  We show how to compute the Ehrhart polynomial of the free sum of two lattice
polytopes containing the origin $P$ and $Q$ in terms of the enumerative
combinatorics of $P$ and $Q$. This generalizes work of Beck, Jayawant,
McAllister, and Braun, and follows from the observation that the weighted
$h^*$-polynomial is multiplicative with respect to the free sum. We deduce that
given a lattice polytope $P$ containing the origin, the problem of computing
the number of lattice points in all rational dilates of $P$ is equivalent to
the problem of computing the number of lattice points in all integer dilates of
all free sums of $P$ with itself.
