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Recurring Decimals To Fractions

Converting Recurring Decimals To Fractions

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.

What is converting recurring decimals to fractions?

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

What is converting recurring decimals to fractions?

$What is converting recurring decimals to fractions?$

Note: these are sometimes called repeating decimals

How to convert recurring decimals to fractions

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

1Equate the recurring decimal to a variable, we will use x, to create Equation $1$

2Multiply 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.

3Subtract Equation $2$ from Equation $1$

4Divide the value by the coefficient of x

5Simplify the fraction

6Clearly state your answer ‘decimal=fraction’

Explain how to convert from a recurring decimal to a fraction in 6 steps

$Explain how to convert from a recurring decimal to a fraction in 6 steps$

$Recurring decimals to fractions worksheet$

Recurring decimals to fractions worksheet

$Recurring decimals to fractions worksheet$

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

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$Recurring decimals to fractions worksheet$

Recurring decimals to fractions worksheet

$Recurring decimals to fractions worksheet$

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

DOWNLOAD FREE

Recurring decimals to fractions examples

Example 1: converting with a simple recurring decimal (in the tenths column only)

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

Equate the recurring decimal to a variable, we will use x, to create Equation $1$

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

2Multiply 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 x by $10$ (see below)

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

3Subtract Equation $2$ from Equation $1$

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

4Divide the value by the coefficient of x

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

We originally stated that x was equal to $0 .\dot{1}$

If x is equal to both $0 .\dot{1}.$ and $\frac{1}{9}$ we can state that $0 .\dot{1}$ and $\frac{1}{9}$ are equal to one another

5Simplify the fraction

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

6Clearly state your answer ‘decimal=fraction’

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

Example 2: converting with a recurring decimal with simplifying

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 x by $100$ (see below)

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

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’

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

Example 3: converting with a recurring decimal (starting in the hundredths column only)

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 x by $10$ (see below)

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

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’

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

Example 4: converting with a recurring decimal (in the tenths and hundredths column)

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 x by $100$ (see below)

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

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’

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

Example 5: converting with a recurring decimal greater than 1

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 x by $10$ (see below)

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

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’

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

Example 6: converting with a more complex recurring decimal

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 x by $100$ (see below)

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

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’

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

Common misconceptions

Multiplying by a power of $10$

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:

Practice converting recurring decimals to fractions questions

1

10

100

It does not matter

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

which allows you 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

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

\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)

\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)

\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)

Converting recurring decimals to fractions GCSE questions

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)

Learning checklist

You have now learned how to:

Convert a recurring decimal to a fraction

The next lessons are

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