# 5.21: Repeating Decimals

**At Grade**Created by: CK-12

**Practice**Repeating Decimals

Have you ever had a math problem that you couldn't figure out?

Well, Josie is having a difficult time trying to figure one out. Josie is in charge of organizing the sixth and seventh grade into six teams for field day. This would seem like an easy task, except that there are 49 students to split up.

Josie wrote the following problem.

\begin{align*}\frac{49}{6}\end{align*}

She figured it would be easier to think of the problem in terms of an improper fraction. But that is where the trouble began. She divided to convert the improper fraction to a mixed number but the answer came out funny.

This is Josie's answer.

\begin{align*}8.1666666\end{align*}

The sixes kept going and going.

Josie is puzzled and isn't sure what this means at all. Do you know?

**This Concept is about repeating decimals. At the end of it, you will know how to help Josie.**

### Guidance

When we can convert a fraction by dividing the numerator by the denominator evenly to form a decimal, we call this a ** terminating decimal**. The word “terminate” means to end. All of the fractions we have been working with are terminating decimals.

\begin{align*}7\frac{1}{4}\end{align*}

Here the 7 is our whole number and so it is placed to the left of the decimal point. We divide 1 by 4 to get the decimal part.

\begin{align*}& \overset{ \quad \ .25}{4 \overline{ ) {1.00 \;}}}\\ & \ \ \underline{-8}\\ & \quad \ \ 20\\ & \quad \underline{-20}\\ & \qquad \ 0 \end{align*}

**This is a terminating decimal. It is called that because once you added the decimal point and the zero placeholders, you were able to divide the dividend by the divisor evenly.**

**What do we call a decimal that is NOT a terminating decimal?**

A decimal that does not end and repeats the same number over and over again is called a ** repeating decimal.** You know that you have a repeating decimal if when you divide the numerator by the denominator, if you keep ending up with the same number.

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

First, this does not have a base ten denominator so we will divide the numerator by the denominator.

\begin{align*}& \overset{ \quad \ .666}{3 \overline{ ) {2.000 \;}}}\\ & \ \underline{-18}\\ & \quad \ \ 20\\ & \quad \underline{-18}\\ & \qquad \ 20 \\ & \quad \ \ \underline{-18}\\ & \qquad \quad 2 \end{align*}

*Look at what happens as we divide!!! The same remainder keeps showing up and our quotient becomes a series of 6’s. It doesn’t matter if we keep adding zeros forever, our decimal will always repeat.***When you have a decimal that is a repeating decimal, we can add a line over the last digit in the quotient. This is a clue that the decimal repeats.**

**Our answer is** \begin{align*}.66\bar{6}\end{align*}.

Divide these fractions and see if you end up with any repeating decimals.

#### Example A

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

**Solution: \begin{align*}.333333\end{align*} is a repeating decimal**

#### Example B

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

**Solution:\begin{align*}.125\end{align*}**

#### Example C

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

**Solution:\begin{align*}5.5\end{align*}**

Now back to Josie and the teams. Here is the original problem once again.

Josie is having a difficult time trying to figure one out. Josie is in charge of organizing the sixth and seventh grade into six teams for field day. This would seem like an easy task, except that there are 49 students to split up.

Josie wrote the following problem.

\begin{align*}\frac{49}{6}\end{align*}

She figured it would be easier to think of the problem in terms of an improper fraction. But that is where the trouble began. She divided to convert the improper fraction to a mixed number but the answer came out funny.

This is Josie's answer.

\begin{align*}8.1666666\end{align*}

The sixes kept going and going.

Josie is puzzled and isn't sure what this means at all. Do you know?

Josie's improper fraction converted to a decimal called a repeating decimal. This means that the values would go on an on indefinitely.

Josie can't evenly divide 49 students onto 6 teams. One team will have an extra player.

If she divides 48 students into 6 teams, there are 8 on each team.

Notice that this is the whole number in the decimal.

She can add the extra student to one of the teams and everything will be fine.

### Vocabulary

Here are the vocabulary words in this Concept.

- Terminating Decimal
- decimal that can be found dividing a numerator and denominator and by adding a decimal point and zero placeholders.

- Repeating Decimal
- a decimal where the digits in the quotient repeat themselves, can be indicated by putting a small line over the second repeating digit.

### Guided Practice

Here is one for you to try on your own.

Is \begin{align*}4 \frac{4}{7}\end{align*} a repeating decimal or a terminating decimal?

**Answer**

To figure this out, let's first convert it to a decimal.

\begin{align*}4 \frac{4}{7} = \frac{32}{7} = 4.5714285\end{align*}

**While this is a long decimal, it is a terminating and not a repeating decimal.**

### Video Review

Here are videos for review.

James Sousa Example of Fractions to a Terminating Decimal

James Sousa Another Example of Fractions to a Terminating Decimal

James Sousa Example of Fractions to a Repeating Decimal

### Practice

Directions:Determine whether each fraction or mixed number is a terminating or repeating decimal.

1. \begin{align*}\frac{14}{3}\end{align*}

2. \begin{align*}\frac{34}{9}\end{align*}

3. \begin{align*}\frac{23}{3}\end{align*}

4. \begin{align*}\frac{17}{4}\end{align*}

5. \begin{align*}\frac{19}{6}\end{align*}

6. \begin{align*}\frac{12}{5}\end{align*}

7. \begin{align*}3\frac{1}{3}\end{align*}

8. \begin{align*}8\frac{1}{2}\end{align*}

9. \begin{align*}9\frac{2}{3}\end{align*}

10. \begin{align*}11\frac{4}{5}\end{align*}

11. \begin{align*}16\frac{1}{4}\end{align*}

12. \begin{align*}\frac{44}{3}\end{align*}

13. \begin{align*}\frac{66}{7}\end{align*}

14. \begin{align*}\frac{18}{4}\end{align*}

15. \begin{align*}\frac{74}{7}\end{align*}

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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.