A theoretically founded equation of state is developed for mixtures of hard spheres with heteronuclear hard dumbbells. It is based on a model previously developed for hard-convex-body fluid mixtures, and further extended to fluid mixtures of homonuclear hard dumbbells. The equation scales the excess compressibility factor for an equivalent hard-sphere fluid mixture to obtain that corresponding to the true mixture. The equivalent mixture is one in which the averaged volume of a sphere is the same as the effective molecular volume of a molecule in the real mixture. Thus, the theory requires two parameters, namely the averaged effective molecular volume of the molecules in the mixture and the scaling factor, which is the effective nonsphericity parameter. Expressions to determine these parameters are derived in terms of the geometrical characteristics of the molecules that form the mixture. The overall results are in closer agreement with simulation data than those obtained with other theories developed for these kinds of mixtures.