0
Infinitely many.
Take any set, S, of fractions. Add these fractions together. The answer, A. A will be rational. If it is an integer, add 1/2 as a new member to S and increase A by 1/2. Then calculate 24-A, which will be a fraction.
If you add 24-A as yet another member to S, then the [new] S will be a set of fractions that sum to 24. Since the starting point for S is arbitrary, there are infinitely many solutions.
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