Tag Archives: recuring decimals to fractions

June 19, 2025 · 7:18 am

Converting recurring (non-terminating) decimals to fractions

The easiest approach is to jump right in with some examples.

Example 1

Convert 0.\overline{5} to a fraction.

Let x=0.\overline{5}

(1)   \begin{equation*}x=0.\overline{5}\end{equation*}

(2)   \begin{equation*}10x=5.\overline{5}\end{equation*}

Subtract equation 1 from equation 2

    \begin{equation*}9x=5\end{equation}

Hence x=\frac{5}{9} so 0.\overline{5}=\frac{5}{9}

Example 2

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

Let x=0.\overline{12}

(3)   \begin{equation*}x=0.\overline{12}\end{equation*}

(4)   \begin{equation*}100x=12.\overline{12}\end{equation*}

Subtract equation 3 from equation 4.

    \begin{equation*}99x=12\end{equation}

    \begin{equation*}x=\frac{12}{99}=\frac{4}{33}\end{equation}

Example 3

Convert 0.1\overline{23} to a fraction

Let x=0.1\overline{23}

(5)   \begin{equation*}x=0.1\overline{23}\end{equation*}

(6)   \begin{equation*}10x=1.\overline{23}\end{equation*}

(7)   \begin{equation*}1000x=123.\overline{23}\end{equation*}

Subtract equation 6 from equation 7

    \begin{equation*}990x=122\end{equation}

    \begin{equation*}x=\frac{122}{990}=\frac{61}{495}\end{equation}

Our aim is to manipulate the recurring decimal to create two numbers each which have only the repeated digits after the decimal point.

One more example.

Example 4

Convert 3.4\overline{56} to a fraction

Let x=3.4\overline{56}

If I multiply by 10, then I will have 34.\overline{56} – only repeated digits after the decimal point.

If I multiply by 1000, then I will have 3456.\overline{56}– only repeated digits after the decimal point.

So I get,

    \begin{equation*}990x=3422\end{equation}

    \begin{equation*}x=\frac{3422}{990}=3\frac{226}{495}\end{equation}

You can also use your Casio classpad to do the conversion. Although I think it is easier just to do it yourself.

Let’s think about example 4,

3.4\overline{56}=3.4+\frac{56}{1000}+\frac{56}{100000}+\frac{56}{10000000}+...

Which is

3.4+\frac{56}{1000\times 100^0}+\frac{56}{1000\times100^1}+\frac{56}{1000\times 100^2}...

3.4+\Sigma_{x=0}^\infty(\frac{56}{1000\times 100^x})

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