Question

How do you convert 0.13 (13 repeating) to a fraction?

Answer 1

`0.bar(13) = 13/99`

Explanation:

First some notation:

Just in case you might not have met it, drawing a bar above a group of digits means that that sequence of digits repeats, so we can write:

`0.131313... = 0.bar(13)`

Multiply by `(100-1)` to get an integer:

`(100-1) 0.bar(13) = (100*0.bar(13)) - (1*0.bar(13)) = 13.bar(13)-0.bar(13) = 13`

Then divide both ends by `(100-1)` and simplify:

`0.bar(13) = 13/(100-1) = 13/99`

Why `(100-1)`?

The multiplier `100` shifts the number two places to the left - the length of the repeating pattern. Then subtracting the original pattern cancels out the repeating tail.