When A and B are subsets of the integers in [1, X] and [1, Y], respectively, with |A| ⩾ α X and |B| ⩾ β Y, we show that the number of rational numbers expressible as a/b with (a, b) in A × B is ≫ (α β)1 + ϵXY for any ϵ > 0, where the implied constant depends on ϵ alone. We then construct examples that show that this bound cannot, in general, be improved to ≫ α β XY. We also resolve the natural generalization of our problem to arbitrary subsets C of the integer points in [1, X] × [1, Y]. Finally, we apply our results to answer a question of Sárközy concerning the differences of consecutive terms of the product sequence of a given integer sequence.