science, and technology.
For example,
• Two and one-half cups of flour for two pie crusts is set in the recipe. Any other ratio of flour to crusts will
not yield the same flaky crusts.
• The scale on an architect’s drawing might be set as ¼ inch to one foot.
• All standard measurement conversion rates (for example, 1 lb./16 oz.) are set rates.
• On any given day, the international currency exchange rate for Euros to US dollars is set.
At other times, the appropriate ratio for an application might be a reflection of data collected over time, and
predictions can be made based upon that information.
For example,
• To predict gasoline consumption (which varies with driving conditions) for your vacation, you might rely
on the rate of miles driven to gasoline consumed the last time you made the drive to Florida.
• To predict how many light switches will be defective out of a week’s run of 30,000 light switches, the
manufacturer will rely on its past statistics of defective switches per switches produced.
METHODOLOGY
For the contexts in which a proportion can be used to solve for an unknown quantity, use the following
methodology. Be sure to note the special cases of unit rates and standard conversions.
Solving an Application Problem with a Proportion
►
Example 1 Todd drives a van with a 30-gallon gas tank. He determined that the
van used 9 gallons of gas for a recent trip of 160 miles. At that rate of
gas consumption, how far can Todd’s van travel on a full tank of gas?
Round the answer to the nearest tenth of a mile.
►
Example 2 It generally takes Glenn 21 minutes to walk his dog 1.2 miles to his
friend’s house. At that same rate, how long will it take him to walk his
dog to his sister’s house 2.3 miles away?
Try It!
Section 4.3 — Applications of Proportions
409
Steps in the Methodology
Step 1
Identify the
unknown.
Example 1
Identify the unknown quantity in the
word problem.
Special Case:
(and its
shortcut)
Calculating a unit rate
(see page 417, Model 8)
total miles traveled
Establish the left side of the proportion with Step 2.
Step 2
Write the
known ratio.
Locate, identify, and write the ratio that
is given in the problem, labeling the two
quantities to distinguish them from each
other.
Alternate interpretation
Option: and presentation of the
given ratio
(see page 410)
*
9 gallons
160 miles
Establish the right side of the proportion with Steps 3, 4, and 5.
Step 3
Use a variable
for the
unknown.
Step 4
Identify the
third known
quantity.
Represent the unknown with a
variable. Match it to the numerator or
denominator of the given ratio.
From the information given in the
problem, identify the third known
component for the proportion.
Step 5
Complete the proportion.
Write the
proportion.
Verify that the quantities are positioned
consistently.
Once verified, you can compute in the
next step without the words.
n= total miles
traveled
9 gallons
=
160 miles
n (miles)
30 gallons
(full tank)
9 gallons
30 gallons
=
160 miles
n (miles)
???
Why do you do this?
Step 6
Solve the
proportion.
Solve the
equation for
the unknown.
Round the
answer, if
requested.
9 × n = 30 × 160
1
9 ×n
1
9
10
=
30 × 160
3
9
10 × 60 1600
=
3
3
n ≈ 533.3
n=
533.33
3 1600.00
−15
)
10
−9
10
−9
10
−9
10
−9
1
Example 2
Chapter 4 — Ratios and Proportions
410
Steps in the Methodology
Step 7
Example 2
Present your answer, with units.
533.3 miles
Present the
answer.
Step 8
Validate your
answer.
Validate your answer. Substitute
the answer in place of the variable
in the original proportion and crossmultiply to test for equality of the
two ratios.
Reminder: When the answer is
rounded, the cross-products will be
close but not exactly the same.
*
Example 1
9 gallons
30 gallons
=
160 miles
533.3 miles
?
9 × 533.3 = 30 × 160
4799.7 ≈ 4800 9
Option: Alternate Interpretation and Presentation of the Given Ratio
In most application problems, when you locate the two components of the given ratio, you have the option of
identifying which should be the numerator of the ratio and which should be its denominator.
In Example 1 of the Methodology, as an alternate presentation, you might write
160 miles
as the given ratio.
9 gallons
The critical step will be Step 3, when you match the unknown with the ratio you have written in Step 2.
In Example 1, 160 miles = n (miles)
9 gallons
30 gallons
When determining a unit rate, however, the numerator and denominator of the given ratio are set by the
specified unit rate labels (see Special Case Model 8). There are no alternate presentations for the unit labels.
???
Why do you do Step 5?
In the next step (Step 6), you will solve a presumably true proportion, and the proportion can be true only if
the quantities in the two ratios are positioned consistently.
As stated in the previous section, in order for two ratios to be equal, they must both present the same order of
comparison.
More specifically, for rates, the contrasting units must be stated in the same order. This process of verifying
the unit comparisons is often referred to as dimensional analysis, especially in the technical fields of science
and engineering, where it is used extensively.
For example,
gallons
gallons
gallons
miles
ns
m
must always match
.
andd not
=
miles
miles
miles
es
ggallons
a
Section 4.3 — Applications of Proportions
411
MODELS
Model 1
Tawana orders floral bouquets for a market. She noticed that in the
previous month, the market sold 220 bouquets of carnations out of
the 400 various bouquets that were sold. She projects that in August
the market will sell 540 bouquets of flowers. How many bouquets of
carnations should she order for August?
Step 1
The unknown is the number of carnation bouquets.
Step 2
Last month,
Step 3
Let n = number of carnation bouquets
Step 4
Third piece of information:
Step 5
Write the proportion.
Step 6
Solve the proportion.
220 carnation bouquets
400 total bouquets
220 carnation
n (carnation)
=
400 total
540 total bouquets
220 carnation
n (carnation)
=
The comparisons match.
400 total
540 total
In this case, you can use
the shortcut for solving the
proportion and reduce the
known ratio.
220 ÷ 20
11
n
⇒
=
400 ÷ 20
20
540
Cross multiply: 11 × 540 = n × 20
Cancel common factors:
11 ×
1
27
540
1
=
n × 20
1
20
20
11 × 27 = n
297 = n
Step 7
Answer = 297 carnation bouquets
Step 8
Validate:
220 carnation ? 297 carnation
=
400 total
540 total
? 297 × 400
220 × 540 =
118, 800 = 118, 800 9
27
× 11
27
+270
297
Chapter 4 — Ratios and Proportions
412
Model 2
On a map of Michigan, ½ inch represents 15 miles. If the distance from Jackson to Mackinaw City
measures 7 ¾ inches on the map, what is the actual distance between the two cities?
Step 1
Step 2
Step 3
Step 4
Step 5
distance (miles) between cities
1
inch
0.5 inch
2
OR use the decimal equivalent of the fraction:
15
miles
15 miles
1
inch
0.5 inch
2
=
OR
=
Let n = miles
15 miles
n (miles)
15 miles
n (miles)
The third given component is 7
1
3
inch
7 inches
2
= 4
15 miles
n (miles)
OR
3
inches.
4
7.75 inches
0.5 inch
=
Labels are consistent.
n (miles)
15 miles
1
Step 6
Using the decimal form:
0.5 × n
1
0.5
=
7.75 × 15
0.5
116.25
.05
n = 232.5
n=
Step 7
Answer: 232.5 miles or 232
Step 8
Validate:
1
miles
2
0.5 inch ? 7.75 inches
=
15 miles
232.5 miles
?
0.5 × 232.5 = 7.75 × 15
116.25 = 116.25 9
7.75
× 15
3875
+7750
116.25
232.5
0.5 116.25
−10
)
16
−15
12
−10
25
−25
0
Model 3
The directions for the marinated pork loin say, “Roast in a 325º oven for 45 minutes per pound.” The
roast Shawn is going to cook weighs 26 ounces. For how many minutes should she cook the roast?
Round to the nearest minute.
Step 1
minutes to cook the roast
Step 2
45 minutes
1 pound
Steps 3-5
26 ounces is the third known quantity. Since it is in ounces, in order to match
the units, write 1 pound as its equivalent, 16 ounces.
Section 4.3 — Applications of Proportions
413
Let n = minutes to cook the roast
45 minutes
n (minutes)
=
16 ounces
26 ounces
45 ×
Step 6
8
13
26
The units match.
45
×13
1
n × 16
= 1
16
16
585
=n
8
73 ≈ n
135
+450
73.1 ≈ 73
8 585.0
−56
)
585
Step 7
Answer: 73 minutes
Step 8
Validate:
25
−24
45 minutes ? 73 minutes
=
16 ounces
26 ounces
?
45 × 26 = 73 × 16
1170 ≈ 1168 9
10
−8
2
Model 4
Robin and Jason agree to share the profits from the sale of their camper in the ratio of 2 to 3.
How much does Robin receive if Jason’s share of the profits is $1470?
Step 1
n = Robin’s share (in dollars)
Steps 3-5
Step 6
The order of names and the given ratio of 2 to 3
imply $2 for Robin for every $3 for Jason.
THINK
Step 2
$2 Robin
$n (Robin)
=
The comparison order is the same.
$3 Jason
$1470 Jason
2×
490
1470
1
n× 3
= 1
3
3
2 × 490 = n
980 = n
1
Step 7
Answer: $980 for Robin
Step 8
Validate:
2 ? $980
=
3 $1470
?
2 × 1470 = 980 × 3
2940 = 2940 9
490
3 1470
−12
)
27
−27
0
$2 Robin
$3 Jason
Chapter 4 — Ratios and Proportions
414
Model 5
In the late afternoon, a six feet tall man casts a shadow 7 feet long. A tree he is standing next to casts
a shadow 16 feet long. To the nearest tenth of a foot, how tall is the tree?
Note: This model is based upon the fact that the ratio of an object’s height to the length of its shadow
is in the same ratio as any other object’s height is to its shadow length at the same time of day.
Step 1
n = height of the tree
Drawing a diagram may help you visualize the ratios
n
6 ft.
Step 2
6 ft. man
7 ft. shadow
7 ft. shadow
16 ft. shadow
Steps 3-5
6 ft. man
n ( ft. tree)
=
The comparison order is the same.
7 ft. shadow 16 ft. shadow
Step 6
6 × 16
n× 7
= 1
7
7
96
=n
7
13.7 ≈ n
13.71
7 96.00
−7
1
)
26
−21
50
−49
10
Step 7
Answer: 13.7 feet tall tree
Step 8
Validate:
−7
3
6 ft. man ? 13.7 ft. tree
=
7 ft. shadow 16 ft. shadow
?
6 × 16 = 13.7 × 7
96 ≈ 95.9 9
Conversions
Conversion is the process of changing from one unit of measurement (in weight, volume, temperature,
time, length, currency) to another.
You can change from one unit of measurement to another within the same system using standard (that is,
constant or unchanging) conversion rates in the process. For example,
⎛12 inches ⎟⎞
, 3 feet convert to 36 inches in the US system of length measurement.
• Using ⎜⎜
⎜⎝ 1 foot ⎟⎟⎠
⎛100 cm ⎞⎟
• Using ⎜⎜
, 2 meters (m) convert to 200 centimeters (cm) in the metric system.
⎜⎝ 1 m ⎟⎟⎠
Section 4.3 — Applications of Proportions
415
You can also change from a unit of measurement in one system to its equivalent unit of measurement in a
different system. For example,
⎛ 2.54 cm ⎞⎟
• Using ⎜⎜
as the standard conversion rate, 4 inches (in) in the US system convert to
⎜⎝ 1 in ⎟⎟⎠
approximately 10.16 centimeters (cm) in the metric system.
•
⎛ 454 g ⎟⎞
Using ⎜⎜
as the standard conversion rate, 908 grams (g) in the metric system convert to
⎜⎝ 1 lb ⎟⎟⎠
approximately 2 pounds (lbs) in the US system.
In all the disciplines that require conversions, you can use proportions to make the conversions. Many, such as
the four examples just mentioned, use a standard conversion rate as the given rate in the process.
Monetary transactions use currency conversion rates that are re-established on a daily basis (for example,
1 US dollar = 1.1727 Canadian dollars on January 19, 2007)
The following two models present applications with conversions.
Model 6
A
►
Standard Conversion
You are driving through Canada and the distance sign reads, “MONTREAL 560 kilometers.”
To the nearest mile, what is the distance in miles, if 1 kilometer (km) equals approximately 0.62 miles?
THINK
In other words, convert 560 kilometers to miles.
Step 1 Solve for miles.
Step 2 The given ratio is a known standard conversion rate.
Steps 3, 4 & 5
The third known quantity is 560 kilometers.
Carefully match to kilometers in the standard conversion rate.
1 km
560 km
=
The units match.
0.62 miles
n miles
Step 6
1 km
0.62 miles
1 × n = 560 × 0.62
n = 560 × 0.62
n ≈ 347 miles
Step 7
Answer: 347 miles
Step 8
Validate:
560
×0.62
1120
+33600
347.20
1 km
? 560 km
=
0.62 miles
347 miles
?
1 × 347 = 0.62 × 560
347 ≈ 347.2 9
416
Chapter 4 — Ratios and Proportions
B
►
Ben weighs 216 pounds (lbs). To determine his body mass index number, his weight must be converted
to kilograms. What is Ben’s weight in kilograms (kg)? Round to the nearest tenth.
Use the standard conversion rate of 1 kg approximately equals 2.2 lbs.
Step 1 weight in kilograms
Step 2 Standard conversion rate is
Steps 3, 4 & 5
Step 6
1 kg
n kg
=
2.2 lbs
216 lbs
1 kg
2.2 lbs
The units match.
98.18 ≈ 98.2
2.2 216.000
−198
)
1 × 216
n × 2.2
=
2.2
2.2
216
=n
2.2
98.2 ≈ n
180
−176
40
−22
Step 7
Answer: 98.2 kilograms
Step 8
Validate:
180
−176
4
1 kg ? 98.2 kg
=
2.2 lbs
216 lbs
?
1 × 216 = 98.2 × 2.2
216 ≈ 216.04 9
Model 7
Currency Conversion
The currency conversion rate on May 21, 2007 for US dollars
(USD) to the Euro (EUR) was 1 USD to 0.741 EUR. How many
Euros (to the nearest whole EUR) would you receive in exchange
for $510 USD?
Step 1 Solve for Euros (EUR).
Step 2 given currency conversion rate:
Steps 3, 4 & 5
Step 6
Step 7
Step 8
1 USD
0.741 EUR
1 USD
510 USD
=
The units match.
0.741 EUR
n EUR
1 × n = 510 × 0.741
n ≈ 378
Answer: 378 Euros
1 USD
? 510 USD
Validate:
=
0.741 EUR
378 EUR
?
1 × 378 = 510 × 0.741
378 ≈ 377.91 9
0.741
×510
000
7410
370500
377.910 ≈ 378
Section 4.3 — Applications of Proportions
417
Determining Unit Rates
Section 4.1 introduced you to the term unit rate, a rate whose denominator is one single unit. Any rate can
be converted to its unit rate form and, in many situations, the unit rate is the preferred way to communicate
information. For example, “130 miles per 10 gallons” can be simply stated as “13 miles per gallon.” Calculating
a unit rate is a special case application of the Methodology, demonstrated in the following models.
Model 8
A
►
Special Case: Calculating a Unit Rate (and its Shortcut)
Write as a unit rate: 90 ounces of cheese dip for 43 guests. Round to the nearest tenth.
Step 1
When calculating a unit rate, the unknown is the unit rate itself.
Solve for ounces per guest:
Step 2
The unit rate requested determines the order
of the units for the given ratio
Steps 3, 4 & 5
n ounces
1 guest
90 ounces
43 guests
90 ounces
n ounces
=
The units match.
43 guests
1 guest
Step 6
Solve:
2.09 ≈ 2.1
43 90.00
−86
)
40
−0
400
−387
90 × 1 = n × 43
90 × 1 n × 43
=
43
43
90
=n
43
2.1 ≈ n
Shortcut (optional)
The third known component, 1 guest,
has been already set in Step 1.
13
Step 7
For a unit rate, both numerator and denominator must be stated in the answer.
Answer : 2.1 ounces per guest
Step 8
or
2.1 ounces
1 guest
? 2.1 ounces
Validate: 90 ounces =
43 guests
1 guest
Shortcut (optional):
Calculating a Unit Rate
or
2.1 ounces
guest
?
90 × 1 = 2.1 × 43
90 ≈ 90.3 9
To calculate a unit rate, simply divide the numerator
of the given rate by its denominator. Skip Steps 3-6.
Chapter 4 — Ratios and Proportions
418
Why can you take this shortcut?
Think about Step 6 in Model 8A. When you cross-multiply the numerator of the given rate by 1 in the
denominator of the unit rate, you get the same numerator number. (Identity Property of Multiplication)
When you divide both sides of the equation by the denominator of the given rate, the calculation to perform at
this point is the same as if you were simply to divide the numerator of the given rate by its denominator.
(For Model 6A: 90 ounces ÷ 43 guests)
B
►
Write 219 miles on 14 gallons of gas as a unit rate. Round to the nearest tenth.
Step 1 Solve for
n miles
1 gallon
Step 2
219 miles
14 gallons
Use the shortcut:
15.64 ≈ 15.6
14 219.00
−14
)
79
−70
90
−84
60
−56
4
Step 7
Answer :
Step 8 Validate:
15.6 miles
15.6 miles
or
or 15.6 miles per gallon or 15.6 mpg
1 gallon
gallon
219 miles ? 15.6 miles
=
14 gallons
1 gallon
?
219 × 1 = 15.6 × 14
219 ≈ 218.4 9
Section 4.3 — Applications of Proportions
419
For practical applications, you can compare rates by determining their unit rates, as in the following Model.
Model 9
Jo types 355 words in 5 minutes, while Frank types 240 words in 4
minutes. Who types faster?
Step 1 Solve for
n words
for each
1 minute
Step 2
Jo
Frank
355 words
5 minutes
240 words
4 minutes
71
5 355
−35
5
−5
60
4 240
−24
0
−0
0
0
)
Use the shortcut for unit rate:
Step 7
Compare:
)
71 words
60 words
(71 wpm) versus
(60 wpm)
minute
minute
Answer: Jo types faster.
Step 8 Validate:
355 words ? 71 words
=
5 minutes
1 minute
?
355 × 1 = 71 × 5
355 = 355 9
240 words ? 60 words
=
4 minutes
1 minute
?
240 × 1 = 60 × 4
240 = 240 9
Many retailers have adopted the practice of unit pricing for the benefit of their customers, allowing them to
comparison shop among different sized packages of the same product and among various brands of the product.
The term unit price refers to the price of one item ($ .55 per candy bar) or of one typically measurable unit
of a product (grapes at $ 1.19 per pound).
To calculate a unit price, you will always use the price of the product as the numerator and the weight,
volume, capacity, length, area, or number of units in a single purchase as the denominator.
A reminder about notation:
•
When you calculate with dollars and cents, the instruction to round to the nearest cent means
to round your answer to its nearest hundredth place.
•
An answer in cents alone, for example 45 cents, may be written in its decimal form as $0.45 (“45
hundredths of a dollar”) or with the ¢ (1/100) sign as 45¢ (“45/100 of a dollar”), but not with both.
notations, as $0.45¢ implies 45/100 of one penny.
Chapter 4 — Ratios and Proportions
420
Model 10
Ready-to-serve pasta salad is on sale for $5.79 for an 8-pound
container. To the nearest cent, what is the price per pound?
Step 1 Solve for dollars (price) per pound—
a unit price:
$n
1 pound
Step 2 Given rate: price (dollars) per pound
$5.79
8 pounds
Use the shortcut for unit rate:
0.723 ≈ 0.72
8 5.790
−56
)
THINK
Round to the hundredth place.
19
−16
30
−24
6
Step 7
Answer :
Step 8 Validate:
$0.72
or
1 pound
$0.72
or $0.72 per pou
und
pound
$5.79 ? $0.72
=
8 pounds 1 pound
?
5.79 × 1 = 0.72 × 8
5.79 ≈ 5.76 9
Section 4.3 — Applications of Proportions
421
Model 11
Which is the best buy—a package of 8 pencils for $1.19 or a package of
13 of the same brand of pencils for $1.85?
Step 1 Solve for
$n
for each package and compare.
pencil
$1.19
8 pencils
Step 2
Use the Shortcut for the unit rate.
THINK
Since the first
two digits of
the unit rates
are the same,
carry out the
divisions one
more place
to make a
comparison.
0.1487 ≈ 0.149
8 1.1900
−8
Answer:
Step 8 Validate:
$1.85
13 pencils
0.1423 ≈ 0.14 2
13 1.8500
−13
)
)
39
−32
55
−52
70
−64
30
−26
60
40
−56
4
−39
1
$0.149
pencil
Step 7
versus
versus
$0.142
pencil
* best buy
The best buy is the package with the lesser cost per pencil,
$1.85 for 13 pencils.
$1.19 ? $0.149
=
8 pencils 1 pencil
?
$1.19 × 1 = $0.149 × 8
1.19 ≈ 1.192 9
$1.85 ? $0.142
=
13 pencils 1 pencil
?
$1.85 × 1 = $0.142 × 13
1.85 ≈ 1.846 9
Chapter 4 — Ratios and Proportions
422
ADDRESSING COMMON ERRORS
Issue
Mismatching
units when
setting up a
proportion for
an application
problem
Incorrect
Process
Joan, who is 5 feet
tall casts a shadow
4 feet long. The
flagpole she is
standing next to
casts a shadow 12
feet long. How tall
is the flagpole?
5 12
=
4
n
5 × n = 12 × 4
48
n=
5
n = 8.6 ft. tall
Resolution
When setting up a
proportion, identify
the ratio given in
the problem and
use word labels to
convey the meaning
of the comparison.
Match the second
ratio in the same
comparison order.
Always match the
unit labels for rates.
Correct Process and Validation
n = height of flagpole
5 ft. height
Joan’s ratio is given:
4 ft. shadow
The shadow length of the flagpole
is given: 12 feet.
Match to denominator of the given
ratio.
Joan
flagpole
5 ft. height
n (height)
=
4 ft. shadow 12 ft. shadow
3
1
5 × 12
=
1
This verification
will assure a true
proportion.
n× 4
1
4
4
5×3 = n
15 = n
15 feet tall
Validation:
?
5 ft.
15 ft.
=
4 ft. shadow
12 ft. shadow
?
5 × 12 = 15 × 4
60 = 60 9
Reversing the
units when
determining a
specified unit
rate
If you can get 6
servings from a
pre-packaged salad
that sells for $8.40,
what is the cost per
serving?
6
8.40
0.714
8.4 60.000
−5 8 8
)
1 20
0
−84
36
360
0
−336
24
$0.71 per serving
First identify the unit
rate specified (or
implied, as for rate
of speed) and write it
with unit labels.
Unknown unit rate is
Match the given rate
to the unit labels of
the unit rate.
Use shortcut
for unit rate:
Unit prices always
use the price as the
numerator and the
number of units as
the denominator.
cost ($)
1 serving
$ 8.40
6 servings
1.40
6 8.40
−6
)
24
−2 4
0
$1.40 per serving
Validation:
? $1.40
$8.40
=
6 servings
1 serving
?
8.40 × 1 = 1.40 × 6
8.40 = 8.40 9
Section 4.3 — Applications of Proportions
423
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
how to solve a proportion
how to identify the unknown in an application problem
how to pull the given information out of an application problem in order to solve it with a proportion
how to verify that the given quantities and the unknown are positioned consistently in the set-up of a
proportion problem
how to solve for a unit rate
how to solve a conversion problem
Section 4.3
ACTIVITY
Applications of Proportions
PERFORMANCE CRITERIA
• Setting up a proportion correctly to solve an application problem
– correct identification of the unknown
– correct identification of given information
– consistent comparison of quantities for the two ratios
• Solving the proportion correctly to solve the application problem
– correct presentation of the answer
– answer rounded to the specified place value
– validation of the answer
CRITICAL THINKING QUESTIONS
1. What are the types of applications that can be solved by using proportions?
2. How do you know that the quantities are positioned correctly when setting up a proportion?
3. How do you determine if you have a unit rate proportion problem?
4. How do you determine if you have a standard conversion problem?
424
Section 4.3 — Applications of Proportions
425
5. What is the shortcut to solving a unit rate problem, and why can you use the shortcut?
6. Why might your answer be the correct solution to your proportion, yet still be an incorrect answer for the
application problem?
7. What everyday situation (different from those in the Models) can you create that can be solved by using a
proportion?
Chapter 4 — Ratios and Proportions
426
TIPS
FOR
SUCCESS
• Always use word labels to set up a proportion problem involving ratios and, more specifically, always use
unit labels to set up a proportion problem involving rates.
• Always verify the order of comparison (the units for rates) before solving the proportion.
• Use a diagram to help visualize the comparisons. (This can be especially useful in shadow problems.)
• Identify a unit rate problem by looking for the key words per…, or for every…, or for each… (single unit)
• To assure a correct answer when solving the proportion, do not skip steps. Always write the equation for
the cross-multiplication and show the division by the multiplier of the variable. In this way, the numbers to
be multiplied and divided will be clearly presented.
• Reduce, when possible, to simplify the computation when solving your proportion.
DEMONSTRATE YOUR UNDERSTANDING
Write the unit rate for each of the following. Round to the nearest hundredth.
Worked Solution
1) Earnings of $825 for 36.5 hours of work.
Validation
Section 4.3 — Applications of Proportions
Worked Solution
2) A landscape nursery bill of $1022 for 28 maple trees.
3) $2.59 for a 32-ounce container of organic chicken broth.
427
Validation
Chapter 4 — Ratios and Proportions
428
Use a proportion to solve each of the following application problems.
Worked Solution
4) In a manufacturing process, it has been found that for every
192 items assembled, 3 are defective. At this rate, if 6400
items are assembled, how many will be defective?
5) If 2½ inches on a map represent 48 miles, what distance
does 6 inches represent?
Validation
Section 4.3 — Applications of Proportions
Worked Solution
6) Lori has volunteered to make the potato salad for a family
reunion picnic. If her recipe that serves 8 requires 3 pounds
of potatoes, how many pounds of potatoes must she prepare
to yield 45 servings? Round to the nearest pound.
7) Chris traveled to Windsor, Ontario on a day that the currency
conversion rate was $0.853 US dollars (USD) for every one
Canadian dollar (CAD). She spent $142.50 Canadian. How
much was that in U.S. dollars? (to nearest cent)
429
Validation
Chapter 4 — Ratios and Proportions
430
Worked Solution
8) If David’s pickup truck consumed 110 gallons of gas to drive
1365 miles to Florida, how many miles per gallon did the
truck average on this trip? Round to the nearest tenth.
9) Martha and Tom shared the commission on the sale of a house
in the ratio of 3 to 7. What was Tom’s commission if Martha
received $3150?
Validation
Section 4.3 — Applications of Proportions
Worked Solution
10) During a sunset, a flagpole casts a shadow 7 ½ feet long
while the 3-foot tall evergreen tree growing next to it casts
a shadow 2 feet long. To the nearest foot, how tall is the
flagpole?
11) The recommended dosage for a particular medication is 0.05
mg per 50 pounds of body weight. What amount should be
administered to a 186-pound patient?
431
Validation
Chapter 4 — Ratios and Proportions
432
Worked Solution
12) Do a unit price comparison. Which is the more economical
purchase—the 25.5-ounce box of toasted oat cereal on sale
this week for $3.00 or the 20-ounce box that regularly sells
for $2.19?
13) The label on the large jug of green tea sweetened with
honey says that it contains 4.75 liters. If 1 gallon equals
approximately 3.8 liters, approximately how many gallons
does the jug contain?
Validation
Section 4.3 — Applications of Proportions
Worked Solution
14) Fifty-nine shares of stock were sold for $1185. To the nearest
cent, what was the cost per share?
433
Validation
Chapter 4 — Ratios and Proportions
434
IDENTIFY
AND
CORRECT
THE
ERRORS
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
1) The 8 pound bag of navel oranges was priced at $5.80. What was the price per pound (to the nearest cent)?
Worked Solution
What is Wrong Here?
Identify Errors
or Validate
The original ratio is
set up incorrectly for
a unit price (price
per pound).
Should be
price
units
Correct Process
$5.80
8 pounds
.725
8 5.800
−5 6
)
20
−16
40
−40
0
Answer:
$.73 per pound
or 73¢ per pound
Validation
Unit price is
price
units
9
$5.80 ? $.73
=
8 lbs.
1 lb.
?
5.80 × 1 = .73 × 8
5.80 ≈ 5.84 9
OR
Validate the
division.
.725
×
8
5.800 9
Section 4.3 — Applications of Proportions
435
2) During a sunset, a 6-foot gentleman casts a shadow 16 feet long. At the same time, a building he’s
standing next to casts a shadow 46.5 feet long. To the nearest foot, how tall is the building?
Worked Solution
What is Wrong Here?
Identify Errors
or Validate
Correct Process
Validation
3) In a citywide survey, prior to the hotly contested primary election, 8 out of every 12 registered voters
surveyed said they would vote on election day. If the city has 16,800 registered voters, how many will
probably vote on election day?
Worked Solution
What is Wrong Here?
Identify Errors
or Validate
Correct Process
Validation
436
Chapter 4 — Ratios and Proportions
ADDITIONAL EXERCISES
Solve each problem by using a proportion. Validate your answers.
1. Write as a unit rate: 1608 miles traveled in 4 days
2. Write as a unit rate: $2.99 per 18-ounce jar of grape jam (rounded to the nearest cent)
3. The ratio of children to adults at the first matinee showing in town of a new animated movie was 5 to 2.
If there were 48 adults in attendance, how many children attended?
4. A train has traveled 240 miles in 3.5 hours. At that rate, how long will it take to travel 625 more miles?
(Round your answer to the nearest tenth.)
5. An 8-pound container of macaroni salad is on sale for $5.79. If you can get 42 servings from the container,
what is the cost per serving? (Round your answer to the nearest cent.)
6. On a map of Florida, ½ inch equals 15 miles. What is the distance from Tampa to Naples on the Gulf Coast
if it measures 4 ¾ inches on the map?
7. Which is the better buy? Fresh green beans pre-packaged in 14-ounce bags for $1.09 each, or bulk green
beans you can bag yourself at $1.29 per pound?
8. Jim’s utility bill stated that he used 846 kilowatt-hours (kwh) of electricity for the 31 days in December.
What was his electric use per day (rounded to the nearest tenth of a kwh)?
9. Betty’s original recipe used 15 ounces of chocolate chips to make twenty large cookies. She still uses the
original proportions of ingredients in that recipe for the cookies she sells in her bakery on Main Street. To
bake today’s batch of twelve dozen cookies, how many ounces of chocolate chips will she need?