In this paper, we prove that if Cohen-Macaulay local/graded rings $R_1$, $R_2$ and $R$ satisfy certain conditions regarding multiplicity and Cohen-Macaulay type, then almost Gorenstein property of $R$ implies Gorenstein properties for all of $R_1$, $R_2$ and $R$. We apply our theorem to tensor products of semi-standard graded rings and some classes of affine semigroup rings, i.e., numerical semigroup rings, edge rings and stable set rings.