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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

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

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

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

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

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

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

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Practice set a

Write the ratio 16 to 100 as a percent.

16%

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Write the ratio 195 to 100 as a percent.

195%

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

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

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

362 100 or 181 50 size 12{ { {"362"} over {"100"} } " or " { {"181"} over {"50"} } } {}

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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 [link] and [link] , let's summarize the conversion techniques.

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

Convert 12% to a decimal.

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.

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Convert 0.75 to a percent.

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.

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

We see in [link] 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 or 3 5 = 0 . 6 = 6 10 = 60 100 = 60% size 12{ { {3} over {5} } =0 "." 6= { {6} over {"10"} } = { {"60"} over {"100"} } ="60"%} {}

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Convert 42% to a fraction.

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

or

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

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Practice set b

Convert 21% to a decimal.

0.21

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Convert 461% to a decimal.

4.61

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Convert 0.55 to a percent.

55%

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Convert 5.64 to a percent.

564%

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

15%

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

137.5%

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

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

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Exercises

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

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

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

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

20%

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

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

62.5%

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

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

28%

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

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

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

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

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

164%

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

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

945%

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

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

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

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

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

80%

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

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25%

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

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65%

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

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12.5%

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

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512.5%

41 8 or   5 1 8 size 12{ { {"41"} over {8} } " or "5 { {1} over {8} } } {}

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9 . 9 _ %

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

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22 . 2 _ %

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

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Exercises for review

( [link] ) Find the quotient. 40 54 ÷ 8 7 21 size 12{ { {"40"} over {"54"} } +8 { {7} over {"21"} } } {} .

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

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( [link] ) 3 8 size 12{ { {3} over {8} } } {} of what number is 2 2 3 size 12{2 { {2} over {3} } } {} ?

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( [link] ) Find the value of 28 15 + 7 10 5 12 size 12{ { {"28"} over {"15"} } + { {7} over {"10"} } - { {5} over {"12"} } } {} .

129 60 or   2 9 60 = 2 3 20 size 12{ { {"129"} over {"60"} } " or "2 { {9} over {"60"} } =2 { {3} over {"20"} } } {}

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( [link] ) Round 6.99997 to the nearest ten thousandths.

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

36 inches

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Questions & Answers

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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