The ratio of the production cross sections times branching fractions $ \left(\sigma \left({\mathrm{B}}{\mathrm{c}}^{\pm}\right)\mathrm{\mathcal{B}}\left({\mathrm{B}}{\mathrm{c}}^{\pm}\to \mathrm{J}/\psi {\pi}^{\pm}\right)\right)/\left(\sigma \left({\mathrm{B}}^{\pm}\right)\mathrm{\mathcal{B}}\left({\mathrm{B}}^{\pm}\to \mathrm{J}/\psi {K}^{\pm}\right)\right) $ is studied in proton-proton collisions at a center-of-mass energy of 7 TeV with the CMS detector at the LHC. The kinematic region investigated requires B${c}^{±}$ and B$^{±}$ mesons with transverse momentum p${T}$ > 15 GeV and rapidity |y| < 1.6. The data sample corresponds to an integrated luminosity of 5.1 fb$^{−1}$. The ratio is determined to be $ \left[0.48\pm 0.05\left(\mathrm{stat}\right)\pm 0.03\left(\mathrm{syst}\right)\pm 0.05\ \left({\tau}{{\mathrm{B}}{\mathrm{c}}}\right)\right]% $ . The B${c}^{±}$  → J/ψπ$^{±}$ π$^{±}$ π$^{∓}$ decay is also observed in the same data sample. Using a model-independent method developed to measure the efficiency given the presence of resonant behaviour in the three-pion system, the ratio of the branching fractions $ \mathrm{\mathcal{B}}\left({\mathrm{B}}{\mathrm{c}}^{\pm}\to \mathrm{J}/\psi {\pi}^{\pm }{\pi}^{\pm }{\pi}^{\mp}\right)/\mathrm{\mathcal{B}}\left({\mathrm{B}}{\mathrm{c}}^{\pm}\to \mathrm{J}/\psi {\pi}^{\pm}\right) $ is measured to be $ 2.55\pm 0.80\left(\mathrm{stat}\right)\pm 0.33{\left(\mathrm{syst}\right)}{-0.01}^{+0.04}\left({\tau}_{B_c}\right) $ , consistent with the previous LHCb result.