If you take two balls randomly out of a jar of colored balls, there is a 50% chance that the balls will both be red.

What is the total percentage of red balls in the jar?

As Charlie pointed out, there is no unique percentage. 3 reds out
of 4 balls is one solution; the next smallest is 15 reds out of
21. This does not yield 75% reds.

The cases in TomM's solution are not broken down correctly.

For example, (r+o)(r+o-1) = 2r(r-1), with r and o mutually prime,
implies r+o *divides* 2(r-1), but not r+o = 2(r-1) or r+o = r-1.