  We present a lattice quantum chromodynamics determination of the ratio of the
scalar and vector form factors for two semileptonic decays of the $B_s$ meson:
$B_s \rightarrow K \ell \nu$ and $B_s \rightarrow D_s \ell \nu$. In conjunction
with future experimental data, our results for these correlated form factors
will provide a new method to extract $|V_{ub}/V_{cb}|$, which may elucidate the
current tension between exclusive and inclusive determinations of these
Cabibbo-Kobayashi-Maskawa mixing matrix parameters. In addition to the form
factor results, we determine the ratio of the differential decay rates, and
forward-backward and polarization asymmetries, for the two decays.
