Let $G$ be a finitely generated abelian group and $H$ be a subgroup. Let subscript $t$ denote the torsion subgroup. If $G/G_t$ is free of rank $n$ and $H/H_t$ is free of rank $m$, it is easy to embed $H/H_t\hookrightarrow G/G_t$ and deduce that $m\le n$. Now the question is that I want to show that $(G/H)/(G/H)_t$ is free of rank $n-m$.
This is harder than it looks and I have not succeeded in finding a proof after many hours.
[EDIT] I'm looking for a group theory proof.