Question

How do you convert 0.123 (123 repeating) to a fraction?

Answer 1

`0.bar(123) = 41/333`

Explanation:

I will use the bar notation to indicate the repeating digits.

First multiply by `(1000-1)` to get an integer:

`(1000-1)*0.bar(123) = 123.bar(123) - 0.bar(123) = 123`

`1000` is chosen to shift the pattern of digits `3` places to the left, i.e. by the length of the repeating section. Then subtracting the original cancels out the repeating tail.

Then divide both sides by `(1000-1)` and simplify:

`0.bar(123) = 123/(1000-1) = 123/(999) = (41*color(red)(cancel(color(black)(3))))/(333*color(red)(cancel(color(black)(3)))) = 41/333`