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Percent

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses percents. By the end of the module students should understand the relationship between ratios and percents and be able to make conversions between fractions, decimals, and percents.

Section Overview

  • Ratios and Percents
  • The Relationship Between Fractions, Decimals, and Percents – Making Conversions

Ratios and Percents

Ratio, Percent

We defined a ratio as a comparison, by division, of two pure numbers or two like denominate numbers. A most convenient number to compare numbers to is 100. Ratios in which one number is compared to 100 are called percents. The word percent comes from the Latin word "per centum." The word "per" means "for each" or "for every," and the word "centum" means "hundred." Thus, we have the following definition.

Percent means “for each hundred," or "for every hundred."

The symbol % is used to represent the word percent.

Sample Set A

Example 1

The ratio 26 to 100 can be written as 26%. We read 26% as "twenty-six percent."

Example 2

The ratio 165100165100 size 12{ { {"165"} over {"100"} } } {} can be written as 165%.

We read 165% as "one hundred sixty-five percent."

Example 3

The percent 38% can be written as the fraction 3810038100 size 12{ { {"38"} over {"100"} } } {}.

Example 4

The percent 210% can be written as the fraction 210100210100 size 12{ { {"210"} over {"100"} } } {} or the mixed number 210100210100 size 12{2 { {"10"} over {"100"} } } {} or 2.1.

Example 5

Since one dollar is 100 cents, 25 cents is 2510025100 size 12{ { {"25"} over {"100"} } } {} of a dollar. This implies that 25 cents is 25% of one dollar.

Practice Set A

Exercise 1

Write the ratio 16 to 100 as a percent.

Solution

16%

Exercise 2

Write the ratio 195 to 100 as a percent.

Solution

195%

Exercise 3

Write the percent 83% as a ratio in fractional form.

Solution

8310083100 size 12{ { {"83"} over {"100"} } } {}

Exercise 4

Write the percent 362% as a ratio in fractional form.

Solution

362100 or 18150362100 or 18150 size 12{ { {"362"} over {"100"} } " or " { {"181"} over {"50"} } } {}

The Relationship Between Fractions, Decimals, and Percents – Making Conversions

Since a percent is a ratio, and a ratio can be written as a fraction, and a fraction can be written as a decimal, any of these forms can be converted to any other.

Before we proceed to the problems in Section 6 and Section 7, let's summarize the conversion techniques.

Table 1: Conversion Techniques – Fractions, Decimals, Percents
To Convert a Fraction To Convert a Decimal To Convert a Percent
To a decimal: Divide the numerator by the denominator To a fraction: Read the decimal and reduce the resulting fraction To a decimal: Move the decimal point 2 places to the left and drop the % symbol
To a percent: Convert the fraction first to a decimal, then move the decimal point 2 places to the right and affix the % symbol. To a percent: Move the decimal point 2 places to the right and affix the % symbol To a fraction: Drop the % sign and write the number “over” 100. Reduce, if possible.

Sample Set B

Example 6

Convert 12% to a decimal.

12% = 12 100 = 0 . 12 12% = 12 100 = 0 . 12 size 12{"12% "= { {"12"} over {"100"} } =" 0" "." "12"} {}

Note that Twelve percent is equal to .12. this diagram shows that the decimal place in 12% moves two spaces to the left to convert to a decimal.

The % symbol is dropped, and the decimal point moves 2 places to the left.

Example 7

Convert 0.75 to a percent.

0 . 75 = 75 100 = 75% 0 . 75 = 75 100 = 75% size 12{0 "." "75"= { {"75"} over {"100"} } ="75"%} {}

Note that .75 percent is equal to 75%. this diagram shows that the decimal place in .75 moves two spaces to the right to convert to a percent.

The % symbol is affixed, and the decimal point moves 2 units to the right.

Example 8

Convert 3535 size 12{ { {3} over {5} } } {} to a percent.

We see in Example 7 that we can convert a decimal to a percent. We also know that we can convert a fraction to a decimal. Thus, we can see that if we first convert the fraction to a decimal, we can then convert the decimal to a percent.

3 5 → .6 5 3.0 3 0 ̲ 0 3 5 → .6 5 3.0 3 0 ̲ 0 or 35=0.6=610=60100=60%35=0.6=610=60100=60% size 12{ { {3} over {5} } =0 "." 6= { {6} over {"10"} } = { {"60"} over {"100"} } ="60"%} {}

Example 9

Convert 42% to a fraction.

42% = 42 100 = 21 50 42% = 42 100 = 21 50 size 12{"42"%= { {"42"} over {"100"} } = { {"21"} over {"50"} } } {}

or

42% = 0 . 42 = 42 100 = 21 50 42% = 0 . 42 = 42 100 = 21 50 size 12{"42"%=0 "." "42"= { {"42"} over {"100"} } = { {"21"} over {"50"} } } {}

Practice Set B

Exercise 5

Convert 21% to a decimal.

Solution

0.21

Exercise 6

Convert 461% to a decimal.

Solution

4.61

Exercise 7

Convert 0.55 to a percent.

Solution

55%

Exercise 8

Convert 5.64 to a percent.

Solution

564%

Exercise 9

Convert 320320 size 12{ { {3} over {"20"} } } {} to a percent.

Solution

15%

Exercise 10

Convert 118118 size 12{ { {"11"} over {8} } } {} to a percent

Solution

137.5%

Exercise 11

Convert 311311 size 12{ { {3} over {"11"} } } {} to a percent.

Solution

27.27¯27.27¯ size 12{"27" "." {overline {"27"}} %} {}%

Exercises

For the following 12 problems, convert each decimal to a percent.

Exercise 12

0.25

Solution

25%

Exercise 13

0.36

Exercise 14

0.48

Solution

48%

Exercise 15

0.343

Exercise 16

0.771

Solution

77.1%

Exercise 17

1.42

Exercise 18

2.58

Solution

258%

Exercise 19

4.976

Exercise 20

16.1814

Solution

1,618.14%

Exercise 21

533.01

Exercise 22

2

Solution

200%

Exercise 23

14

For the following 10 problems, convert each percent to a deci­mal.

Exercise 24

15%

Solution

0.15

Exercise 25

43%

Exercise 26

16.2%

Solution

0.162

Exercise 27

53.8%

Exercise 28

5.05%

Solution

0.0505

Exercise 29

6.11%

Exercise 30

0.78%

Solution

0.0078

Exercise 31

0.88%

Exercise 32

0.09%

Solution

0.0009

Exercise 33

0.001%

For the following 14 problems, convert each fraction to a per­cent.

Exercise 34

1515 size 12{ { {1} over {5} } } {}

Solution

20%

Exercise 35

3535 size 12{ { {3} over {5} } } {}

Exercise 36

5858 size 12{ { {5} over {8} } } {}

Solution

62.5%

Exercise 37

116116 size 12{ { {1} over {"16"} } } {}

Exercise 38

725725 size 12{ { {7} over {"25"} } } {}

Solution

28%

Exercise 39

16451645 size 12{ { {"16"} over {"45"} } } {}

Exercise 40

27552755 size 12{ { {"27"} over {"55"} } } {}

Solution

49.09¯49.09¯ size 12{"49" "." {overline {"09"}} } {}%

Exercise 41

158158 size 12{ { {"15"} over {8} } } {}

Exercise 42

41254125 size 12{ { {"41"} over {"25"} } } {}

Solution

164%

Exercise 43

645645 size 12{6 { {4} over {5} } } {}

Exercise 44

99209920 size 12{9 { {9} over {"20"} } } {}

Solution

945%

Exercise 45

12001200 size 12{ { {1} over {"200"} } } {}

Exercise 46

611611 size 12{ { {6} over {"11"} } } {}

Solution

54.54¯54.54¯ size 12{"54" "." {overline {"54"}} } {}%

Exercise 47

35273527 size 12{ { {"35"} over {"27"} } } {}

For the following 14 problems, convert each percent to a fraction.

Exercise 48

80%

Solution

4545 size 12{ { {4} over {5} } } {}

Exercise 49

60%

Exercise 50

25%

Solution

1414 size 12{ { {1} over {4} } } {}

Exercise 51

75%

Exercise 52

65%

Solution

13201320 size 12{ { {"13"} over {"20"} } } {}

Exercise 53

18%

Exercise 54

12.5%

Solution

1818 size 12{ { {1} over {8} } } {}

Exercise 55

37.5%

Exercise 56

512.5%

Solution

418 or   518418 or   518 size 12{ { {"41"} over {8} } " or "5 { {1} over {8} } } {}

Exercise 57

937.5%

Exercise 58

9.9_%9.9_%

Solution

110110 size 12{ { {1} over {"10"} } } {}

Exercise 59

55.5_%55.5_%

Exercise 60

22.2_%22.2_%

Solution

2929 size 12{ { {2} over {9} } } {}

Exercise 61

63.6_%63.6_%

Exercises for Review

Exercise 62

((Reference)) Find the quotient. 4054÷87214054÷8721 size 12{ { {"40"} over {"54"} } +8 { {7} over {"21"} } } {}.

Solution

445445 size 12{ { {4} over {"45"} } } {}

Exercise 63

((Reference)) 3838 size 12{ { {3} over {8} } } {} of what number is 223223 size 12{2 { {2} over {3} } } {}?

Exercise 64

((Reference)) Find the value of 2815+710−5122815+710−512 size 12{ { {"28"} over {"15"} } + { {7} over {"10"} } - { {5} over {"12"} } } {}.

Solution

12960 or  2960=232012960 or  2960=2320 size 12{ { {"129"} over {"60"} } " or "2 { {9} over {"60"} } =2 { {3} over {"20"} } } {}

Exercise 65

((Reference)) Round 6.99997 to the nearest ten thousandths.

Exercise 66

((Reference)) On a map, 3 inches represent 40 miles. How many inches represent 480 miles?

Solution

36 inches

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