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

1 Answer
Mar 9, 2016

#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.