A measurement of the ratio of branching fractions: $\frac{\mathcal{B}(B^{\pm}\rightarrow D K^{\pm})}{\mathcal{B}(B^{\pm}\rightarrow D\pi^{\pm})}$ for $D\rightarrow K\pi$, $KK$, $K\pi\pi\pi$ and $K^0_S\pi\pi$

Using the 36.5~pb$^{-1}$ of data collected by the LHCb experiment in 2010, the ratio of two $B_u$ hadronic branching fractions: ${\cal B}(B^{\pm}\rightarrow D K^{\pm})\ /\ {\cal B}(B^{\pm}\rightarrow D \pi^{\pm})$ is measured for events in which the $D$ meson is reconstructed in one of four final states. In the Cabibbo-favoured decays $D^0\rightarrow K^{-}\pi^{+}$ and $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$, this ratio is measured as $(6.30\pm0.38\pm0.40)\%$. It is determined to be $(9.31\pm1.89\pm0.53)\%$ in the $CP$ eigenstate mode $D\rightarrow K^{+} K^{-}$ and $(12\,^{+6}_{-5}\pm1)\%$ in the $D\rightarrow K^0_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ channel. From this work, two observables are measured: \begin{eqnarray*} R_{CP+} &=& 1.48 \pm 0.31 (stat.) \pm 0.12 (syst.)\\ A_{CP+} &=& 0.07 \pm 0.18 (stat.) \pm 0.07 (syst.) \end{eqnarray*}