Question

The repeating decimal, `0\bar (.27)`, is converted to the fraction `0.\bar(27) = 3/x`. What is the value of `x` in the fraction?

Answer 1

`x=11`

Explanation:

`E1: 0.bar(27)=3/x`

The key here is to get rid of the repeating decimal. I do that by multiplying by a term, on both sides, to get one set of repeating decimal isolated. In this case, I'll multiply by 100 to achieve:

`E2: 27.bar(27)=300/x`

And now subtract E1 from E2:

`E2 - E1: 27=300/x-3/x=297/x`

And now I can solve:

`27=297/x`

`x=297/27=11`

Checking:

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