It's fairly easy to take a fraction and turn it into a repeating decimal by hand. This is simple division. But being able to go in the other direction and turn a repeating decimal into a fraction would be a cool party trick.
Yay! Party trick!
First I will need a random repeating decimal from the audience.
"0.57575757575757..."
Thank you. This would be 19 / 33.
"Wow, Blake! How did you do that?"
Simply take the number that repeats, in this case 57, and divide it by however many 9's as there are digits in the original number. Since 57 is 2 digits, then I would divide 57 by 99.
57 / 99 simplifies down to 19 / 33.
If my repeating number was .189189189189189..., then I would divide 189 by 999 since it has 3 digits (which would simplify down to 7 / 37).
Simple!
Just make sure if you perform this as a "magic trick" of sorts, to simplify your fraction down before you blurt it out, to conceal the magic.