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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

Have you ever eaten a part of a pie and you couldn't figure out how much you ate?

While Daniel ate peach pie, Natalia ate blueberry. Her pie was cut into nine pieces and she ate only two of them.

What fraction of the pie did she eat? What is this fraction as a decimal?

In this Concept, you will learn about repeating decimals. Then you will see how this applies to Natalia.

### Guidance

By now you’ve gotten the hang of converting fractions to decimals. So far, we have been working with what are known as terminating decimals, or decimals that have an end like 0.75 or 0.5.

One reason that we sometimes use fractions instead of decimals is because some decimals are repeating decimals, or decimals that go on forever. If you try to find a decimal for 13\begin{align*}\frac{1}{3}\end{align*} by dividing, you can divide forever because 13\begin{align*}\frac{1}{3}\end{align*} written as a decimal =0.3333333333....\begin{align*}= 0.3333333333 ....\end{align*} It goes on and on. That’s why we usually just simply write a line above the number that repeats. For 13\begin{align*}\frac{1}{3}\end{align*}, we write: 0.3¯\begin{align*}0.\overline{3}\end{align*}. Let’s check out some problems involving repeating decimals.

Write 56\begin{align*}\frac{5}{6}\end{align*} using decimals

First, we rewrite 56\begin{align*}\frac{5}{6}\end{align*} as the division problem 5÷6\begin{align*}5 \div 6\end{align*}. We already know that we will have to go on the right side of the decimal point, so we are going to begin by dividing 6 into 5.0.

Six goes into 5.0 .8 times, but we have the remainder of .2. Six goes into 0.2 .03 times and we have a remainder of .02. Since 6 always goes into 20 three times, (36=18)\begin{align*}(3 \cdot 6 = 18)\end{align*} and there will always be a remainder of 2, we can see that it will never evenly divide.

If you keep dividing, you will get 0.83333333333.... forever and ever.

Our final answer is 0.83¯\begin{align*}0.8\overline{3}\end{align*}.

Well, there are some mixed numbers where the fraction part is a repeating decimal. Let’s look at one like this.

Write 223\begin{align*}2 \frac{2}{3}\end{align*} using decimals.

Just as we did with the terminating decimals, we are going to leave the whole number, 2 to the side before we are ready to add it to the final answer. So, we are simply solving for the decimal equivalent of 23\begin{align*}\frac{2}{3}\end{align*}. We write the division problem 2.0÷3\begin{align*}2.0 \div 3\end{align*}. How many times does 3 go into 2.0? It goes into 2.0 0.6 times.

We have 0.20 as the remainder. How many times does 3 go into 0.20? The answer is 0.06 times.

Are you noticing a pattern here? It is obvious that there will always be a remainder whether we divide 3 into 2.0, 0.2, 0.02, 0.002, or 0.0002 and on and on. Clearly 23\begin{align*}\frac{2}{3}\end{align*} is a repeating decimal.

For our final answer we write 2.6¯\begin{align*}2.\overline{6}\end{align*}.

Try a few of these on your own. Convert each to a repeating decimal.

#### Example A

16\begin{align*}\frac{1}{6}\end{align*}

Solution: .16¯\begin{align*}.1\overline{6}\end{align*}

#### Example B

446\begin{align*}4 \frac{4}{6}\end{align*}

Solution:4.6¯\begin{align*}4.\overline{6}\end{align*}

#### Example C

49\begin{align*}\frac{4}{9}\end{align*}

Solution:.4¯\begin{align*}.\overline{4}\end{align*}

Here is the original problem once again.

While Daniel ate peach pie, Natalia ate blueberry. Her pie was cut into nine pieces and she ate only two of them.

What fraction of the pie did she eat? What is this fraction as a decimal?

First, let's write the fraction of two out of nine.

29\begin{align*}\frac{2}{9}\end{align*}

Now let's divide the numerator by the denominator.

0.222222222\begin{align*}0.222222222\end{align*}

We have repeating decimal. We can rewrite this repeating decimal using different notation as it was presented in the Concept.

.2¯\begin{align*}.\overline{2}\end{align*}

### Vocabulary

Fraction
a part of a whole written using a numerator and a denominator and a fraction bar
Decimal
a part of a whole written using a decimal point and place value
Mixed Number
a number written with a whole number and a fraction.
Terminating Decimal
a decimal with an ending.
Repeating Decimal
a decimal that does not end but repeats and repeats

### Guided Practice

Here is one for you to try on your own.

Write 216\begin{align*}2 \frac{1}{6}\end{align*} as a decimal.

To do this, first, we have to convert the mixed number to an improper fraction.

136\begin{align*}\frac{13}{6}\end{align*}

Now we divide the numerator by the denominator.

2.1666666\begin{align*}2.1666666\end{align*}

We have a repeating decimal. We can rewrite this using decimal notation.

2.16¯\begin{align*}2.1\overline6\end{align*}

### Video Review

http://www.youtube.com/watch?v=UHQrykNrlOM - This is a James Sousa video on writing fractions as repeating decimals.

### Practice

Directions: Write each fraction or mixed number as a repeating decimal or terminating decimal.

1. 23\begin{align*}\frac{2}{3}\end{align*}

2. 56\begin{align*}\frac{5}{6}\end{align*}

3. 513\begin{align*}5 \frac{1}{3}\end{align*}

4. 13\begin{align*}\frac{1}{3}\end{align*}

5. 34\begin{align*}\frac{3}{4}\end{align*}

6. 16\begin{align*}\frac{1}{6}\end{align*}

7. 18\begin{align*}\frac{1}{8}\end{align*}

8. 37\begin{align*}\frac{3}{7}\end{align*}

9. 26\begin{align*}\frac{2}{6}\end{align*}

10. 423\begin{align*}4 \frac{2}{3}\end{align*}

11. 713\begin{align*}7 \frac{1}{3}\end{align*}

12. 656\begin{align*}6 \frac{5}{6}\end{align*}

13. 812\begin{align*}8 \frac{1}{2}\end{align*}

14. 923\begin{align*}9 \frac{2}{3}\end{align*}

15. 1115\begin{align*}11 \frac{1}{5}\end{align*}

### Vocabulary Language: English

Decimal

Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).
fraction

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
Mixed Number

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.