I remember from a long time ago (about 40 years, believe it or not) learning how to convert a repeating decimal into a fraction... like .33333... = 1/3. That one is obvious, but what about something like .571428571428... = 4/7?

Well, if you have a repeating decimal, you can transform that into a (non-minimal) fraction by deviding the x positions long prepeating part by an as long sequence of 9s. In your example: 0.3333... has the single position repeating sequence of "3", so you devide 3/9, which is obviously equal to 1/3. 0.5714285714... has a 6 position repeating pattern, so you devide 571428/999999 which simplifies to 4/7. To simplify, you should use euklids algorithm like posted by wfaulk.

cu,
sven