  We study metabelian groups $G$ given by full rank finite presentations
$\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ of
metabelian groups. We prove that $G$ is a product of a free metabelian subgroup
of rank $\max\{0, |A|-|R|\}$ and a virtually abelian normal subgroup, and that
if $|R| \leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while
it is decidable if $|R|\geq |A|$. We further prove that if $|R| \leq |A|-1$
then in any direct decomposition of $G$ all, but one, factors are virtually
abelian. Since finite presentations have full rank asymptotically almost
surely, finitely presented metabelian groups satisfy all the aforementioned
properties asymptotically almost surely.
