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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.

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

Divide 1 by 4 to find the decimal value.

\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*}4)1.00¯¯¯¯¯¯¯¯¯¯¯¯0.25   8   20200

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

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

\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*}4)2.000¯¯¯¯¯¯¯¯¯¯¯¯¯  0.66618  20   1820 18  2

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

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.

\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*}3)10.000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  3.333910910910 91¯¯¯¯¯¯¯¯¯

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

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

Example 2

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

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

\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*}9)4.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  0.444436  40   3640 36  40   36   4 

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

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

Example 3

Determine if the fraction is a repeating or terminating decimal.

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

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

\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*}3)1.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  0.3333   9  10 910   9  10 91

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

Example 4

Determine if the fraction is a repeating or terminating decimal.

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

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

\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*}8)1.000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  0.125  8  20   1640 40  0

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

Example 5

Determine if the fraction is a repeating or terminating decimal.

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

First, convert the fraction part to a decimal.

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

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

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

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

Review

Determine if the fractions are repeating or terminating decimals.

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

Review (Answers)

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

Resources

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Vocabulary

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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