5.2 Percent: Converting Between Fractions, Decimals, and Percents
The concept of percent permeates most common uses of mathematics in everyday life. We pay
taxes based on percents, many people earn income as a percent sales commission, investments
and banks compare alternatives using percent, and sports team records are represented as a
percent. A percent is a ratio compared with 100. When we pay 7.25% sales tax, we pay $7.25
tax for every $100 in sales price. As a ratio, this can be written as:
$7.25 tax
= 0.0725
$100 price
Since the denominator of a percent is 100, we can easily convert percents to decimals by
dividing by 100. Recall this involves moving the decimal point two place values to the left.
Similarly, we can convert percents to reduced fractions. Given the percent 25%, we convert to
fraction:
25% =
Example 1
Convert each percent to the indicated quantity.
a.
b.
c.
d.
Solution
25
25 • 1 1
=
=
100 25 • 4 4
28% to a fraction
19.8% to a decimal
3.5% to a fraction
325% to a decimal
a.
Writing the percent as a fraction and simplifying:
28
4•7
7
28% =
=
=
100 4 • 25 25
b.
Writing the percent as a ratio and dividing:
19.8
19.8% =
= 0.198
100
c.
Writing the percent as a fraction, eliminating decimals, and simplifying:
3.5 10
35
5•7
7
3.5% =
•
=
=
=
100 10 1000 5 • 200 200
372
d.
Writing the percent as a ratio and dividing:
325
325% =
= 3.25
100
Since percents are used so frequently to compare different ratios, we often want to convert ratios
7
(or fractions) to percents. Suppose we want to convert the fraction
to a percent. Recall that a
8
x
percent is a ratio with a denominator of 100. Thus we want to find the value of x such that
100
7
is equivalent to the original fraction . We solve the proportion:
8
x
7
=
100 8
x
7
200 •
= 200 •
100
8
2x = 175
175
x=
= 87.5%
2
The fraction
Example 2
7
is equivalent to 87.5%.
8
Convert each fraction or mixed number to a percent.
a.
3
4
b.
4
c.
2
3
d.
2
1
2
5
6
373
Solution
a.
Solving the proportion:
x
3
=
100 4
x
3
100 •
= 100 •
100
4
x = 75%
3
Thus = 75% .
4
b.
First write 4
c.
Solving the proportion:
x
2
=
100 3
x
2
300 •
= 300 •
100
3
3x = 200
200
2
x=
= 66 %
3
3
2
2
Thus = 66 % .
3
3
1
9
as the fraction . Solving the proportion:
2
2
x
9
=
100 2
x
9
100 •
= 100 •
100
2
x = 450%
1
Thus 4 = 450% .
2
374
d.
5
17
as the fraction
. Solving the proportion:
6
6
x
17
=
100 6
x
17
300 •
= 300 •
100
6
3x = 850
First write 2
x=
Thus 2
850
1
= 283 %
3
3
5
1
= 283 % .
6
3
Converting a decimal to a percent is extremely easy. Given the decimal 0.824, we are looking for
x
the ratio
which is equivalent to 0.824. Setting up the equation:
100
x
= 0.824
100
x
100 •
= 100 • 0.824
100
x = 82.4%
Thus 0.824 = 82.4%. Note that, in effect, we are merely multiplying the decimal by 100 to
convert it to a percent.
Example 3
Convert each decimal to a percent.
a.
b.
c.
d.
0.569
0.75
1.671
0.0084
375
Solution
a.
Solving the equation:
x
= 0.569
100
x
100 •
= 100 • 0.569
100
x = 56.9%
Thus 0.569 = 56.9%.
b.
Solving the equation:
x
= 0.75
100
x
100 •
= 100 • 0.75
100
x = 75%
Thus 0.75 = 75%.
c.
Solving the equation:
x
= 1.671
100
x
100 •
= 100 • 1.671
100
x = 167.1%
Thus 1.671 = 167.1%.
d.
Solving the equation:
x
= 0.0084
100
x
100 •
= 100 • 0.0084
100
x = 0.84%
Thus 0.0084 = 0.84%.
The remainder of examples in this section are devoted to applications of percents.
376
Example 4
Mr. Jones pays $12,480 income tax for $46,520 in taxable income. Find his
income tax rate, expressed as a percent rounded to the nearest hundredth.
Solution
His tax rate is given as the ratio of
income tax
. So, we want to find the ratio
taxable income
$12, 480
x
which is equivalent to
. Setting up and solving the proportion:
$46,520
100
x
12, 480
=
100 46,520
x
12480
465200 •
= 465200 •
100
46520
4652x = 124800
124800
! 26.83%
4652
Mr. Jones’ income tax rate is approximately 26.83%.
x=
In the previous example note that, in solving the proportion, we did not use the LCM of 46520
and 100. Instead, we used a convenient number which we knew would clear fractions in the
equation. This is common practice in solving percent proportion applications, since the numbers
are often fairly large and finding the LCM can be time consuming.
Example 5
At one point of the season, the San Francisco Giants had won 26 games and
lost 14 games. Express their win to games played ratio as a percent.
Solution
First note that they have played 26 + 14 = 40 games. Their win to games played
26
x
ratio is
, which we want to express as a percent
. Setting up and solving
40
100
the proportion:
x
26
=
100 40
x
26
200 •
= 200 •
100
40
2x = 130
130
x=
= 65%
2
The Giants winning percent is 65%. For baseball fans, note that this is equivalent
to 0.650, which is how these winning percents are represented in the media.
377
Example 6
A car salesperson earns $662.50 commission on the sale of a car with a purchase
price of $26,500. If her commission rate is a percent based on the purchase price,
find the salesperson’s commission rate expressed as a percent.
Solution
Her commission rate is given as the ratio
commission
. We want to find the
purchase price
$662.50
x
which is equivalent to
. Setting up and solving the
$26, 500
100
proportion:
x
662.50
=
100 26500
x
662.50
26500 •
= 26500 •
100
26500
265x = 662.50
ratio
662.5
= 2.5%
265
The salesperson’s commission rate is 2.5%.
x=
In the next section we will solve a variety of additional percent problems.
Terminology
percent
Exercise Set 5.2
Convert each percent to a fraction. Be sure to reduce all answers.
1.
3.
5.
7.
9.
11.
13.
15.
35%
48%
90%
2.5%
6.5%
1.25%
175%
500%
2.
4.
6.
8.
10.
12.
14.
16.
378
75%
56%
24%
19.8%
4.4%
0.8%
240%
320%
Convert each percent to a decimal.
17.
19.
21.
23.
25.
27.
84%
3.5%
175%
600%
0.9%
0.06%
18.
20.
22.
24.
26.
28.
50%
6.5%
240%
850%
0.5%
0.08%
Convert each fraction or mixed number to a percent.
29.
31.
1
2
3
8
30.
32.
1
4
5
1
16
4
9
5
6
1
3
3
5
2
7
1
4
5
8
3
4
9
2
16
7
9
4
7
1
4
6
4
3
11
33. 6
34. 4
35.
36.
37.
39.
41.
43.
38.
40.
42.
44.
Convert each decimal to a percent.
45.
47.
49.
51.
53.
55.
57.
0.348
0.65
3.28
5.064
0.0569
0.00468
0.0005419
46.
48.
50.
52.
54.
56.
58.
379
0.215
0.25
1.16
3.048
0.0498
0.00517
0.0004906
Solve each of the following percent applications.
59. A small business pays $13,860 tax for $93,500 in taxable income. Find their tax rate,
expressed as a percent rounded to the nearest hundredth.
60. A small business pays $22,691 tax for $121,640 in taxable income. Find their tax rate,
expressed as a percent rounded to the nearest hundredth.
61. You pay $56.25 sales tax on a purchase of $1250. Find the sales tax rate, expressed
as a percent.
62. You pay $55.90 sales tax on a purchase of $860. Find the sales tax rate, expressed
as a percent.
63. A salesperson earns $1,467 commission on sales of $32,600. If the commission is
based on the sales, find the salesperson’s commission rate expressed as a percent.
64. A salesperson earns $2,250 commission on sales of $125,000. If the commission is
based on the sales, find the salesperson’s commission rate expressed as a percent.
65. A company has 23 full-time employees and 17 part-time employees. What percent of
its employees are full-time employees?
66. A small hot dog stand makes 125 jumbo dogs and 75 chili dogs for a baseball game.
What percent of its hot dogs made are chili dogs?
67. During one week, Frank has cleaning orders for 275 shirts and 125 pants. What
percent of his cleaning orders are pants?
68. During one day, Todd makes 18 business-related phone calls and 22 personal phone
calls. What percent of his phone calls are personal calls?
69. During a playoff game, Michael Jordan made 19 shots out of 32 shot attempts.
Express his shots made as a percent of shot attempts.
70. During a playoff game, Kobe Bryant made 15 shots out of 25 shot attempts. Express
his shots made as a percent of shot attempts.
71. At one point of the season, the Chicago Cubs had won 32 games and lost 38 games.
Express their win to games played ratio as a percent, rounded to the tenths place.
72. At the end of a season, the Atlanta Braves had won 102 games and lost 60 games.
Express their win to games played ratio as a percent, rounded to the tenths place.
73. An experimental asthma drug is tested on 429 people, and 391 show improvement.
What percent of the tested people showed improvement? Round your answer to the
nearest tenth of a percent.
74. An experimental chemotherapy drug is tested on 1200 people, and 965 show decrease
in cancer levels. What percent of the tested people showed decrease in cancer levels?
Round your answer to the nearest tenth of a percent.
75. A large computer manufacturer reduces prices of its computers from $1400 to $1190.
Express the ratio of amount of price decrease to original price as a percent.
76. A cellular phone company reduces prices of its phones from $120 to $103.20. Express
the ratio of amount of price decrease to original price as a percent.
380
77. An investment earns $380 profit on an investment of $2,000. Express the profit to
investment ratio as a percent.
78. An investment earns $900 profit on an investment of $5,000. Express the profit to
investment ratio as a percent.
381