Question

How do you convert 0.27 (27 being repeated) to a fraction?

Answer 1

`0.bar(27) = 3/11`

Explanation:

The standard notation for a repeating decimal is to put a bar over the repeating digits, that is, `0.27272727... = 0.bar(27)`

Let `x = 0.bar(27)`

`=> 100x = 27.bar(27)`

`=> 100x - x = 27.bar(27)-0.bar(27)`

`=>99x = 27`

`=> x = 27/99 = 3/11`

This trick works in general for repeating decimals. The idea is to multiply by a power of `10` which will result in the same repeating decimal after multiplication, that is, `10^n` where there are `n` digits in the sequence which repeats.