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8.6 Decimals: converting a fraction to a decimal

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to convert a fraction to a decimal. By the end of the module students should be able to convert a fraction to a decimal.

Now that we have studied and practiced dividing with decimals, we are also able to convert a fraction to a decimal. To do so we need only recall that a fraction bar can also be a division symbol. Thus, 3 4 size 12{ { {3} over {4} } } {} not only means "3 objects out of 4," but can also mean "3 divided by 4."

Sample set a

Convert the following fractions to decimals. If the division is nonterminating, round to two decimal places.

3 4 size 12{ { {3} over {4} } } {} . Divide 3 by 4.

.75 4 3.00 2 8   ̲ 20 20 ̲ 0

Thus, 3 4 = 0 . 75 size 12{ { {3} over {4} } =0 "." "75"} {} .

1 5 size 12{ { {1} over {5} } } {} Divide 1 by 5.

.2 5 1.0 1.0 ̲ 0

Thus, 1 5 = 0 . 2 size 12{ { {1} over {5} } =0 "." 2} {}

5 6 size 12{ { {5} over {6} } } {} . Divide 5 by 6.

Long division. 5 divided by 6 ends in a recurring remainder. The quotient is .833. The recurring remainder indicates that the division is nonterminating.

5 6 = 0 . 833 ⋯ size 12{ { {5} over {6} } =0 "." "833" dotsaxis } {} We are to round to two decimal places.

Thus, 5 6 = 0 . 83 size 12{ { {5} over {6} } =0 "." "83"} {} to two decimal places.

5 1 8 size 12{5 { {1} over {8} } } {} . Note that 5 1 8 = 5 + 1 8 size 12{5 { {1} over {8} } =5+ { {1} over {8} } } {} .

Convert 1 8 size 12{ { {1} over {8} } } {} to a decimal.

.125 8 1.000   8     ̲ 20   16   ̲ 40 40 ̲ 0

1 8 = . 125 size 12{ { {1} over {8} } = "." "125"} {}

Thus, 5 1 8 = 5 + 1 8 = 5 + . 125 = 5 . 125 size 12{5 { {1} over {8} } =5+ { {1} over {8} } =5+ "." "125"=5 "." "125"} {} .

0 . 16 1 4 size 12{0 "." "16" { {1} over {4} } } {} . This is a complex decimal.

Note that the 6 is in the hundredths position.

The number 0 . 16 1 4 size 12{0 "." "16" { {1} over {4} } } {} is read as "sixteen and one-fourth hundredths."

0 . 16 1 4 = 16 1 4 100 = 16 ⋅ 4 + 1 4 100 = 65 4 100 1 = 65 13 4 ⋅ 1 100 20 = 13 ⋅ 1 4 ⋅ 20 = 13 80 size 12{0 "." "16" { {1} over {4} } = { {"16" { {1} over {4} } } over {"100"} } = { { { {"16" cdot 4+1} over {4} } } over {"100"} } = { { { {"65"} over {4} } } over { { {"100"} over {1} } } } = { { {"65"} cSup { size 8{"13"} } } over {4} } cdot { {1} over { {"100"} cSub { size 8{"20"} } } } = { {"13" cdot 1} over {4 cdot "20"} } = { {"13"} over {"80"} } } {}

Now, convert 13 80 size 12{ { {"13"} over {"80"} } } {} to a decimal.

.1625 80 13.0000 8 0       ̲ 5 00     4 80     ̲ 200   160   ̲ 400 400 ̲ 0

Thus, 0 . 16 1 4 = 0 . 1625 size 12{0 "." "16" { {1} over {4} } =0 "." "1625"} {} .

Practice set a

Convert the following fractions and complex decimals to decimals (in which no proper fractions appear). If the divison is nonterminating, round to two decimal places.

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

0.25

1 25 size 12{ { {1} over {"25"} } } {}

0.04

1 6 size 12{ { {1} over {6} } } {}

0.17

15 16 size 12{ { {"15"} over {"16"} } } {}

0.9375

0 . 9 1 2 size 12{0 "." 9 { {1} over {2} } } {}

0.95

8 . 0126 3 8 size 12{8 "." "0126" { {3} over {8} } } {}

8.0126375

Exercises

For the following 30 problems, convert each fraction or complex decimal number to a decimal (in which no proper fractions appear).

1 2 size 12{ { {1} over {2} } } {}

0.5

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

7 8 size 12{ { {7} over {8} } } {}

0.875

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

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

0.6

2 5 size 12{ { {2} over {5} } } {}

1 25 size 12{ { {1} over {"25"} } } {}

0.04

3 25 size 12{ { {3} over {"25"} } } {}

1 20 size 12{ { {1} over {"20"} } } {}

0.05

1 15 size 12{ { {1} over {"15"} } } {}

1 50 size 12{ { {1} over {"50"} } } {}

0.02

1 75 size 12{ { {1} over {"75"} } } {}

1 3 size 12{ { {1} over {3} } } {}

0 . 3 ¯ size 12{0 "." {overline {3}} } {}

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

3 16 size 12{ { {3} over {"16"} } } {}

0.1875

9 16 size 12{ { {9} over {"16"} } } {}

1 27 size 12{ { {1} over {"27"} } } {}

0 . 0 37 ¯ size 12{0 "." 0 {overline {"37"}} } {}

5 27 size 12{ { {5} over {"27"} } } {}

7 13 size 12{ { {7} over {"13"} } } {}

0 . 538461 ¯ size 12{0 "." {overline {"538461"}} } {}

9 14 size 12{ { {9} over {"14"} } } {}

7 2 3 size 12{7 { {2} over {3} } } {}

7 . 6 ¯ size 12{7 "." {overline {6}} } {}

8 5 16 size 12{8 { {5} over {"16"} } } {}

1 2 15 size 12{1 { {2} over {"15"} } } {}

1 . 1 3 ¯ size 12{1 "." 1 {overline {3}} } {}

65 5 22 size 12{"65" { {5} over {"22"} } } {}

101 6 25 size 12{"101" { {6} over {"25"} } } {}

101.24

0 . 1 1 2 size 12{0 "." 1 { {1} over {2} } } {}

0 . 24 1 8 size 12{0 "." "24" { {1} over {8} } } {}

0.24125

5 . 66 2 3 size 12{5 "." "66" { {2} over {3} } } {}

810 . 3106 5 16 size 12{"810" "." "3106" { {5} over {"16"} } } {}

810.31063125

4 . 1 1 9 size 12{4 "." 1 { {1} over {9} } } {}

For the following 18 problems, convert each fraction to a decimal. Round to five decimal places.

1 9 size 12{ { {1} over {9} } } {}

0.11111

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

3 9 size 12{ { {3} over {9} } } {}

0.33333

4 9 size 12{ { {4} over {9} } } {}

5 9 size 12{ { {5} over {9} } } {}

0.55556

6 9 size 12{ { {6} over {9} } } {}

7 9 size 12{ { {7} over {9} } } {}

0.77778

8 9 size 12{ { {8} over {9} } } {}

1 11 size 12{ { {1} over {"11"} } } {}

0.09091

2 11 size 12{ { {2} over {"11"} } } {}

3 11 size 12{ { {3} over {"11"} } } {}

0.27273

4 11 size 12{ { {4} over {"11"} } } {}

5 11 size 12{ { {5} over {"11"} } } {}

0.45455

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

7 11 size 12{ { {7} over {"11"} } } {}

0.63636

8 11 size 12{ { {8} over {"11"} } } {}

9 11 size 12{ { {9} over {"11"} } } {}

0.81818

10 11 size 12{ { {"10"} over {"11"} } } {}

Calculator problems

For the following problems, use a calculator to convert each fraction to a decimal. If no repeating pattern seems to exist, round to four decimal places.

16 125 size 12{ { {"16"} over {"125"} } } {}

0.128

85 311 size 12{ { {"85"} over {"311"} } } {}

192 197 size 12{ { {"192"} over {"197"} } } {}

0.9746

1 1469 size 12{ { {1} over {"1469"} } } {}

4 21 , 015 size 12{ { {4} over {"21","015"} } } {}

0.0002

81 , 426 106 , 001 size 12{ { {"81","426"} over {"106","001"} } } {}

16 , 501 426 size 12{ { {"16","501"} over {"426"} } } {}

38.7347

Exercises for review

( [link] ) Round 2,105,106 to the nearest hundred thousand.

( [link] ) 8 5 size 12{ { {8} over {5} } } {} of what number is 3 2 size 12{ { {3} over {2} } } {} ?

15 16 size 12{ { {"15"} over {"16"} } } {}

( [link] ) Arrange 1 9 16 size 12{1 { {9} over {"16"} } } {} , 1 5 8 size 12{1 { {5} over {8} } } {} , and 1 7 12 size 12{1 { {7} over {"12"} } } {} in increasing order.

( [link] ) Convert the complex decimal 3 . 6 5 4 size 12{3 "." 6 { {5} over {4} } } {} to a fraction.

3 29 40 size 12{3 { {"29"} over {"40"} } } {} or 3.725

( [link] ) Find the quotient. 30 ÷ 1 . 1 size 12{"30" div 1 "." 1} {} .

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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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