If you have a number x of the form:
a.bc
Where a is the integer part, b is the non-repeating fractional part, and c is the repeating fractional part, then the resulting fraction has value:
a.b + c / ((10^num_digits(c) - 1) * 10^num_digits(b))
So for 0.333333... we have:
a = 0
b = 0
c = 3
Giving:
0.0 + 3 / ((10^1 - 1) * 10^0)
1 / 3
The more complex .20151515151515... we have:
a = 0
b = 20
c = 15
Giving:
0.20 + 15 / ((10^2 - 1) * 10^2)
133 / 660