I want to prove , using the fundamental theorem for finitely generated abelian groups, that if $G$ is a finitely generated abelian group and $G_t$ is its torsion subgroup such that $G/G_t$ is of rank $m$ and $H$ is a subgroup such that $H/H_t$ is of rank $n$, then $(G/H)/(G/H)_t$ is of rank $m-n$. There is a solution Rank of $(G/H)/(G/H)_t$ where $G$ is finitely generated abelian and $H$ is a subgroup. but its proof uses notions with which I am not familiar.
I could solve it by assuming that every subgroup of $G$ is a direct sum of factors each of which is a subgroup of the corresponding factors in the decomposition of $G$, but this is not true in general. What should I do?