Repeating Decimal to Fraction

To Convert a Repeating Decimal to a Fraction, simply follow these steps:

• Note the number of digits in the repeating pattern. Does only one digit repeat? Two? Three?
• If only one digit repeats, put it over $9$.
• If two digits repeat, put them over $99$.
• If three digits repeat, put them over $999$.
• You get the idea...

Examples

$$0.\overline5 = \frac{5}{9}$$

$$0.\overline{29} = \frac{29}{99}$$

$$0.\overline{712} = \frac{712}{999}$$

What is this Sorcery?

To see why this works, see if you can follow the steps below:

$$x = 0.\overline{43}$$

$100x = 43.\overline{43}$                                   Multiply both sides by $100$.

  $99x = 43$                                   Subtract $x=0.\overline{43}$ from both sides.

$x = \frac{43}{99}$                                   Divide both sides by $99$.