5.5 Applications of the Percent Equation
This section will introduce a number of general applications of percents. The next chapter will
study specific consumer applications involving interest, loans, markup, and discount. Percents
are used in a variety of applications including finance, sports, even your grade in this course!
Example 1
Your instructor informs you on the first day syllabus that the course will have
600 points total, and that you need to earn 80% of those points to earn a B grade.
a.
b.
Solution
How many points do you need in the course to earn a B grade?
If you earn 510 points, what percent have you earned?
a.
You need to earn 80% of the 600 total points possible to earn a B grade. We
use the percent equation with percent = 80% = 0.80, base = 600, and
percentage unknown. Using the percent equation:
percentage = percent • base
x = 0.80 • 600
x = 480
You will need to earn 480 points to earn a “B” grade. Note that we could
also use the proportion method to solve this problem, using the proportion:
80
x
=
100 600
80
x
600 •
= 600 •
100
600
x = 480
We obtain the same answer. In general, we will choose to use the percent
equation approach to solve problems in this section (as well as next
chapter), as it is usually the easiest approach and provides good algebra
practice.
b.
We need to find the percent = p, given the percentage = 510 and the
base = 600. Using the percent equation:
percent • base = percentage
p • 600 = 510
510
p=
600
p = 0.85 = 85%
You have earned 85% of the total points available (a “B” grade).
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Example 2
In California the sales tax rate for purchases is 7.25%.
a.
b.
Solution
If you buy a $1,860 computer, how much will the California state sales
tax be?
If you pay $1,624 sales tax for the purchase of a car, what was the
price of the car?
a.
We are given the percent = 7.25% = 0.0725. Since this percent uses the
price of the computer as the base, the base = 1860 and the percentage is
unknown. Using the percent equation:
percentage = percent • base
x = 0.0725 • 1860
x = 134.85
The state sales tax will be $134.85.
b.
We are given the percent = 7.25% = 0.0725. Since this percent uses the
price of the car as the base, the base is unknown and the percentage = 1624.
Using the percent equation:
percent • base = percentage
0.0725x = 1624
1624
x=
0.0725
x = 22, 400
The price of the car was $22,400.
Example 3
In the year 2000 NBA playoffs, Shaquille O’Neal played center for the
Los Angeles Lakers.
a.
b.
During one game he made 15 field goals out of 27 attempts. What was
his field goal percent? Round your answer to the nearest percent.
During one series he made 52% of his free throw attempts. If he made
31 free throws, how many attempts did he have? Round your answer to
the nearest whole number.
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Solution
a.
The base = 27, the percentage = 15, and the percent is unknown. Using the
percent equation:
percent • base = percentage
p • 27 = 15
15
p=
27
p ! 0.56 = 56%
Shaquille’s field goal percent was 56%.
b.
The percent = 52% = 0.52. Since this percent is based on the attempts, the
percentage = 31 and the base is unknown. Using the percent equation:
percent • base = percentage
0.52x = 31
31
x=
0.52
x ! 60
Shaquille attempted 60 free throws.
Example 4
1
When applying for the loan on a house, you are required to pay 1 % of the loan
2
amount as up-front interest (these are called points).
a.
b.
Solution
a.
If the loan is for $142,000, how much must you pay in points?
If you pay $1,350 in points on a loan, how much is the loan amount?
1
The percent = 1 % = 1.5% = 0.015. Since this is based on the loan amount,
2
the base = 142000 and the percentage is unknown. Using the percent
equation:
percentage = percent • base
x = 0.015 • 142000
x = 2130
You must pay $2,130 in points.
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b.
1
The percent = 1 % = 1.5% = 0.015. Since this is based on the loan amount,
2
the base is unknown and the percentage = 1350. Using the percent equation:
percent • base = percentage
0.015x = 1350
1350
x=
0.015
x = 90000
The loan amount is $90,000.
Example 5
A real estate broker earns a commission which is 6% of the sales price of the
homes she sells.
a.
b.
Solution
If she sells two homes for $85,000 and $57,000 one month, what will
her commission be?
If her commission is $17,160 for one quarter, what were her total home
sales that quarter?
a.
Her total home sales are $85,000 + $57,000 = $142,000, which is what her
commission is based on. The percent = 6% = 0.06, the base = $142,000, and
the percentage is unknown. Using the percent equation:
percentage = percent • base
x = 0.06 • 142000
x = 8520
Her sales commission will be $8,520.
b.
The percent = 6% = 0.06, the percentage is $17,160, and the base is
unknown. Using the percent equation:
percent • base = percentage
0.06x = 17160
17160
x=
0.06
x = 286000
Her total home sales were $286,000 that quarter.
The exercises in this section will pursue a number of other applications of percents.
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Terminology
points
Exercise Set 5.5
Solve the following application problems. Be sure to answer the question.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
You earn 700 out of 800 possible points in your algebra class. What is your percent of
points earned?
On a 120 point exam, your instructor tells you that 70% is the score cutoff for a “C”
grade. What is the point cutoff for a “C” grade?
You earn 888 points in a course, which is 74% of the possible points in the course.
What are the possible points in the course?
Stella earns 51 out of 60 points possible on a quiz. What is her percent score on the
quiz?
On a 250 point final exam, you need to earn 78% of the points to keep your “B” grade.
How many points do you need to earn to keep your “B” grade?
You earn 492 points in a course, which is 82% of the possible points in the course.
What are the possible points in the course?
The sales tax rate in a certain state is 5.75%. If you buy a $18,600 car, how much will
the state sales tax be?
The sales tax rate in a certain state is 5.75%. If you buy a $32,000 yacht, how much
will the state sales tax be?
The sales tax rate in a certain state is 6.25%. If you pay $150 sales tax on living room
furniture, what was the purchase price of the furniture?
The sales tax rate in a certain state is 6.25%. If you pay $487.50 sales tax on a new
motorcycle, what was the purchase price of the motorcycle?
If you pay $169 state sales tax for a $2600 purchase, what is the state sales tax rate?
If you pay $105 state sales tax for a $1400 purchase, what is the state sales tax rate?
A baseball player has a career 0.320 average, meaning he gets a hit 32% of the time he
has an at bat (comes to the plate). During one season he has 600 at bats. How many
hits would we expect him to have?
A baseball player has a career 0.320 average, meaning he gets a hit 32% of the time he
has an at bat (comes to the plate). During one season he has 200 hits. How many at
bats would we expect him to have?
Jeff Hornacek makes 90% of his free throw attempts. During one season he makes
378 free throws. How many attempts did he have that season?
Jeff Hornacek makes 90% of his free throw attempts. If he attempts 540 free throws in
a season, how many free throws did he make?
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17. The Atlanta Braves finish a baseball season with 104 wins and 56 losses. What percent
of the games played did they win?
18. Shaquille O’Neal finishes a game with 7 offensive rebounds and 13 defensive
rebounds. What percent of his rebounds were offensive rebounds?
1
19. When applying for the loan on a house, you must pay 1 % of the loan amount as
2
points. If the loan is for $82,400, how much will you need to pay in points?
3
20. When applying for the loan on a house, you must pay 1 % of the loan amount as
4
points. If the loan is for $135,000, how much will you need to pay in points?
3
21. When applying for the loan on a house, you must pay 1 % of the loan amount as
4
points. If you pay $2,485 in points on a loan, how much is the loan amount?
1
22. When applying for the loan on a house, you must pay 1 % of the loan amount as
2
points. If you pay $996 in points on a loan, how much is the loan amount?
23. When purchasing a house, the total closing costs are $5,940 for a $132,000 loan. What
percent of the loan amount are the total closing costs?
24. When purchasing a house, the total closing costs are $8,700 for a $150,000 loan. What
percent of the loan amount are the total closing costs?
25. A real estate broker earns a commission which is 6% of the sales price of the homes
she sells. If she sells two homes for $52,000 and $68,500, what will her commission
be?
26. A real estate broker earns a commission which is 6% of the sales price of the homes
she sells. If her commission is $15,480 on the sale of a large home, what was the
selling price of the home?
27. A sales representative earns a commission which is 1.5% of the sales in his district. If
he earns $384 commission during one week, how much were his sales that week?
28. A sales representative earns a commission which is 1.5% of the sales in his district. If
his sales are $128,000 one month, how much commission will he earn?
29. A sales clerk receives a monthly salary of $1250 plus a commission of 4% of the
monthly sales above $7,500. If the clerk has sales of $19,800, what is his monthly
income?
30. A fashion merchandiser receives a monthly salary of $1480 plus a commission of 5%
of the monthly sales above $25,000. If the merchandiser has sales of $43,600, what is
her monthly income?
31. A sales clerk receives a monthly salary of $1250 plus a commission of 4% of the
monthly sales above $7,500. If the clerk has a monthly income of $2062, what are his
sales that month?
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32. A fashion merchandiser receives a monthly salary of $1480 plus a commission of 5%
of the monthly sales above $25,000. If the merchandiser has a monthly income of
$2855, what are her sales that month?
33. You miss 18 points on a test and receive a percent score of 85%. How many points
were possible for the test?
34. You miss 33 points on a test and receive a percent score of 78%. How many points
were possible for the test?
35. A teacher has 28% of her salary withheld for income tax. If her take-home pay is
$3240, how much was her gross pay (amount before taxes)?
36. An investment advisor has 36% of his salary withheld for income tax. If his take-home
pay is $4160, how much was his gross pay (amount before taxes)?
37. Carlos invests $450 of his $3600 wage as a computer programmer in his company
401(k) program. What percent of his wage does he save?
38. Rosie saves $156 of her $2400 monthly paycheck for her children’s college savings
account. What percent of her paycheck does she save?
39. The average selling price of a sport utility vehicle is $28,000, and the average
manufacturer’s profit is $2100. What percent of the selling price does this profit
represent? This is referred to as the manufacturer’s profit margin.
40. The average selling price of a four door sedan is $17,600, and the average
manufacturer’s profit is $440. What is the manufacturer’s profit margin? Refer to
Exercise 39 for the definition of profit margin.
41. The customary tip in a restaurant is 15% of the amount of the meal. If a meal for four
people is $90, how much tip should be left?
42. You compute the tip for a meal to be $10.20. If the customary tip is 15% of the
amount of the meal, how much was the amount of the meal?
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