# 6.2 Converting a decimal to a fraction

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

## Section overview

• Converting an Ordinary Decimal to a Fraction
• Converting a Complex Decimal to a Fraction

## Converting an ordinary decimal to a fraction

We can convert a decimal fraction to a fraction, essentially, by saying it in words, then writing what we say. We may have to reduce that fraction.

## Sample set a

Convert each decimal fraction to a proper fraction or a mixed number. Reading: six tenths→ $\frac{6}{10}$ .

Reduce: $\frac{3}{5}$ . Reading: nine hundred three thousands→ $\frac{\text{903}}{\text{1000}}$ . Reading: eighteen and sixty-one hundredths→ $\text{18}\frac{\text{61}}{\text{100}}$ . Reading: five hundred eight and five ten thousandths→ $\text{508}\frac{5}{\text{10},\text{000}}$ .

Reduce: $\text{508}\frac{1}{2,\text{000}}$ .

## Practice set a

Convert the following decimals to fractions or mixed numbers. Be sure to reduce.

16.84

$\text{16}\frac{\text{21}}{\text{25}}$

0.513

$\frac{\text{513}}{1,\text{000}}$

6,646.0107

$6,\text{646}\frac{\text{107}}{\text{10},\text{000}}$

1.1

$1\frac{1}{\text{10}}$

## Complex decimals

Numbers such as $0\text{.}\text{11}\frac{2}{3}$ are called complex decimals . We can also convert com­plex decimals to fractions.

## Sample set b

Convert the following complex decimals to fractions.

$0\text{.}\text{11}\frac{2}{3}$

The $\frac{2}{3}$ appears to occur in the thousands position, but it is referring to $\frac{2}{3}$ of a hundredth. So, we read $0\text{.}\text{11}\frac{2}{3}$ as "eleven and two-thirds hundredths."

$\begin{array}{ccc}0.11\frac{2}{3}=\frac{\text{11}\frac{2}{3}}{\text{100}}& =& \frac{\frac{\text{11}\cdot 3+2}{3}}{\text{100}}\\ & =& \frac{\frac{\text{35}}{3}}{\frac{\text{100}}{1}}\hfill \\ & =& \frac{\text{35}}{3}÷\frac{\text{100}}{1}\hfill \\ & =& \frac{\stackrel{7}{\overline{)35}}}{3}\cdot \frac{1}{\underset{\text{20}}{\overline{)100}}}\hfill \\ & =& \frac{7}{\text{60}}\hfill \end{array}$

$4\text{.}\text{006}\frac{1}{4}$

Note that $4\text{.}\text{006}\frac{1}{4}=4+\text{.}\text{006}\frac{1}{4}$

$\begin{array}{ccc}\hfill 4+.006\frac{1}{4}& =& 4+\frac{6\frac{1}{4}}{\text{1000}}\hfill \\ & =& 4+\frac{\frac{\text{25}}{4}}{\frac{\text{1000}}{1}}\hfill \\ & =& 4+\frac{\stackrel{1}{\overline{)25}}}{4}\cdot \frac{1}{\underset{40}{\overline{)1000}}}\hfill \\ & =& 4+\frac{1\cdot 1}{4\cdot \text{40}}\hfill \\ & =& 4+\frac{1}{\text{160}}\hfill \\ & =& 4\frac{1}{\text{160}}\hfill \end{array}$

## Practice set b

Convert each complex decimal to a fraction or mixed number. Be sure to reduce.

$0\text{.}8\frac{3}{4}$

$\frac{7}{8}$

$0\text{.}\text{12}\frac{2}{5}$

$\frac{\text{31}}{\text{250}}$

$6\text{.}\text{005}\frac{5}{6}$

$6\frac{7}{1,\text{200}}$

$\text{18}\text{.}1\frac{3}{\text{17}}$

$\text{18}\frac{2}{\text{17}}$

## Exercises

For the following 20 problems, convert each decimal fraction to a proper fraction or a mixed number. Be sure to reduce.

0.7

$\frac{7}{\text{10}}$

0.1

0.53

$\frac{\text{53}}{\text{100}}$

0.71

0.219

$\frac{\text{219}}{1,\text{000}}$

0.811

4.8

$4\frac{4}{5}$

2.6

16.12

$\text{16}\frac{3}{\text{25}}$

25.88

6.0005

$6\frac{1}{2,\text{000}}$

1.355

16.125

$\text{16}\frac{1}{8}$

0.375

3.04

$3\frac{1}{\text{25}}$

21.1875

8.225

$8\frac{9}{\text{40}}$

1.0055

9.99995

$9\frac{\text{19},\text{999}}{\text{20},\text{000}}$

22.110

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

$0\text{.}7\frac{1}{2}$

$\frac{3}{4}$

$0\text{.}\text{012}\frac{1}{2}$

$2\text{.}\text{16}\frac{1}{4}$

$2\frac{\text{13}}{\text{80}}$

$5\text{.}\text{18}\frac{2}{3}$

$\text{14}\text{.}\text{112}\frac{1}{3}$

$\text{14}\frac{\text{337}}{3,\text{000}}$

$\text{80}\text{.}\text{0011}\frac{3}{7}$

$1\text{.}\text{40}\frac{5}{\text{16}}$

$1\frac{\text{129}}{\text{320}}$

$0\text{.}8\frac{5}{3}$

$1\text{.}9\frac{7}{5}$

$2\frac{1}{\text{25}}$

$1\text{.}7\frac{\text{37}}{9}$

## Exercises for review

( [link] ) Find the greatest common factor of 70, 182, and 154.

14

( [link] ) Find the greatest common multiple of 14, 26, and 60.

( [link] ) Find the value of $\frac{3}{5}\cdot \frac{\text{15}}{\text{18}}÷\frac{5}{9}$ .

$\frac{9}{\text{10}}$

( [link] ) Find the value of $5\frac{2}{3}+8\frac{1}{\text{12}}$ .

( [link] ) In the decimal number 26.10742, the digit 7 is in what position?

thousandths

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