How do you convert 0.13 (13 repeating) to a fraction?

1 Answer
May 12, 2016

#0.bar(13) = 13/99#

Explanation:

First some notation:

Just in case you might not have met it, drawing a bar above a group of digits means that that sequence of digits repeats, so we can write:

#0.131313... = 0.bar(13)#

Multiply by #(100-1)# to get an integer:

#(100-1) 0.bar(13) = (100*0.bar(13)) - (1*0.bar(13)) = 13.bar(13)-0.bar(13) = 13#

Then divide both ends by #(100-1)# and simplify:

#0.bar(13) = 13/(100-1) = 13/99#

Why #(100-1)#?

The multiplier #100# shifts the number two places to the left - the length of the repeating pattern. Then subtracting the original pattern cancels out the repeating tail.