# Repeating Decimals

How to write repeating decimals as a fraction: example and its solutions (2 ways).

## Example

0.123 = 0.1232323...

The numbers under the bar are repeating.

Split the decimal.

0.1 + 0.023 + 0.00023 + 0.0000023 + ...

0.023 + 0.00023 + 0.0000023 + ...

= (23/1000) + (23/1000)⋅(1/100) + (23/1000)⋅(1/100)^{2} + ...

This is an infinite geometric series.

Use the infinite geometric series formula.*a*_{1} = 23/1000, *r* = 1/100

0.123 = 0.1 + **(23/1000) / (1 - 1/100)**

Multiply 1000 to both of the numerator and the denominator.

Add and simplify the expression.

Then 0.123 = 61/495.

## Example: Another Solution

Let's see another way to solve this example.

Write 0.123 = 0.1232323... .

The bar is on 2 digits.

So write

100⋅0.123 = 12.3232323...

on the next line.

(Multiplying 10^{2} = 100 on both sides)

Subtract these two.

Then 99⋅0.123 = 12.2.

The repeating parts are all cancelled.

Divide both sides by 99.

Then 0.123 = 12.2/99.

Simplify the fraction.

Then you can get the same answer: 61/495.