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9.1 Percent: solving percent problems

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

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

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

5 8 % = 5 8   of 1% size 12{ { {5} over {8} } %= { {5} over {8} } `"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 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 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 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 size 12{ { {7} over {"11"} } %= { {7} over {"11"} } `"of 1%"= { {7} over {"11"} } cdot { {1} over {"100"} } = { {7} over {"1100"} } } {}

Sample set a

Convert 2 3 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

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

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

Practice set a

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

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

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

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

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

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

Conversions involving nonterminating fractions

We must be careful when changing a fraction of 1% to a decimal. The number 2 3 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

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

  1. Convert 2 3 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 2 3 size 12{ { {2} over {3} } } {} to one place ( 2 + 1 = 3 ) size 12{ \( "2 "+" 1 "=" 3" \) } {} .

    2 3 % = 0 . 7 size 12{ { {2} over {3} } %=0 "." 7%} {} % to one decimal place. 2 3 = 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.

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

Convert 5 4 11 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 4 11 size 12{ { {4} over {"11"} } } {} to two places.

    5 4 11 % = 5 . 36 size 12{5 { {4} over {"11"} } %=5 "." "36"%} {} % to two decimal places. 4 11 = 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.

    5 4 11 % = 0 . 0536 size 12{5 { {4} over {"11"} } %=5 "." "36"%} {} to four decimal places.

Convert 28 5 9 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 5 9 size 12{ { {5} over {9} } } {} to two places.

    28 5 9 % = 28 . 56% size 12{"28" { {5} over {9} } %="28" "." "56"%} {} to two decimal places. 5 9 = 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.

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

Practice set b

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

0.008

Convert 51 5 11 % to a decimal rounded to ten thousandths.

0.5145

Exercises

Make the conversions as indi­cated.

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

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

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

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

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

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

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

5 400 or 1 80 size 12{ { {5} over {"400"} } " or " { {1} over {"80"} } } {}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

0.0627

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

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

0.242

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

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

0.1194

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

Exercises for review

( [link] ) Write 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 size 12{"8 " cdot " 8 " cdot " 8 " cdot " 8 " cdot " 8"} {} using exponents.

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

( [link] ) Convert 4 7 8 size 12{4 { {7} over {8} } } {} to an improper fraction.

( [link] ) Find the sum. 7 10 + 2 21 + 1 7 size 12{ { {7} over {"10"} } + { {2} over {"21"} } + { {1} over {7} } } {} .

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

( [link] ) Find the product. ( 4 . 21 ) ( 0 . 006 ) size 12{ \( 4 "." "21" \) \( 0 "." "006" \) } {} .

( [link] ) Convert 8.062 to a percent.

806 . 2 size 12{"806" "." 2%} {} %
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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