Question

How do you convert 0.012 (12 repeating) to a fraction?

Answer 1

`0.0bar(12) = 2/165`

Explanation:

First multiply by `10(100-1) = 1000-10 = 990` to get an integer:

`990*0.0bar(12) = (1000-10) 0.0bar(12) = 12.bar(12) - 0.bar(12) = 12`

Divide both ends by `990` to get:

`0.0bar(12) = 12/990 = (color(red)(cancel(color(black)(6)))*2)/(color(red)(cancel(color(black)(6)))*165) = 2/165`

Why multiply by `10(100-1)` ?

The factor of `10` shifts the repeating pattern one place to the left so that it starts just after the decimal point.

The factor of `(100-1)` shifts the resulting value two more places to the left then subtracts the original value. This has the effect of cancelling out the fractional repeating tail.