Download Learning Objective: Solve Problems by Writing Proportions Example 1

Professor Busken – 6.3 Problem Solving Using Proportions
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Learning Objective: Solve Problems by Writing Proportions
Example 1: A Google map indicates that 1 inch corresponds to 15 miles. How many
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miles corresponds to 4 inches?
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Solution:
1. Understand. You may want to reread the problem a couple of times. We need
to know how many miles is represented by a length of four and one-quarter inches.
1 inch corresponds to 15 miles
2 inches corresponds to 30 miles
3 inches corresponds to 45 miles
4 inches corresponds to 60 miles
5 inches corresponds to 75 miles
We expect the answer to be between 60 and 75 miles.
2. Translate. Let the variable x represent the unknown quantity. In this case,
x = the number of miles represented on the map by 4
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or 4.25 inches.
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3. Solve. Since 1 inch is to 15 miles as 4.25 inches is to x miles, we can write the
proportion
1 inch
4.25 inches
=
15 miles
x miles
Now solve this equation by setting the cross products of the proportion equal to
each other. We neglect the units while we solve the equation, but we will remember
to report our units for x as miles when we are finished.
4.25
1
=
15
x
x · 1 = 15 · 4.25
x = 63.75
4. Interpret. This answer seems appropriate since its a value between 60 and 75. The
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conclusion is that 4 inches corresponds to 63.75 miles.
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1. Try this one on your own. A Google map indicates that 1 inch corresponds to
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18 miles. How many miles corresponds to 2 inches?
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Professor Busken – 6.3 Problem Solving Using Proportions
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Example 2: Kathy’s Humvee Beach Cruiser averages 588 miles on a 21 gallon tank
of gasoline. After spring break, Kathy is leaving San Felipe in Baja, Mexico to drive
back to San Diego, California in order to make it on time for Mr. Busken’s class. Will
Kathy’s Humvee be capable of traveling the driving distance between San Felipe and
San Diego (236 miles) if she has exactly 6 gallons of gas when she leaves San Felipe?
Solution:
1. Understand. You may want to reread the problem a couple of times. We need to
know the number of miles Kathy can expect to travel on 6 gallons of gas, assuming
she gets around 588 miles on a full tank (21 gallons).
•
•
•
We
21 gallons gets 588 miles
One-half of 21 gallons, or 10.5 gallons gets 12 · 588 = 296 miles
One-half of 10.50 gallons, or 5.25 gallons gets 21 · 296 = 147 miles
expect that 6 gallons of gas will get Kathy between 296 and 147 miles distance.
2. Translate. Let the variable x represent the unknown quantity. In this case,
x = the number of miles Kathy’s Humvee can travel on 6 gallons of gasoline.
3. Solve. Since the ratio of 21 gallons to 588 miles is the same as the ratio of 6 gallons
to x miles, we can write the proportion
21 gallons
6 gallons
=
588 miles
x miles
We then solve this equation by setting the cross products of the proportion equal to
each other. We neglect the units while we solve the equation, but we will remember
to report our units for x as miles when we are finished.
21
6
=
588
x
x · 21 = 588 · 6
21x = 3528
Now multiply.
Now divide both sides 21.
21x
3528
=
21
21
x = 168
4. Interpret. This answer seems appropriate since its a value between 296 and 147
miles. The conclusion is that 6 gallons gets 168 miles. However, the driving distance
between San Felipe and San Diego (236 miles). Therefore, 6 gallons will not be
enough.
Professor Busken – 6.3 Problem Solving Using Proportions
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2. An 80-pound bag of ready to use concrete mix fills 8 cubic feet. Joe needs 280 cubic
feet of concrete, so how many bags does he need?
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3. The daily supply of oxygen for one person is provided by 625 square feet of lawn. A
total of 3750 square feet of lawn would provide the daily supply of oxygen for how
many people?
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4. Yosemite National Park is 420 miles from Mr. Busken’s house. If his car averages
203 miles on 7 gallons, how much gas money will he need to budget for a round trip?
Assume gas is $4.60 per gallon and add 8 gallons to your answer to account for the
driving Mr. Busken does around the 750,000 acre park during his vacation.
4.
Professor Busken – 6.3 Problem Solving Using Proportions
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For each proportion, determine the value of the unknown number, x.
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6.
7.
18
x
=
30
5
5
1
6
=
5.
6
8
x
1.2
x
=
9
1.8
6.
7.