6.4
Three Types of Percent Problems
6.4
OBJECTIVES
1. Find the unknown amount in a percent problem
2. Find the unknown rate in a percent problem
3. Find the unknown base in a percent problem
From your work in Section 6.3, you may have observed that there are three basic types of
percent problems. These depend on which of the three parts—the amount, the rate, or the
base—is missing in the problem statement. The solution for each type of problem depends
on the percent relationship.
Rules and Properties:
Percent Relationship
Amount rate base
We will illustrate the solution of each type of problem in the following examples. Let’s start
with a problem in which we want to find the amount.
Example 1
Finding an Unknown Amount
NOTE Type 1: Finding an
unknown amount.
What is 18% of 300?
We know the rate, 18%; and the base, 300; the amount is the unknown. Using the
percent relationship, we can translate the problem to an equation.
Rate Base
Amount 0.18 300
Write 18% as the decimal 0.18 by the
rule of Section 6.1. Then multiply to
find the amount.
54
So 54 is 18% of 300.
CHECK YOURSELF 1
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Find 65% of 200.
1. If the rate is less than 100%, the amount will be less than the base.
20 is 40% of 50
and
20 50
2. If the rate is greater than 100%, the amount will be greater than the base.
75 is 150% of 50
and
75 50
Let’s consider a second type of percent problem involving an unknown rate.
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CHAPTER 6
PERCENTS
Example 2
Finding an Unknown Percent
NOTE Type 2: Finding an
unknown percent.
30 is what percent of 150?
We know the amount, 30, and the base, 150; the rate (what percent) is the unknown.
Again using the percent relationship to translate to an equation, we have
Base Amount
Rate 150 30
NOTE This will leave the rate
alone on the left.
We divide both sides by 150 to find the rate.
Rate 30
1
1
0.20 20
20%
150
5
100
30 is 20% of 150.
CHECK YOURSELF 2
75 is what percent of 300?
1. If the amount is less than the base, the rate will be less than 100%.
2. If the amount is greater than the base, the rate will be greater than 100%.
Let’s look at a percent problem involving an unknown base in Example 3.
Example 3
Finding an Unknown Base
NOTE Type 3: Finding an
unknown base.
28 is 40% of what number?
We know the amount, 28, and the rate, 40%. The base (what number) is the unknown.
From the percent relationship we have
Rate
NOTE Notice that 40% is
Amount
0.40 base 28
written as 0.40.
Base 28
70
0.40
So 28 is 40% of 70.
CHECK YOURSELF 3
70 is 35% of what number?
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We divide both sides by 0.40 to find the base.
THREE TYPES OF PERCENT PROBLEMS
SECTION 6.4
497
We have now seen solution methods for the three basic types of percent problems: finding the amount, the rate, and the base. As you will see in the remainder of this section, our
work in Chapter 5 with proportions will allow us to solve each type of problem in an identical fashion. In fact, many students find percent problems easier to approach with the
proportion method.
First, we will write what is called the percent proportion.
Rules and Properties:
The Percent Proportion
Amount
R
Base
100
In symbols,
R
NOTE On the right,
is the
100
rate, and this proportion is
equivalent to our earlier
percent relationship.
R
A
B
100
Because in any percent problem we know two of the three quantities (A, B, or R), we can
always solve for the unknown term. Consider in Example 4 the use of the percent proportion.
Example 4
Solving a Problem Involving an Unknown Amount
NOTE This is an unknown-
________ is 30% of 150.
amount problem.
A
R
B
Substitute the values into the percent proportion.
R
A
30
150
100
The amount A is the unknown
term of the proportion.
B
We solve the proportion with the methods of Section 5.4.
100A 150 30
100A 4500
Divide by the coefficient, 100.
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1
4500
100 A
100
100
1
A 45
The amount is 45. This means that 45 is 30% of 150.
CHECK YOURSELF 4
Use the percent proportion to answer this question: What is 24% of 300?
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CHAPTER 6
PERCENTS
The same percent proportion will work if you want to find the rate.
Example 5
Solving a Problem Involving an Unknown Rate
NOTE This is an unknown-rate
______% of 400 is 72.
problem.
R
B
A
Substitute the known values into the percent proportion.
A
72
R
400
100
R, the rate, is the unknown
term in this case.
B
Solving, we get
400R 7200
1
400R
7200
400
400
1
R 18
The rate is 18%. So 18% of 400 is 72.
CHECK YOURSELF 5
Use the percent proportion to answer this question: What percent of 50 is 12.5?
Finally, we use the same proportion to find an unknown base.
Example 6
Solving a Problem Involving an Unknown Base
NOTE This is an unknown-base
40% of ______ is 200.
problem.
R
B
A
Substitute the known values into the percent proportion.
R
200
40
B
100
In this case B, the base, is the unknown
term of the proportion.
Solving gives
40B 200 100
1
20,000
40B
40
40
1
B 500
The base is 500, and 40% of 500 is 200.
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A
THREE TYPES OF PERCENT PROBLEMS
SECTION 6.4
499
Remember that a percent (the rate) can be greater than 100.
CHECK YOURSELF 6
288 is 60% of what number?
Example 7
Solving a Percent Problem
NOTE The rate is 125%. The
base is 300.
NOTE When the rate is greater
than 100%, the amount will be
greater than the base.
What is 125% of 300?
In the percent proportion, we have
A
125
300
100
So 100A 300 125.
Dividing by 100 yields
A
37,500
375
100
So 375 is 125% of 300.
CHECK YOURSELF 7
Find 150% of 500.
We next look at two examples of solving percent problems involving fractions of a
percent.
Example 8
Solving a Percent Problem
34 is 8.5% of what number?
Using the percent proportion yields
NOTE The amount is 34, the
rate is 8.5%. We want to find
the base.
34
8.5
B
100
Solving, we have
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8.5B 34 100
or
NOTE Divide by 8.5.
B
3400
400
8.5
So 34 is 8.5% of 400.
CHECK YOURSELF 8
12.5% of what number is 75?
CHAPTER 6
PERCENTS
Example 9
Estimating Percentages
Find 19.3% of 500.
Round the rate to 20% as a fraction,
1
. An estimate of the amount is then
5
1
500 100
5
Rounded rate
Base
Estimate of amount
CHECK YOURSELF 9
Estimate the amount.
20.2% of 800
CHECK YOURSELF ANSWERS
1. 130
4.
2. 25%
R
3. 200
5. A
A
24
300
100
12.5
R
50
100
B
B
100A 7200; A 72
50R 1250; R 25%
6. A
R
7. 750
8. 600
9. 160
288
60
B
100
60B 28,800; B 480
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500
Name
6.4 Exercises
Section
Date
Solve each of the following problems involving percent.
1. What is 35% of 600?
2. 20% of 400 is what number?
3. 45% of 200 is what number?
4. What is 40% of 1200?
5. Find 40% of 2500.
7. What percent of 50 is 4?
9. What percent of 500 is 45?
11. What percent of 200 is 340?
1.
2.
3.
4.
5.
6.
8. 51 is what percent of 850?
7.
8.
10. 14 is what percent of 200?
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
6. What is 75% of 120?
12. 392 is what percent of 2800?
13. 46 is 8% of what number?
14. 7% of what number is 42?
15. Find the base if 11% of the base is 55.
16. 16% of what number is 192?
17. 58.5 is 13% of what number?
18. 21% of what number is 73.5?
19. Find 110% of 800.
20. What is 115% of 600?
21. What is 108% of 4000?
22. Find 160% of 2000.
23. 210 is what percent of 120?
24. What percent of 40 is 52?
25. 360 is what percent of 90?
26. What percent of 15,000 is 18,000?
27. 625 is 125% of what number?
ANSWERS
28. 140% of what number is 350?
33.
29. Find the base if 110% of the base is 935.
30. 130% of what number is 1170?
31. Find 8.5% of 300.
32. 8 % of 800 is what number?
1
4
34.
35.
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36.
3
4
33. Find 11 % of 6000.
34. What is 3.5% of 500?
35. What is 5.25% of 3000?
36. What is 7.25% of 7600?
37. 60 is what percent of 800?
38. 500 is what percent of 1500?
37.
38.
39.
40.
39. What percent of 180 is 120?
40. What percent of 800 is 78?
501
ANSWERS
41.
42.
41. What percent of 1200 is 750?
42. 68 is what percent of 800?
43. 10.5% of what number is 420?
44. Find the base if 11 % of the base
1
2
is 46.
43.
45. 58.5 is 13% of what number?
46. 6.5% of what number is 325?
45.
47. 195 is 7.5% of what number?
48. 21% of what number is 73.5?
46.
Estimate the amount in each of the following problems.
47.
49. Find 25.8% of 4000.
50. What is 48.3% of 1500?
51. 74.7% of 600 is what number?
52. 9.8% of 1200 is what number?
53. Find 152% of 400.
54. What is 118% of 5000?
44.
48.
49.
50.
55. It is customary when eating in a restaurant to leave a 15% tip.
51.
(a) Outline a method to do a quick approximation for the amount of tip to leave.
(b) Use this method to figure a 15% tip on a bill of $47.76.
52.
53.
54.
55.
56.
56. The dean of Enrollment Management at a college states, “Last year was not a good year.
Answers
1. 210
15. 500
27. 500
2
3
51. 450
39. 66 %
502
3. 90
5. 1000
7. 8%
9. 9%
11. 170%
13. 575
17. 450
19. 880
21. 4320
23. 175%
25. 400%
29. 850
31. 25.5
33. 705
35. 157.5
37. 7.5%
41. 62.5%
53. 600
43. 4000
55.
45. 450
47. 2600
49. 1000
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Our enrollments were down 25%. But this year we increased our enrollment by 30% over
last year. I think we have turned the corner.” Evaluate the dean’s analysis.