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 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 .
$$x=0.58\overline{3}$$  (1) 
$$10x=5.8\overline{3}$$  (2) 
$$10x=5.83\overline{3}$$  (3) 
$$9x=5.25$$  (4) 
$$900x=525$$  (5) 
$$x=\frac{525}{900}$$  (6) 
$$x=\frac{7}{12}$$  (7) 
An important application of this algorithm is that it supplies a proof for the fact that $0.\overline{9}=1$:
$x$  $=0.\overline{9}$  
$10x$  $=9.\overline{9}$  
$9x$  $=9$  
$x$  $=1$ 
Title  converting a repeating decimal to a fraction 

Canonical name  ConvertingARepeatingDecimalToAFraction 
Date of creation  20130322 16:55:22 
Last modified on  20130322 16:55:22 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  10 
Author  Wkbj79 (1863) 
Entry type  Algorithm 
Classification  msc 11A99 
Classification  msc 1100 