Question

How do you convert 0.013 (13 repeating) to a fraction?

Answer 1

`0.0bar(13) = 13/990`

Explanation:

It is common to denote a repeating decimal by putting a bar over the repeating digits. In this example, then, we would denote the number in question as `0.0bar(13)`

Let `x = 0.0bar(13)`

`=> 100x = 1.3bar(13)`

`=>100x - x = 1.3bar(13)-0.0bar(13)`

`=>99x = 1.3 = 13/10`

`=>x = (13/10)/99 = 13/990`

The multiply-then-subtract method used above is a common trick for finding the fractional representation of a repeating decimal. Simply multiply by `10^n` where `n` is the number of digits in the repeating segment, subtract the repeating decimal, and then solve for the variable representing the decimal (divide by `10^n-1`).