Critical comparisons of a pair of alternate approaches to free energy calculations are performed. The "single topology" method incorporated in widely-used programs such as AMBER is compared to the "dual topology" approach used by widely-used programs such as CHARMM. For each topological approach, both free energy perturbation (FEP), and thermodynamic integration (TI) methods are characterized. All calculations are performed for the "zero sum" model system of "ethane-to-ethane" in water. The symmetry of the change in this system allows one to unambiguously evaluate the approach to convergence for a series of simulations. The results of these calculations indicate that (1) using FEP, the single topology approach is more efficient than the dual topology approach for all but very long simulations; (2) using TI, the single topology approach is considerably more efficient than the dual topology approach for relatively short simulations, while the two methods are of comparable efficiency for long Simulations; and (3) TI and FEP are comparably efficient with the single topology technique, but TI is more efficient when the dual topology method is used. The results here indicate that >700 ps of total sampling is required to obtain a converged free energy result for this relatively simple model system.