  At present, two outstanding discrepancies between experiment and the standard
model are the measurements of the hadronic branching fractions $R_b$ and $R_c$.
We note that an independent measurement of these branching fractions may be
obtained from the width of hadronic $Z$ decays with a prompt photon,
$\Gamma_{qq\gamma}$, along with the total hadronic decay rate, $\Gamma_{had}$,
and an additional theoretical assumption. Such an analysis requires no flavor
tagging. We consider several plausible theoretical assumptions and find that
the current value of $\Gamma_{qq\gamma}$ favors larger $R_b$ and smaller $R_c$
relative to standard model predictions, in accord with the direct measurements.
If $\Gamma_{qq\gamma}$ and $\Gamma_{had}$ are combined with the direct
measurements, generation-blind corrections to all up-type and all down-type
quark widths are most favored. An updated measurement of $\Gamma_{qq\gamma}$
with the currently available LEP data is likely to provide an even stronger
constraint on both the branching fraction discrepancies and their possible
non-standard model sources.
