A recent extension to mixtures of Verlet’s closure is applied in conjunction with the Ornstein-Zernike relation to solve the structure and thermodynamics of mixtures of hard-spheres and homonuclear hard-dumbbells. This integral equation (IE), which is proven to be very accurate when compared with simulation data, is used to explore the possibility of phase separation in an asymmetric mixture. While our results do not show evidence of such phase separation in the asymmetric binary hard-sphere mixture studied by Biben and Hansen [T. Biben and J. P. Hansen, Phys. Rev. Lett. 66, 2215 (1991)], an equivalent mixture of hard-dumbbells and large hard-spheres seems to exhibit a certain tendency to phase separate as far as the integral equation results are concerned. Finally, given the ability of this integral equation to reproduce the hard-core system, we have incorporated these results into a previous Reference Hypernetted Chain scheme to treat a mixture of N-2 and Ar modeled by means of site-site Lennard-Jones potentials. In consonance with the results for pure fluids, the parameterization of a hard-core reference system with the same molecular shape leads to excellent results both for the structure and thermodynamics of real systems.