The next generation of long-baseline neutrino experiments will be capable of precision measurements of neutrino oscillation parameters, precision neutrino-nucleus scattering, and unprecedented sensitivity to physics beyond the Standard Model. Reduced uncertainties in neutrino fluxes are necessary to achieve high precision and sensitivity in these future precise neutrino measurements. New measurements of hadron-nucleus interaction cross sections are needed to reduce uncertainties of neutrino fluxes. We report measurements of the differential cross-section as a function of scattering angle for proton-carbon interactions with a single charged particle in the final state at beam momenta of 20, 30, and 120 GeV/c. These measurements are the result of a beam test for EMPHATIC, a hadron-scattering and hadron-production experiment. The total, elastic and inelastic cross-sections are also extracted from the data and compared to previous measurements. These results can be used in current and future long-baseline neutrino experiments, and demonstrate the feasibility of future measurements by an upgraded EMPHATIC spectrometer.