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In order to access this I need to be confident with:

Place value

Multiplying and dividing by 10, 100, 1000

Simplifying a fraction

Addition and subtraction

This topic is relevant for:

Here we will learn about **converting recurring decimals** to fractions including how to define a recurring decimal.

There are also converting recurring decimals to fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

Converting **recurring decimals to fractions** is representing a recurring decimal as a fraction without changing its value.

A **recurring decimal **is a decimal number that has a digit (or group of digits) that repeats forever. The part that repeats can also be shown by placing dots over the first and last digits of the repeating pattern.

E.g.

\[\begin{aligned}
&0 . \dot{3} = 0.333333… \\\\
&0 . \dot{2}\dot{4} = 0.24242424… \\\\
&0 . \dot{1}\dot{2}\dot{3} = 0.123123123…
\end{aligned}\]

are all recurring decimals

*Note: these are sometimes called repeating decimals *

In order to convert from a recurring decimal to a fraction:

1**Equate the recurring decimal to a variable, we will use **

2**Multiply both sides of Equation 1 by a power of 10 so the recurring parts of the decimals align in regards to their place value, this creates Equation 2**

- If the decimal has
**one repeating digit**, then multiply by**10** - If the decimal has
**two repeating digits**, then multiply by**100** - If the decimal has
**three repeating digits**, then multiply by**1000**etc.

3**Subtract Equation 2 from Equation 1**

4**Divide the value by the coefficient of x**

5**Simplify the fraction**

6**Clearly state your answer ‘decimal=fraction’**

Get your free recurring decimals to fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free recurring decimals to fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONConvert 0.\dot{1} to a fraction

**Equate the recurring decimal to a variable, we will use**x,

\[0 . \dot{1}=x \hspace{3cm} \text{Equation 1}\]

2**Multiply both sides of Equation 1 by a power of 10 so the recurring element of the decimals align (this creates Equation 2)**

You now need to multiply 0 . \dot{1} by a power of 10 (e.g 10,100,1000) so the recurring elements of the decimal align in regards to their place value.

Because 0 . \dot{1} has one repeating digit, we will multiply by 10

\[0 . \dot{1}\times10=1.\dot{1}\]

Remember because you are multiplying the whole of Equation 1 you also need to multiply the variable

\[\begin{aligned}
0 . \dot{1}&=x \hspace{3cm} \text{Equation 1} \\\\
0 . \dot{1} \times 10&=x \times 10 \\\\
1.\dot{1} &=10 x \hspace{2.5cm} \text{Equation 2}
\end{aligned}
\]

3**Subtract Equation 2 from Equation 1**

\[\begin{aligned}
1 . \dot{1}&=10x \hspace{3cm} \text{Equation 2}\\\\
0. \dot{1}&=x \hspace{3.5cm} \text{Equation 1} \\\\
1&=9 x
\end{aligned}
\]

4**Divide the value by the coefficient of x**

\[\begin{aligned}
1&=9x\\\\
\frac{1}{9}&=x
\end{aligned}
\]

We originally stated that

If

5**Simplify the fraction**

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

6**Clearly state your answer ‘decimal=fraction’**

\[0 . \dot{1}= \frac{1}{9}\]

Convert 0 .\dot{1}\dot{2} to a fraction

**Equate the recurring decimal to a variable to create Equation 1 **

\[0 .\dot{1}\dot{2}=x \hspace{3cm} \text{Equation 1} \]

**Multiply both sides of Equation 1 by a power of 10 so the recurring element of the decimals align (this creates Equation 2 )**

As 0.\dot{1}\dot{2} has two repeating digits, we will multiply by 100

\[0 . \dot{1}\dot{2} \times100=12 .\dot{1}\dot{2}\]

Remember because you are multiplying the whole of Equation 1 by 100 you also need to multiply the variable

\[\begin{aligned}
0.\dot{1}\dot{2}&=x \hspace{3.8cm} \text{Equation 1}
\\\\
0.\dot{1}\dot{2}\times100&=x\times100\\\\
12.\dot{1}\dot{2}&=100x \hspace{3cm} \text{Equation 2}
\end{aligned}
\]

**Subtract Equation 2 from Equation 1 **

\[\begin{aligned}
12.\dot{1}\dot{2}&=100x \hspace{3cm} \text{Equation 2}\\\\
0.\dot{1}\dot{2}&=x \hspace{3.8cm} \text{Equation 1} \\\\
12&=99x
\end{aligned}
\]

**Divide the value by the coefficient of x **

\[\begin{aligned}
\ 12&=99x\\\\
\frac{12}{99}&=x
\end{aligned}
\]

**Simplify the fraction**

\[\frac{12}{99}=x
\]

Here you can divide the numerator and the denominator by 3 because 3 is the highest common factor of both 12 and 99

\[\begin{array}{l}
\frac{12 \div 3}{99 \div 3} \\\\
=\frac{4}{33}
\end{array}
\]

**Clearly state your answer ‘decimal=fraction’**

\[0 . \dot{1}\dot{2}=\frac{4}{33}
\]

Convert 0.0\dot{1} to a fraction

**Equate the recurring decimal to a variable to create Equation 1 **

\[0.0\dot{1} = x \hspace{3cm} \text{Equation 1} \]

**Multiply both sides of Equation 1 by a power of 10 to create Equation 2 **

As 0.0\dot{1} has one repeating digit, we will multiply by 10

\[0 . 0\dot{1} \times10=0.\dot{1}\]

Remember because you are multiplying the whole of Equation 1 by 10 you also need to multiply the variable

\[\begin{aligned}
0.0\dot{1}&=x \hspace{3.7cm} \text{Equation 1}
\\\\
0.0\dot{1}\times10&=x\times10\\\\
0.\dot{1}&=10x \hspace{3.2cm} \text{Equation 2}
\end{aligned}
\]

**Subtract Equation 2 from Equation 1 **

\[\begin{aligned}
0.\dot{1}&=10x \hspace{3cm} \text{Equation 2}\\\\
0.0\dot{1}&=x \hspace{3.6cm} \text{Equation 1} \\\\
0.1&=9x
\end{aligned}
\]

**Divide the value by the coefficient of x **

\[\begin{aligned}
\ 0.1&=9x\\\\
\frac{0.1}{9}&=x
\end{aligned}
\]

**Simplify the fraction**

\[\frac{0.1}{9}=x
\]

Here you will notice that we have a decimal as the numerator so we need to multiply the numerator by 10. To make sure we do not change the value of the denominator we also need to multiply the denominator by 10.

\[\begin{array}{l}
\frac{0.1\times10}{9\times10} \\\\
=\frac{1}{90}
\end{array}
\]

We cannot simplify the fraction \frac{1}{90}

**Clearly state your answer ‘decimal=fraction’**

\[0 . 0\dot{1}=\frac{1}{90}
\]

Convert 0.\dot{2}\dot{3} to a fraction

**Equate the recurring decimal to a variable to create Equation 1 **

\[0.\dot{2}\dot{3} = x \hspace{3cm} \text{Equation 1} \]

**Multiply both sides of Equation 1 by a power of 10 to create Equation 2 **

As 0.\dot{2}\dot{3} has two repeating digits, we will multiply by 100

\[0 . \dot{2}\dot{3} \times100=23 .\dot{2}\dot{3} \]

Remember because you are multiplying the whole of Equation 1 by 100 you also need to multiply the variable

\[\begin{aligned}
0.\dot{2}\dot{3} &=x \hspace{3.8cm} \text{Equation 1}
\\\\
0.\dot{2}\dot{3} \times100&=x\times100\\\\
23.\dot{2}\dot{3} &=100x \hspace{3cm} \text{Equation 2}
\end{aligned}
\]

**Subtract Equation 2 from Equation 1 **

\[\begin{aligned}
23.\dot{2}\dot{3} &=100x \hspace{3cm} \text{Equation 2}\\\\
0.\dot{2}\dot{3} &=x \hspace{3.8cm} \text{Equation 1} \\\\
23&=99x
\end{aligned}
\]

**Divide the value by the coefficient of x **

\[\begin{aligned}
\ 23&=99x\\\\
\frac{23}{99}&=x
\end{aligned}
\]

**Simplify the fraction**

This fraction cannot be simplified as the only factor that 23 and 99 shares into 1

**Clearly state your answer ‘decimal=fraction’**

\[0 . \dot{2}\dot{3} =\frac{23}{99}
\]

Convert 8.\dot{7} to a fraction

**Equate the recurring decimal to a variable to create Equation 1 **

\[8.\dot{7} = x \hspace{3cm} \text{Equation 1} \]

**Multiply both sides of Equation 1 by a power of 10 to create Equation 2 **

As 8.\dot{7} has one repeating digits, we will multiply by 10

\[8 . \dot{7} \times10=87 .\dot{7}\]

Remember because you are multiplying the whole of Equation 1 by 10 you also need to multiply the variable

\[\begin{aligned}
8.\dot{7}&=x \hspace{3.8cm} \text{Equation 1}
\\\\
8.\dot{7}\times10&=x\times10\\\\
87.\dot{7}&=10x \hspace{3.3cm} \text{Equation 2}
\end{aligned}
\]

**Subtract Equation 2 from Equation 1 **

\[\begin{aligned}
87.\dot{7}&=10x \hspace{3cm} \text{Equation 2}\\\\
8.\dot{7}&=x \hspace{3.6cm} \text{Equation 1} \\\\
79&=9x
\end{aligned}
\]

**Divide the value by the coefficient of x **

\[\begin{aligned}
\ 79&=9x\\\\
\frac{79}{9}&=x
\end{aligned}
\]

**Simplify the fraction**

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

**Clearly state your answer ‘decimal=fraction’**

\[8 . \dot{7}=\frac{79}{9}
\]

Convert 4.0\dot{4}\dot{6} to a fraction

**Equate the recurring decimal to a variable to create Equation 1 **

\[4.0\dot{4}\dot{6} = x \hspace{3cm} \text{Equation 1} \]

**Multiply both sides of Equation 1 by a power of 10 to create Equation 2 **

As 4.0\dot{4}\dot{6} has two repeating digits, we will multiply by 100

\[4.0\dot{4}\dot{6} \times100=404.6\dot{4}\dot{6} \]

Remember because you are multiplying the whole of Equation 1 by 100 you also need to multiply the variable

\[\begin{aligned}
4.0\dot{4}\dot{6} &=x \hspace{3.7cm} \text{Equation 1}
\\\\
4.0\dot{4}\dot{6} \times100&=x\times100\\\\
404.6\dot{4}\dot{6} &=100x \hspace{3cm} \text{Equation 2}
\end{aligned}
\]

**Subtract Equation 2 from Equation 1 **

\[\begin{aligned}
404.6\dot{4}\dot{6} &=100x \hspace{3cm} \text{Equation 2}\\\\
4.0\dot{4}\dot{6}&=x \hspace{3.8cm} \text{Equation 1} \\\\
400.6&=99x
\end{aligned}
\]

**Divide the value by the coefficient of x **

\[\begin{aligned}
400.6&=99x\\\\
\frac{400.6}{99}&=x
\end{aligned}
\]

**Simplify the fraction**

\[\frac{400.6}{99}=x\]

As we have a decimal as the numerator, we need to multiply the numerator (and denominator) by 10

\[\begin{aligned}
\frac{400.6\times10}{99\times10}
\\\\
=\frac{4006}{990}
\end{aligned}
\]

We can simplify the fraction by dividing the numerator and the denominator by their highest common factor, 2

\[\begin{aligned}
\frac{4006\div2}{990\div2}
\\\\
=\frac{2003}{495}
\end{aligned}
\]

**Clearly state your answer ‘decimal=fraction’**

\[4.0\dot{4}\dot{6} =\frac{2003}{495}
\]

of 10**Multiplying by a power**

You must make sure you are multiplying by the correct power of 10 (e.g 10, 100, 1000) so the recurring aspect of decimal can be eliminated.

E.g

Multiplying 0 . \dot{1}\dot{3} by 10 does not help us eliminate the recurring decimal by subtraction (in step 3). In this example you need to multiply by 100 because you can now eliminate the recurring aspect of the decimal by subtraction.

Incorrect

\[ 0 . \dot{1}\dot{3} \times 10 = 1.\dot{3}\dot{1} \]

1 . \dot{3}\dot{1}-0 . \dot{1}\dot{3} does not eliminate the recurring decimal

Correct

\[ 0 . \dot{1}\dot{3} \times 100 = 13.\dot{1}\dot{3} \]

13.\dot{1}\dot{3} - 0 . \dot{1}\dot{3} does eliminate the recurring decimal

**Fractions with a decimals**

Fractions should not include a decimal

E.g

\frac{0.1}{2}

In this example you can multiply the numerator and denominator by 10 to be left with \frac{1}{20}

**Not simplifying the fraction**

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

Recurring decimals to fractions is part of our series of lessons to support revision on comparing fractions, decimals and percentages. You may find it helpful to start with the main comparing fractions, decimals and percentages lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

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

1

10

100

It does not matter

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

which allows you to eliminate the recurring decimal

2. What should you multiply 0 . \dot{1}\dot{4} by to help to eliminate the recurring decimal?

1

10

100

It does not matter

0 . \dot{1}\dot{4}\times100=14.\dot{1}\dot{4}

which allows you to eliminate the recurring decimal

3. What should you multiply 0 .0 \dot{1}\dot{5} by to help to eliminate the recurring decimal?

1

10

100

It does not matter

0 .0 \dot{1}\dot{5}\times100=1.5 \dot{1}\dot{5}

which allows you to eliminate the recurring decimal

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

\frac{2}{9}

\frac{2}{10}

\frac{2}{99}

\frac{2}{999}

Equate to x to create Equation 1

Multiply the recurring decimal by 10 to create Equation 2

Eliminate the recurring decimal by subtracting the equations

Divide by 9 (cannot be simplified)

5. What is 0. \dot{0}\dot{2} as a fraction?

\frac{2}{9}

\frac{2}{100}

\frac{2}{99}

\frac{2}{999}

Equate to x to create Equation 1

Multiply the recurring decimal by 100 to create Equation 2

Eliminate the recurring decimal by subtracting the equations

Divide by 99 (cannot be simplified)

6. What is 0. \dot{0}\dot{0}\dot{2} as a fraction?

\frac{2}{9}

\frac{2}{1000}

\frac{2}{99}

\frac{2}{999}

Equate to x to create Equation 1

Multiply the recurring decimal by 1000 to create Equation 2

Eliminate the recurring decimal by subtracting the equations

Divide by 999 (cannot be simplified)

1. Convert 0.\dot{2}\dot{7} to a fraction, give your answer in its simplest form

**(4 Marks)**

Show answer

Multiping 0.\dot{2}\dot{7} by 100 or 27.\dot{2}\dot{7}

**(1)**

Subtracting equations to create 27=99x

**(1)**

\frac{27}{99}

**(1)**

\frac{3}{11}

**(1)**

2. Convert 0.\dot{0}\dot{7} to a fraction

**(3 Marks)**

Show answer

Multiping 0.\dot{0}\dot{7} by 100 or 7.\dot{0}\dot{7}

**(1)**

Subtracting equations to create 7=99x

**(1)**

\frac{7}{99}

**(1)**

3. Prove 0.\dot{3}\dot{9} =\frac{13}{33}

**(4 Marks)**

Show answer

Multiping 0.\dot{3}\dot{9} by 100 or 39.\dot{3}\dot{9}

**(1)**

Subtracting equations to create 39=99x

**(1)**

\frac{39}{99}

**(1)**

\frac{39}{99}=\frac{13}{33}

**(1)**

4. Prove 0.1\dot{2}\dot{6} =\frac{25}{198}

**(4 Marks)**

Show answer

Multiping 0.1\dot{2}\dot{6} by 100 or 12.6\dot{2}\dot{6}

**(1)**

Subtracting equations to create 12.5=99x

**(1)**

\frac{125}{990}

**(1)**

\frac{125}{990}=\frac{25}{198}

**(1)**

You have now learned how to:

- Convert a recurring decimal to a fraction

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