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Here you will learn about converting a repeating decimal to a fraction including how to define a repeating decimal.

Students will first learn about converting repeating decimals (often called recurring decimals) to fractions as part of the number system in 8 th grade.

Converting a **repeating decimal to fraction** is representing the repeating decimals as a fraction without changing its value.

A **repeating decimal (recurring decimal)** is a decimal number that has a digit (or group of digits) right of the decimal point that repeats forever. The part that repeats can also be shown by placing a line over the first to the last digit of the repeating pattern. Repeating decimals are also known as non-terminating decimals.

For example,

0.\overline{3}=0.333333\ldots 4.\overline{24}=4.242424\ldots 10.\overline{123}=10.123123\ldots 6.5\overline{8}=6.588888\ldotsare all repeating decimals.

To convert a repeating decimal to a fraction, you can form an equation. E.g.

x=1.\overline{3} \hspace{1cm} ①Keep track by labeling equations ①, ②, ③, etc.

You need to be able to multiply both sides of this equation confidently by different powers of 10. E.g.

10x=1.\overline{3} \hspace{1cm} ② 100x=13.\overline{3} \hspace{0.65cm} ③How does this relate to 8 th grade math?

**Grade 8 – The Number System (8.NS.A.1)**Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Use this quiz to check your grade 4 to 6 students’ understanding of converting fractions, decimals and percents. 10+ questions with answers covering a range of 4th, 5th and 6th grade converting fractions, decimals and percents topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 4 to 6 students’ understanding of converting fractions, decimals and percents. 10+ questions with answers covering a range of 4th, 5th and 6th grade converting fractions, decimals and percents topics to identify areas of strength and support!

DOWNLOAD FREEIn order to convert from a repeating decimal to a fraction, you need to:

**Equate the repeating decimal to a variable, \textbf{x}, to create \bf{①}.****Multiply both sides of \bf{①} by a power of \bf{10} to create \bf{②}.****Subtract \bf{②} from \bf{①}.****Divide the value by the coefficient of \textbf{x}.****Simplify the fraction.****State the answer.**

Convert 0.\overline{1} to a fraction.

**Equate the repeating decimal to a variable, \textbf{x}, to create \bf{①}.**

2**Multiply both sides of \bf{①} by a power of \bf{10} to create \bf{②}. **

As 0.\overline{1} has one repeating digit, multiply by 10. Remember, because you are multiplying the whole of ① by 10 , you also need to multiply the variable x by 10.

\begin{aligned}& 0.\overline{1}=x \hspace{0.8cm} ① \\\\ &0.\overline{1}\times 10=x\times10 \\\\ &1.\overline{1}=10x \hspace{0.5cm} ② \end{aligned}3**Subtract \bf{②} from \bf{①}. **

Subtract the first equation from the second equation.

\begin{aligned}&1.\overline{1}=10x \hspace{1cm} ② \\\\ -&0.\overline{1}=x \hspace{1.35cm} ① \\\\ &1.0=9x \end{aligned}4**Divide the value by the coefficient of \textbf{x}. **

As x is equal to 0.\overline{1} and \cfrac{1}{9}, it can be stated that 0.\overline{1} and \cfrac{1}{9} are equal to one another.

5**Simplify the fraction.**

This fraction cannot be simplified as the only factor that 1 and 9 share is 1.

6**State your answer.**

Convert 0.\overline{12} to a fraction.

**Equate the repeating decimal to a variable, \textbf{x}, to create \bf{①}. **

0.\overline{12}=x \hspace{0.8cm} ①

**Multiply both sides of \bf{①} by a power of \bf{10} to create \bf{②}. **

As 0.\overline{12} has two repeating digits, multiply by 100. Remember, because you are multiplying the whole of ① by 100, you also need to multiply the variable x by 100.

\begin{aligned}& 0.\overline{12}=x \hspace{1.2cm} ① \\\\ &0.\overline{12}\times 100=x\times100 \\\\ &12.\overline{12}=100x \hspace{0.5cm} ② \end{aligned}

**Subtract \bf{②} from \bf{①}. **

Subtract the first equation from the second equation.

\begin{aligned}&12.\overline{12}=100x \hspace{1cm} ② \\\\ -&0.\overline{12}=x \hspace{1.7cm} ① \\\\ &12=99x \end{aligned}

**Divide the value by the coefficient of \textbf{x}. **

\begin{aligned}12&=99x \\\\ \cfrac{12}{99}&=x\end{aligned}

**Simplify the fraction.**

3 is the greatest common factor of both 12 and 99 so, divide the numerator and the denominator by 3.

\cfrac{12\div3}{99\div3}=\cfrac{4}{33}

**State your answer.**

0.\overline{12}=\cfrac{4}{33}

Convert 0.0\overline{1} to a fraction.

**Equate the repeating decimal to a variable, \textbf{x}, to create \bf{①}. **

0.0\overline{1}=x \hspace{0.8cm} ①

**Multiply both sides of \bf{①} by a power of \bf{10} to create \bf{②}. **

As 0.0\overline{1} has one repeating digit, multiply by 10. Remember, because you are multiplying the whole of ① by 10 , you also need to multiply the variable x by 10.

\begin{aligned}& 0.0\overline{1}=x \hspace{0.8cm} ① \\\\ &0.0\overline{1}\times 10=x\times10 \\\\
&0.1\overline{1}=10x \hspace{0.5cm} ② \end{aligned}

**Subtract \bf{②} from \bf{①}. **

Subtract the first equation from the second equation.

\begin{aligned}&0.1\overline{1}=10x \hspace{1cm} ② \\\\ -&0.0\overline{1}=x \hspace{1.35cm} ① \\\\ &0.1=9x \end{aligned}

**Divide the value by the coefficient of \textbf{x}. **

\begin{aligned}0.1&=9x\\\\ \cfrac{0.1}{9}&=x\end{aligned}

**Simplify the fraction.**

\cfrac{0.1}{9}=x

Here, notice that there is a decimal as the numerator, so multiply the numerator and the denominator by 10 to remove the decimal.

\cfrac{0.1\times10}{9\times10}=\cfrac{1}{90}

You cannot simplify the fraction \cfrac{1}{90}.

**State the answer.**

0.0\overline{1}=\cfrac{1}{90}

Convert 0.\overline{23} to a fraction.

**Equate the repeating decimal to a variable, \textbf{x}, to create \bf{①}. **

0.\overline{23}=x \hspace{0.8cm} ①

**Multiply both sides of \bf{①} by a power of \bf{10} to create \bf{②}. **

As 0.\overline{23} has two repeating digits, multiply by 100. Remember, because you are multiplying the whole of ① by 100 , you also need to multiply the variable x by 100.

\begin{aligned}& 0.\overline{23}=x \hspace{1.5cm} ① \\\\ &0.\overline{23}\times 100=x\times100 \\\\ &23.\overline{23}=100x \hspace{0.8cm} ② \end{aligned}

**Subtract \bf{②} from \bf{①}. **

Subtract the first equation from the second equation.

\begin{aligned}&23.\overline{23}=100x \hspace{1cm} ② \\\\ -&0.\overline{23}=x \hspace{1.7cm} ① \\\\ &23.00=99x \end{aligned}

**Divide the value by the coefficient of \textbf{x}. **

\begin{aligned}23&=99x \\\\ \cfrac{23}{99}&=x\end{aligned}

**Simplify the fraction.**

This fraction cannot be simplified as the only factor that 23 and 99 share is 1.

**State the answer.**

0.\overline{23}=\cfrac{23}{99}

Convert 8.\overline{7} to a fraction.

**Equate the repeating decimal to a variable, \textbf{x}, to create \bf{①}. **

8.\overline{7}=x \hspace{0.8cm} ①

**Multiply both sides of \bf{①} by a power of \bf{10} to create \bf{②}. **

As 8.\overline{7} has one repeating digit, you will multiply by 10. Remember, because you are multiplying the whole of ① by 10 , you also need to multiply the variable x by 10.

\begin{aligned}& 8.\overline{7}=x \hspace{1.5cm} ① \\\\ &8.\overline{7}\times 10=x\times10 \\\\ &87.\overline{7}=10x \hspace{1cm} ② \end{aligned}

**Subtract \bf{②} from \bf{①}. **

Subtract the first equation from the second equation.

\begin{aligned}&87.\overline{7}=10x \hspace{1cm} ② \\\\ -&8.\overline{7}=x \hspace{1.55cm} ① \\\\ &79=9x \end{aligned}

**Divide the value by the coefficient of \textbf{x}. **

\begin{aligned}79&=9x \\\\ \cfrac{79}{9}&=x\end{aligned}

**Simplify the fraction.**

This fraction cannot be simplified as the only factor that 79 and 9 share is 1.

**State the answer.**

8.\overline{7}=\cfrac{79}{9}

Convert 4.0\overline{46} to a fraction.

**Equate the repeating decimal to a variable, \textbf{x}, to create \bf{①}. **

4.0\overline{46}=x \hspace{0.8cm} ①

**Multiply both sides of \bf{①} by a power of \bf{10} to create \bf{②}. **

As 4.0 \overline{46} has two repeating digits, you will multiply by 100. Remember, because you are multiplying the whole of ① by 100 , you also need to multiply the variable x by 100.

\begin{aligned}& 4.0\overline{46}=x \hspace{1.85cm} ① \\\\ &4.0\overline{46}\times 100=x\times100 \\\\ &404.6\overline{46}=100x \hspace{1cm} ② \end{aligned}

**Subtract \bf{②} from \bf{①}. **

Subtract the first equation from the second equation.

\begin{aligned}&404.6\overline{46}=100x \hspace{0.65cm} ② \\\\ -&4.0\overline{46}=x \hspace{1.55cm} ① \\\\ &400.6=99x \end{aligned}

**Divide the value by the coefficient of \textbf{x}. **

\begin{aligned}400.6&=99x \\\\ \cfrac{400.6}{99}&=x \end{aligned}

**Simplify the fraction.**

\cfrac{400.6}{99}=x

There is a decimal as the numerator, so you will multiply the numerator (and denominator) by 10.

\cfrac{400.6\times10}{99\times10}=\cfrac{4006}{990}

You can simplify the fraction by dividing the numerator and the denominator by their greatest common factor, 2.

\cfrac{4006\div2}{990\div2}=\cfrac{2003}{495}

**State the answer.**

4.0\overline{46}=\cfrac{2003}{495}

- Students should have a firm grasp on what recurring or repeating decimals are before beginning to convert them. Make sure students have a clear definition to refer back to.

- Encourage students to identify the repeating pattern in the decimals as the repeating digits. Make sure to emphasize that the recurring part of the decimal repeats indefinitely.

- Use of a converting repeating decimals to fractions worksheet will provide students with plenty of opportunities to practice.

- Post step-by-step instructions within the classroom or have students write steps down in a math journal to refer back to when needed.

**Multiplying by an incorrect power of \bf{10}**

You must make sure you are multiplying by the correct power of 10 \; (10, 100, 1000) so the repeating part of the decimal can be eliminated. For example,

Multiplying 0.\overline{13} by 10 does not help us eliminate the repeating decimal by subtraction. In this example you need to multiply by 100 because you can now eliminate the repeating aspect of the decimal by subtraction.

- Incorrect:

0 . \overline{13} \times 10=1 . \overline{31}

1.\overline{31}-0.\overline{13}=1.\overline{18} does not eliminate the repeating decimal.. - Correct:

0.\overline{13}\times100=13.\overline{13}

13.\overline{13}-0.\overline{13}=13 does eliminate the repeating decimal.

- Incorrect:

**Leaving the fraction with a decimal in the numerator**

Fractions should not include a decimal. For example, \cfrac{0.1}{2}.

In this example, multiply the numerator and denominator by 10 to be left with \cfrac{1}{20}.

**Not simplifying the fraction**

Always check to see if you have simplified the fraction into its simplest form.

The easiest way to do this is by using a divisibility check to see if the numerator and denominator are divisible by another number besides 1.

1. What should you multiply 0 . \overline{2} by to help eliminate the repeating decimal?

1

10

2

5

If the decimal has one repeating digit, then multiply by 10.

0.\overline{2}\times10=2.\overline{2}

2. What should you multiply 0.\overline{14} by to help eliminate the repeating decimal?

1

10

100

14

If the decimal has two repeating digits, then multiply by 100.

0.\overline{14}\times100=14.\overline{14}

3. What should you multiply 0.0\overline{15} by to help eliminate the repeating decimal?

1

10

100

1000

If the decimal has two repeating digits, then multiply by 100.

0.0\overline{15}\times100=1.5\overline{15}

4. What is 0.\overline{2} as a fraction?

\cfrac{2}{9}

\cfrac{2}{10}

\cfrac{2}{99}

\cfrac{2}{999}

Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of 10.

\begin{aligned}&0.\overline{2}=x \hspace{1cm} ① \\\\ &0.\overline{2}\times10=x\times10 \\\\ &2.\overline{2}=10x \hspace{0.65cm} ② \end{aligned}

Subtract ① from ②.

\begin{aligned}&2.\overline{2}=10x \hspace{1cm} ② \\\\ -&0.\overline{2}=x \hspace{1.35cm} ① \\\\ &2.0=9x \end{aligned}

Divide both sides of the equation by 9

\cfrac{2}{9}=x

5. What is 0.\overline{34} as a fraction?

\cfrac{34}{100}

\cfrac{34}{99}

\cfrac{3}{33}

\cfrac{34}{999}

Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of 10.

\begin{aligned}&0.\overline{34}=x \hspace{2.0cm} ① \\\\ &0.\overline{34}\times100=x\times100 \\\\ &34.\overline{34}=100x \hspace{1.35cm} ② \end{aligned}

Subtract ① from ②.

\begin{aligned}&34.\overline{34}=100x \hspace{1cm} ② \\\\ -&0.\overline{34}=x \hspace{1.7cm} ① \\\\ &34.00=99x \end{aligned}

Divide both sides of the equation by 99 .

\cfrac{34}{99}=x

This does not simplify further.

6. What is 6.\overline{7} as a fraction?

6

\cfrac{60}{99}

\cfrac{60}{100}

\cfrac{61}{9}

Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of 10.

\begin{aligned}&6.\overline{7}=x \hspace{1cm} ① \\\\ &6.\overline{7}\times10=x\times10 \\\\ &67.\overline{7}=10x \hspace{0.65cm} ② \end{aligned}

Subtract ① from ②.

\begin{aligned}&67.\overline{7}=10x \hspace{1cm} ② \\\\ -&6.\overline{7}=x \hspace{1.55cm} ① \\\\ &61.0=9x \end{aligned}

Divide both sides of the equation by 9 .

\cfrac{61}{9}=x

This does not simplify further.

No, both terms can be used synonymously.

A surd is an irrational number expressed in radical form. They cannot be simplified to whole numbers or fractions.

Terminating decimals are decimals that contain a finite number of digits after the decimal point.

Yes. These are called irrational numbers and include pi (\pi=3.14159265\ldots) and Euler’s number (e=2.7182818284590\ldots).

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