Question

How do you convert 0.54 (54 repeating) to a fraction?

Answer 1

`0.bar(54)=6/11`

Explanation:

We will use the notation of a bar over repeating digits, that is, `0.545454... = 0.bar(54)`.

Let `x = 0.bar(54)`

`=> 100x = 54.bar(54)`

`=> 100x - x = 54.bar(54)-0.bar(54)`

`=> 99x = 54`

`=> x = 54/99 = 6/11`

This strategy works in general. Given a repeating decimal, let `x` represent the initial value, multiply by `10^n` where `n` is the number of digits repeating, subtract `x` (the original value), and solve for `x`.

We can also notice a pattern in the above: we always end up with the repeating portion divided by `10^n-1`, where `n` is the number of digits in the repeating portion. This gives us a shortcut:

`0.bar(a_1a_2a_3...a_n) = (a_1a_2a_3...a_n)/(10^n-1)`