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# Repeating Decimals

## Identify repeating decimals by dividing the numerator of a fraction by the denominator.

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Repeating Decimals
License: CC BY-NC 3.0

Jose has 10 bars of chocolate that he needs to give to 3 of his friends. How many bars of chocolate does each friend receive?

In this concept, you will learn to write fractions and mixed numbers as repeating decimals.

### Writing Fractions and Mixed Numbers as Repeating Decimals

A terminating decimal is a decimal number that does not go on forever. The word “terminate” means to end. Most of the fractions you have been working with are terminating decimals.

Here is a fraction with a terminating decimal.

14\begin{align*}\frac{1}{4}\end{align*}

Divide 1 by 4 to find the decimal value.

4)1.00¯¯¯¯¯¯¯¯¯¯¯¯0.25   8   20200\begin{align*}\begin{array}{rcl} && \overset{ \quad 0.25}{4 \overline{ ) {1.00 \;}}}\\ && \ \ \ \underline{-8}\\ && \quad \ \ \ 20 \\ && \quad \underline{ -20} \\ && \qquad 0 \end{array}\end{align*}

You use zero placeholders, but ultimately, the decimal will divide evenly.

A decimal that does not end and repeats the same number or numbers over and over again is called a repeating decimal. When you divide the numerator by the denominator and keep ending up with the same number, you might have a repeating decimal.

Convert 23\begin{align*}\frac{2}{3}\end{align*} to a decimal.

First, this does not have a base ten denominator. Divide the numerator by the denominator.

4)2.000¯¯¯¯¯¯¯¯¯¯¯¯¯  0.66618  20   1820 18  2\begin{align*}\begin{array}{rcl} && \overset{ \ \ 0.666}{4 \overline{ ) {2.000 }}}\\ && \underline{ -\; 18}\\ && \quad \ \ 20 \\ && \ \ \ \underline{ -18 }\\ && \qquad 20\\ && \quad \ \underline{ -18 }\\ && \qquad \ \ 2 \end{array}\end{align*}

The same remainder keeps showing up and the quotient becomes a series of 6’s. It does not matter if you keep adding zero placeholders. A repeating decimal is indicated by adding a line over the last digit or series of digits in the quotient that repeats itself.

The decimal value of 23\begin{align*}\frac{2}{3}\end{align*} is 0.6¯\begin{align*}0.\bar{6}\end{align*} .

### Examples

#### Example 1

Earlier, you were given a problem about Jose and his chocolate bars.

Jose wants to give 10 chocolate bars to 3 of his friends. Divide 10 by 3 to find how many chocolate bar each friend receives.

Divide 10 by 3.

3)10.000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  3.333910910910 91¯¯¯¯¯¯¯¯¯\begin{align*}& \overset{ \ \ 3.333}{3 \overline{ ) {10.000 \;}}}\\ & \underline{ \; \;\; -9}\\ & \quad 10 \\ & \underline{ \; \; \;-9}\\ & \quad 10 \\ & \underline{ \; \; \;-9}\\ & \quad 10 \\ & \ -9 \\ & \overline{ \; \; \; \; \;1}\end{align*}

The answer is a repeating decimal 3.3¯\begin{align*}3. \bar{3}\end{align*}.

Jose can give each friend 3.3¯\begin{align*}3.\bar{3}\end{align*} bars of chocolate.

#### Example 2

Is 49\begin{align*}\frac{4}{9}\end{align*} a repeating decimal or a terminating decimal?

Convert the fraction to a decimal. Divide 4 by 9.

9)4.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  0.444436  40   3640 36  40   36   4\begin{align*}\begin{array}{rcl} && \overset{ \ \ 0.4444}{9 \overline{ ) {4.0000 \;}}}\\ && \underline{- \; 36}\\ && \quad \ \ 40 \\ && \ \ \ \underline{-36}\\ && \qquad 40\\ && \quad\ \underline{-36} \\ && \qquad \ \ 40\\ && \quad \ \ \ \underline{-36} \\ && \qquad \ \ \ 4 \end{array}\end{align*}

The same remainder keeps showing up and the quotient will go on and on as a series of 4s.

The decimal value of 49\begin{align*}\frac{4}{9}\end{align*} is a repeating decimal, 0.4¯\begin{align*}0.\bar{4}\end{align*}.

#### Example 3

Determine if the fraction is a repeating or terminating decimal.

13\begin{align*}\frac{1}{3}\end{align*}

Convert the fraction to a decimal. Divide 1 by 3.

3)1.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  0.3333   9  10 910   9  10 91\begin{align*}\begin{array}{rcl} && \overset{ \ \ 0.3333}{3 \overline{ ) {1.0000 \;}}}\\ && \ \ \ \underline{ -9}\\ && \quad \ \ 10 \\ && \quad \ \underline{-9} \\ && \qquad 10\\ && \quad \ \ \ \underline{ -9} \\ && \qquad \ \ 10\\ && \qquad \ \underline{ -9} \\ && \qquad \quad 1 \end{array}\end{align*}

The decimal value of 13\begin{align*}\frac{1}{3}\end{align*} is a repeating decimal, 0.3¯\begin{align*}0. \bar{3}\end{align*}.

#### Example 4

Determine if the fraction is a repeating or terminating decimal.

18\begin{align*}\frac{1}{8}\end{align*}

Convert the fraction to a decimal. Divide 1 by 8.

8)1.000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  0.125  8  20   1640 40  0\begin{align*}\begin{array}{rcl} && \overset{ \ \ 0.125}{8 \overline{ ) {1.000 \;}}}\\ && \ \ \underline{ \;-8}\\ &&\quad \ \ 20 \\ && \ \ \ \underline{ -16} \\ && \qquad 40\\ && \quad \ \underline{ -40} \\ && \qquad \ \ 0 \end{array}\end{align*}

The decimal value of 18\begin{align*}\frac{1}{8}\end{align*} is a terminating decimal, 0.125.

#### Example 5

Determine if the fraction is a repeating or terminating decimal.

512\begin{align*}5 \frac{1}{2}\end{align*}

First, convert the fraction part to a decimal.

12=0.5\begin{align*}\frac{1}{2} = 0.5\end{align*}

Then, place the whole number to the left of the decimal point.

512=5.5\begin{align*}5 \frac{1}{2}=5.5\end{align*}

The decimal value of 512\begin{align*}5 \frac{1}{2}\end{align*} is a terminating decimal, 5.5.

### Review

Determine if the fractions are repeating or terminating decimals.

1. 143\begin{align*}\frac{14}{3}\end{align*}
2. 349\begin{align*}\frac{34}{9}\end{align*}
3. 233\begin{align*}\frac{23}{3}\end{align*}
4. 174\begin{align*}\frac{17}{4}\end{align*}
5. 196\begin{align*}\frac{19}{6}\end{align*}
6. 125\begin{align*}\frac{12}{5}\end{align*}
7. 313\begin{align*}3 \frac{1}{3}\end{align*}
8. 812\begin{align*}8 \frac{1}{2}\end{align*}
9. 923\begin{align*}9 \frac{2}{3}\end{align*}
10. 1145\begin{align*}11 \frac{4}{5}\end{align*}
11. 1614\begin{align*}16 \frac{1}{4}\end{align*}
12. 443\begin{align*}\frac{44}{3}\end{align*}
13. 667\begin{align*}\frac{66}{7}\end{align*}
14. 184\begin{align*}\frac{18}{4}\end{align*}
15. 747\begin{align*}\frac{74}{7}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.21.

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### Vocabulary Language: English

TermDefinition
Repeating Decimal A repeating decimal is a decimal number that ends with a group of digits that repeat indefinitely. 1.666... and 0.9898... are examples of repeating decimals.
Terminating Decimal A terminating decimal is a decimal number that ends. The decimal number 0.25 is an example of a terminating decimal.

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