This is an interesting question. I plan to consider it more in depth, but here's the simple answer (that I think is true in all situations)

First, the numbers being divided have to be integers (if they are decimals, then these rules might not work).

Fractions that result in terminating decimals involve numbers being divided by: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100... (the numbers go on). The more simple answer would be that the divisor MUST BE either 1, 2, 5 OR some multiple of only these three terms OR the fraction can be reduced such that the divisor follows one of the first two rules. (If the divisor is a multiple of any number other than these, such as 3, 7, 11, then the decimal is not terminating.)

Fractions that result in a reacuring decimal (such as .3333... or .8888...) occur only when the divisor is 3 or 9, or the fraction can be reduced in such a way that the divsor will become 3 or 9. No other divisor (that I can think of) will result in a reacuring decimal.

By a process of elimination, any fraction that does not follow either of the first two rules contains a decimal that is repeating. Examples would be a number divided by 12, 23, 98, etc.