Question

How do you convert 0.12345 (345 repeating) to a fraction?

Answer 1

`0.12bar(345) = 4111/33300`

Explanation:

Multiply by `100(1000-1) = 100000-100` to get an integer:

`(100000-100)0.12bar(345)=12345.bar(345)-12.bar(345)=12333`

The factor `100` shifts the number so that the repeating portion starts just after the decimal point. The factor `(1000-1)` shifts it left by a further `3` places (the length of the repeating pattern) and subtracts the original to cancel the fractional tail.

Then divide both ends by `(100000-100)` to get:

`0.12bar(345) = 12333/(100000-100) = 12333/99900 = (color(red)(cancel(color(black)(3)))*4111)/(color(red)(cancel(color(black)(3)))*33300) = 4111/33300`