Repeating Decimals

Repeating Decimals

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

Definition

A repeating decimal is a decimal number that has repeating digits.

A repeating decimal is a decimal number
that has repeating digits.

For example, 0.1232323... is a repeating decimal
because [23] is repeating.

A bar above the digits is used
to simply write the repeating part.

So 0.123 = 0.1232323....

Example

Write the given repeating decimal as a fraction. 0.1(bar)23

0.123 = 0.1232323...

The bar is above [two digits].
So [two digits] are repeating.

So multiply 10[2] = 100
on both sides of 0.123 = 0.1232323....

So write 100⋅0.123 = 12.3232323....

And write 0.123 = 0.1232323...
in the next line.

100⋅0.123 = 12.3232323...
0.123 = 0.1232323...

Subtract these two equations.

Then 99⋅0.123 = 12.2.

The repeating parts are all cancelled.
(gray parts)

Divide both sides by 99.

Then 0.123 = 12.2/99.

Move the decimal points of both numbers
1 digit to the right.

(= Multiply 10
to both of the numerator and the denominator.)

Reduce 122 to, 122/2, 61
and reduce 990 to, 990/2, 495.

So (given) = 61/495.