How do you convert 2.136 (36 repeating) as a fraction?

1 Answer
May 1, 2016

#2.1bar(36) = 47/22#

Explanation:

There is a notation for repeating decimals that places a bar above the repeating pattern.

Using that notation, the original decimal representation is written:

#2.1bar(36)#

To make this into an integer, multiply by #10(100-1) = 1000-10#. The factor #10# here is to shift the repeating section to just after the decimal point. the factor #(100-1)# shifts the number by an additional #2# places (the length of the repeating pattern) and subtracts the original number to cancel out the repeating part:

#(1000-10)*2.1bar(36) = 2136.bar(36) - 21.bar(36) = 2115#

Then divide both sides by #(1000-10)# and simplify to find:

#2.1bar(36) = 2115/(1000-10) = 2115/990 = (color(red)(cancel(color(black)(3*3*5)))*47)/(color(red)(cancel(color(black)(3*3*5)))*22) = 47/22#