Question

How do you convert 3.2 (2 being repeated) to a fraction?

Answer 1

`3.bar(2) = 29/9`

Explanation:

Using the notation of a bar over a set of digits to denote their infinite repetition,

let `x = 3.bar(2)`

`=> 10x = 32.bar(2)`

`=> 10x - x = 32.bar(2)-3.bar(2) = 29`

`=> 9x = 29`

`:. x = 29/9`

This technique works in general. Set `x` as your desired value, multiply by `10^n` where `n` is the number of digits repeating, and then subtract `x` to eliminate the infinitely repeating portion. It then is just a matter of dividing by `10^n-1` and reducing the resulting fraction.