We present the first simultaneous measurement of the ratio of branching fractions, $R=\mathcal{B}(t\ensuremath{\rightarrow}Wb)/\mathcal{B}(t\ensuremath{\rightarrow}Wq)$, with $q$ being a $d$, $s$, or $b$ quark, and the top-quark pair production cross section ${\ensuremath{\sigma}}_{t\overline{t}}$ in the lepton plus jets channel using $0.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collision data at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ collected with the D0 detector. We extract $R$ and ${\ensuremath{\sigma}}_{t\overline{t}}$ by analyzing samples of events with 0, 1, and $\ensuremath{\ge}2$ identified $b$ jets. We measure $R={0.97}_{\ensuremath{-}0.08}^{+0.09}(\mathrm{stat}+\mathrm{syst})$ and ${\ensuremath{\sigma}}_{t\overline{t}}={8.18}_{\ensuremath{-}0.84}^{+0.90}(\mathrm{stat}+\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.50(\mathrm{lumi})\text{ }\text{ }\mathrm{pb}$, in agreement with the standard model prediction.