converting a repeating decimal to a fraction

The following algorithm can be used to convert a repeating decimal to a fraction:

1. Set the repeating decimal equal to $x$ .
2. Multiply both sides of the equation by $10^n$ , where $n$ is the number of digits that appear under the bar.
3. If applicable, rewrite the second equation so that its repeating part lines up with the repeating part in the original equation.
4. Subtract the original equation from the most recently obtained equation. (The repeating part should cancel at this step.)
5. If applicable, multiply both sides by a large enough power of $10$ so that the equation is of the form $ax=b$ , where $a$ and $b$ are integers.
6. Divide both sides of the equation by the coefficient of $x$ .
7. Reduce the fraction to lowest terms.

Below, this algorithm is demonstrated for $0.58\overline{3}$ with the steps indicated on the far right.

\begin{equation} x=0.58\overline{3} \end{equation} \begin{equation} 10x=5.8\overline{3} \end{equation} \begin{equation} 10x=5.83\overline{3} \end{equation} \begin{equation} 9x=5.25 \end{equation} \begin{equation} 900x=525 \end{equation} \begin{equation} x=\frac{525}{900} \end{equation} \begin{equation} x=\frac{7}{12} \end{equation} An important application of this algorithm is that it supplies a proof for the fact that $0.\overline{9}=1$ :