Given a number of different fractions, create a new fraction whose numerator is the sum of all those fractions' numerators and whose denominator is the sum of the denominators. Call it y. Call the smallest of the original fractions x and the largest z.
Prove that for all cases, x < y < z.
(In reply to Counterexamples
by Steve Herman)
I agree with your first counterexample (the word "positive" is missing in the problem statement). But your second is not valid; the problem says "different fractions".