How do you convert 3.2 (2 being repeated) to a fraction?

1 Answer
Mar 7, 2016

3.bar(2) = 29/93.¯2=299

Explanation:

Using the notation of a bar over a set of digits to denote their infinite repetition,

let x = 3.bar(2)x=3.¯2

=> 10x = 32.bar(2)10x=32.¯2

=> 10x - x = 32.bar(2)-3.bar(2) = 2910xx=32.¯23.¯2=29

=> 9x = 299x=29

:. x = 29/9

This technique works in general. Set x as your desired value, multiply by 10^n where n is the number of digits repeating, and then subtract x to eliminate the infinitely repeating portion. It then is just a matter of dividing by 10^n-1 and reducing the resulting fraction.