  We construct free resolutions for quotient rings $R/\langle \mathcal{I}',
\mathcal{I}\mathcal{J}, \mathcal{J}'\rangle$, give conditions for the quotient
to be realized as a fiber product, and give criteria for the construction to be
minimal. We then specialize this result to fiber products over a field $k$ and
recover explicit formulas for Betti numbers, graded Betti numbers, and
Poincar\'{e} series.
