Question

How do you convert `0.916` repeating to a fraction?

Answer 1

`0.91bar(6) = 11/12`

Explanation:

If you meant `0.91bar(6) = 0.916666...` then first multiply by `100(10-1) = 1000-100`. The factor `100` is to shift the number `2` places to the left to leave the repeating portion just after the decimal point. The factor `(10-1)` is to shift it one further place to the left (the length of the repeating pattern), then subtract the original to cancel out the repeating tail...

`(1000-100)0.91bar(6) = 916.bar(6) - 91.bar(6) = 825`

Then dividing both ends by `(1000-100)` we find:

`0.91bar(6) = 825/(1000-100) = 825/900 = (color(red)(cancel(color(black)(75)))xx11)/(color(red)(cancel(color(black)(75)))xx12) = 11/12`