# Repeating Decimal

How to write a repeating decimal as a fraction: definition, 1 example, and its solution.

## Definition

A repeating deciaml is a decimal number

that has a repeating part.

See 0.123.

The numbers under the bar, 23,

is the repeating part.

So 0.123 = 0.1232323... .

## Example0.123 → Fraction

See 0.123.

23 is under the bar.

So 0.123 = 0.1232323... .

There are 2 digits under the bar.

So multiply, 10^{2}, 100

to both sides.

Power

Then 100⋅0.123 = 12.3232323... .

When multiplying 100 to the right side,

move the decimal point

2 digits to the right.

Write 0.123 = 0.1232323...

in the next line.

Subtract these two equations.

100⋅0.123 - 0.123

= 99⋅0.123

12.3232323... - 0.1232323...

= 12.2

The repeating parts are all cancelled.

Divide both sides by 99.

Then 0.123 = 12.2/99.

Move the decimal points

1 digit to the right.

12.2/99 = 122/990

Reduce 122 to, 122/2, 61

and reduce 990 to, 990/2, 495.

So 61/495 is the answer.