Write down a few simple positive fractions.
Now create a new fraction, such that its numerator equals the sum of all the numerators you wrote down and the denominator equals the sum of all the denominators.
Example: you wrote down 1/3, 2/7, 4/15 so the new fraction is 7/25.
Prove: the new fraction is smaller than the largest on the initial list
and bigger than the smallest.
(In reply to Treating as probabilities
Charlie's comment is completely intuitive and correct, but as stated it only applies if each of simple positive fractions are less than or equal to 1.
Simple fractions, of course, can be greater than one. For example, 13/7 is a simple fraction, because both the numerator and denominator are integers, but it can never be a probability.
Ady's theorem is still true for any simple positive fractions, even those greater than one, because (as charlie pointed out) the final fraction is a weighted average.
In fact, it is true for a11 simple fractions, not just positive fractions, as long as the denominators are all positive. For instance, starting with 1/2 and -2/3 results in -1/5, which is between the two. Again, this is just a weighted average of the two. However, starting with 1/2 and 2/(-3) results in 3/(-1), which is not between the two. The problem in this case is the negative denominator.