  When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$
respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the
number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is
$\gg (\alpha \beta)^{1+\epsilon}XY$ for any $\epsilon > 0$, where the implied
constant depends on $\epsilon$ alone. We then construct examples that show that
this bound cannot in general be improved to $\gg \alpha \beta XY$. We also
resolve the natural generalisation of our problem to arbitrary subsets $C$ of
the integer points in $[1,X] \times [1,Y]$. Finally, we apply our results to
answer a question of S\'ark\"ozy concerning the differences of consecutive
terms of the product sequence of a given integer sequence.
