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Ratios and Rates: Fractions of One 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 fractions of one percent. By the end of the module students should understand the meaning of a fraction of one percent and be able to make conversions involving fractions of one percent.

Section Overview

  • Conversions Involving Fractions of One Percent
  • Conversions Involving Nonterminating Fractions

Conversions Involving Fractions of One Percent

Percents such as 1212 size 12{ { {1} over {2} } } {}%, 3535 size 12{ { {3} over {5} } } {} %, 5858 size 12{ { {5} over {8} } } {} %, and 711711 size 12{ { {7} over {"11"} } } {}%, where 1% has not been attained, are fractions of 1%. This implies that

1 2 % = 1 2   of 1% 1 2 % = 1 2   of 1% size 12{ { {1} over {2} } %= { {1} over {2} } `"of 1%"} {}

3 5 % = 3 5   of 1% 3 5 % = 3 5   of 1% size 12{ { {3} over {5} } %= { {3} over {5} } `"of 1%"} {}

5 8 % = 5 8   of 1% 5 8 % = 5 8   of 1% size 12{ { {5} over {8} } %= { {5} over {8} } `"of 1%"} {}

7 11 % = 7 11   of 1% 7 11 % = 7 11   of 1% size 12{ { {7} over {"11"} } %= { {7} over {"11"} } `"of 1%"} {}

Since "percent" means "for each hundred," and "of" means "times," we have

1 2 % = 1 2 of 1% = 1 2 ⋅ 1 100 = 1 200 1 2 % = 1 2 of 1% = 1 2 ⋅ 1 100 = 1 200 size 12{ { {1} over {2} } %= { {1} over {2} } `"of 1%"= { {1} over {2} } cdot { {1} over {"100"} } = { {1} over {"200"} } } {}

3 5 % = 3 5 of 1% = 3 5 ⋅ 1 100 = 3 500 3 5 % = 3 5 of 1% = 3 5 ⋅ 1 100 = 3 500 size 12{ { {3} over {5} } %= { {3} over {5} } `"of 1%"= { {3} over {5} } cdot { {1} over {"100"} } = { {3} over {"500"} } } {}

5 8 % = 5 8 of 1% = 5 8 ⋅ 1 100 = 5 800 5 8 % = 5 8 of 1% = 5 8 ⋅ 1 100 = 5 800 size 12{ { {5} over {8} } %= { {5} over {8} } `"of 1%"= { {5} over {8} } cdot { {1} over {"100"} } = { {5} over {"800"} } } {}

7 11 % = 7 11 of 1% = 7 11 ⋅ 1 100 = 7 1100 7 11 % = 7 11 of 1% = 7 11 ⋅ 1 100 = 7 1100 size 12{ { {7} over {"11"} } %= { {7} over {"11"} } `"of 1%"= { {7} over {"11"} } cdot { {1} over {"100"} } = { {7} over {"1100"} } } {}

Sample Set A

Example 1

Convert 2323 size 12{ { {2} over {3} } } {}% to a fraction.

2 3 % = 2 3 of 1 % = 2 1 3 ⋅ 1 100 50 = 1⋅1 3⋅50 = 1 150 2 3 % = 2 3 of 1 % = 2 1 3 ⋅ 1 100 50 = 1⋅1 3⋅50 = 1 150

Example 2

Convert 5858 size 12{ { {5} over {8} } } {} % to a decimal.

5 8 % = 5 8 of 1 % = 5 8 ⋅ 1 100 = 0.625⋅0.01 = 0.00625 5 8 % = 5 8 of 1 % = 5 8 ⋅ 1 100 = 0.625⋅0.01 = 0.00625

Practice Set A

Exercise 1

Convert 1414 size 12{ { {1} over {4} } } {} % to a fraction.

Solution

14001400 size 12{ { {1} over {"400"} } } {}

Exercise 2

Convert 3838 size 12{ { {3} over {8} } } {} % to a fraction.

Solution

38003800 size 12{ { {3} over {"800"} } } {}

Exercise 3

Convert 313313 size 12{3 { {1} over {3} } } {} % to a fraction.

Solution

130130 size 12{ { {1} over {"30"} } } {}

Conversions Involving Nonterminating Fractions

We must be careful when changing a fraction of 1% to a decimal. The number 2323 size 12{ { {2} over {3} } } {}, as we know, has a nonterminating decimal representation. Therefore, it cannot be expressed exactly as a decimal.

When converting nonterminating fractions of 1% to decimals, it is customary to express the fraction as a rounded decimal with at least three decimal places.

Converting a Nonterminating Fraction to a Decimal

To convert a nonterminating fraction of 1% to a decimal:

  1. Convert the fraction as a rounded decimal.
  2. Move the decimal point two digits to the left and remove the percent sign.

Sample Set B

Example 3

Convert 2323 size 12{ { {2} over {3} } } {} % to a three-place decimal.

  1. Convert 2323 size 12{ { {2} over {3} } } {} to a decimal.

    Since we wish the resulting decimal to have three decimal digits, and removing the percent sign will account for two of them, we need to round 2323 size 12{ { {2} over {3} } } {} to one place (2 + 1 = 3)(2 + 1 = 3) size 12{ \( "2 "+" 1 "=" 3" \) } {}.

    23%=0.723%=0.7 size 12{ { {2} over {3} } %=0 "." 7%} {}% to one decimal place. 23=0.6666…23=0.6666… size 12{ left ( { {2} over {3} } =0 "." "6666" dotslow right )} {}

  2. Move the decimal point two digits to the left and remove the % sign. We'll need to add zeros to locate the decimal point in the correct location.

    23% = 0.00723% = 0.007 size 12{ { {2} over {3} } "% "=" 0" "." "007"} {} to 3 decimal places

Example 4

Convert 54115411 size 12{5 { {4} over {"11"} } } {}% to a four-place decimal.

  1. Since we wish the resulting decimal to have four decimal places, and removing the percent sign will account for two, we to round 411411 size 12{ { {4} over {"11"} } } {} to two places.

    5411%=5.36%5411%=5.36 size 12{5 { {4} over {"11"} } %=5 "." "36"%} {}% to two decimal places. 411=0.3636…411=0.3636… size 12{ left ( { {4} over {"11"} } =0 "." "3636" dotslow right )} {}

  2. Move the decimal point two places to the left and drop the percent sign.

    5411%=0.05365411%=0.0536 size 12{5 { {4} over {"11"} } %=5 "." "36"%} {} to four decimal places.

Example 5

Convert 28592859 size 12{"28" { {5} over {9} } } {}% to a decimal rounded to ten thousandths.

  1. Since we wish the resulting decimal to be rounded to ten thousandths (four decimal places), and removing the percent sign will account for two, we need to round 5959 size 12{ { {5} over {9} } } {} to two places.

    2859%=28.56%2859%=28.56% size 12{"28" { {5} over {9} } %="28" "." "56"%} {} to two decimal places. 59=0.5555…59=0.5555… size 12{ left ( { {5} over {9} } =0 "." "5555" dotslow right )} {}

  2. Move the decimal point to the left two places and drop the percent sign.

    2859%=0.28562859%=0.2856 size 12{"28" { {5} over {9} } %=0 "." "2856"} {} correct to ten thousandths.

Practice Set B

Exercise 4

Convert 7979 size 12{ { {7} over {9} } } {}% to a three-place decimal.

Solution

0.008

Exercise 5

Convert 51511%51511% to a decimal rounded to ten thousandths.

Solution

0.5145

Exercises

Make the conversions as indi­cated.

Exercise 6

Convert 3434 size 12{ { {3} over {4} } } {} % to a fraction.

Solution

34003400 size 12{ { {3} over {"400"} } } {}

Exercise 7

Convert 5656 size 12{ { {5} over {6} } } {}% to a fraction.

Exercise 8

Convert 1919 size 12{ { {1} over {9} } } {}% to a fraction.

Solution

19001900 size 12{ { {1} over {"900"} } } {}

Exercise 9

Convert 15191519 size 12{ { {"15"} over {"19"} } } {} % to a fraction.

Exercise 10

Convert 5454 size 12{ { {5} over {4} } } {} % to a fraction.

Solution

5400 or 1805400 or 180 size 12{ { {5} over {"400"} } " or " { {1} over {"80"} } } {}

Exercise 11

Convert 7373 size 12{ { {7} over {3} } } {}% to a fraction.

Exercise 12

Convert 167167 size 12{1 { {6} over {7} } } {}% to a fraction.

Solution

1370013700 size 12{ { {"13"} over {"700"} } } {}

Exercise 13

Convert 25162516 size 12{2 { {5} over {"16"} } } {}% to a fraction.

Exercise 14

Convert 25142514 size 12{"25" { {1} over {4} } } {}% to a fraction.

Solution

101400101400 size 12{ { {"101"} over {"400"} } } {}

Exercise 15

Convert 50125012 size 12{"50" { {1} over {2} } } {}% to a fraction.

Exercise 16

Convert 72357235 size 12{"72" { {3} over {5} } } {}% to a fraction.

Solution

363500363500 size 12{ { {"363"} over {"500"} } } {}

Exercise 17

Convert 99189918 size 12{"99" { {1} over {8} } } {}% to a fraction.

Exercise 18

Convert 1362313623 size 12{"136" { {2} over {3} } } {}% to a fraction.

Solution

41304130 size 12{ { {"41"} over {"30"} } } {}

Exercise 19

Convert 5213452134 size 12{"521" { {3} over {4} } } {}% to a fraction.

Exercise 20

Convert 10151015 size 12{"10" { {1} over {5} } } {}% to a decimal.

Solution

51500=0.10251500=0.102 size 12{ { {"51"} over {"500"} } =0 "." "102"} {}

Exercise 21

Convert 12341234 size 12{"12" { {3} over {4} } } {}% to a decimal.

Exercise 22

Convert 378378 size 12{3 { {7} over {8} } } {}% to a decimal.

Solution

31800=0.0387531800=0.03875 size 12{ { {"31"} over {"800"} } =0 "." "03875"} {}

Exercise 23

Convert 71167116 size 12{7 { {1} over {"16"} } } {}% to a decimal.

Exercise 24

Convert 3737 size 12{ { {3} over {7} } } {}% to a three-place decimal.

Solution

0.0040.004 size 12{0 "." "004"} {}

Exercise 25

Convert 1919 size 12{ { {1} over {9} } } {}% to a three-place decimal.

Exercise 26

Convert 63116311 size 12{6 { {3} over {"11"} } } {}% to a four-place decimal.

Solution

0.0627

Exercise 27

Convert 927927 size 12{9 { {2} over {7} } } {}% to a four-place decimal.

Exercise 28

Convert 2452124521 size 12{"24" { {5} over {"21"} } } {}% to a three-place decimal.

Solution

0.242

Exercise 29

Convert 4582745827 size 12{"45" { {8} over {"27"} } } {}% to a three-place decimal.

Exercise 30

Convert 111617111617 size 12{"11" { {"16"} over {"17"} } } {} % to a four-place decimal.

Solution

0.1194

Exercise 31

Convert 517517 size 12{5 { {1} over {7} } } {}% to a three-place decimal.

Exercises for Review

Exercise 32

((Reference)) Write 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 88 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 size 12{"8 " cdot " 8 " cdot " 8 " cdot " 8 " cdot " 8"} {} using exponents.

Solution

8585 size 12{8 rSup { size 8{5} } } {}

Exercise 33

((Reference)) Convert 478478 size 12{4 { {7} over {8} } } {} to an improper fraction.

Exercise 34

((Reference)) Find the sum. 710+221+17710+221+17 size 12{ { {7} over {"10"} } + { {2} over {"21"} } + { {1} over {7} } } {}.

Solution

197210197210 size 12{ { {"197"} over {"210"} } } {}

Exercise 35

((Reference)) Find the product. (4.21)(0.006)(4.21)(0.006) size 12{ \( 4 "." "21" \) \( 0 "." "006" \) } {}.

Exercise 36

((Reference)) Convert 8.062 to a percent.

Solution

806.2806.2 size 12{"806" "." 2%} {}%

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