The mixing rule estimates of the radial distribution functions g(alpha beta)(r), from the van der Waals one-fluid and mean density approximations for Lennard-Jones fluid mixtures, are examined by comparing with Monte Carlo simulations. In addition, these mixing rule estimates of g(alpha beta)(r) are used to calculate the Helmholtz energy for Lennard-Jones homonuclear chain mixtures based on the thermodynamic perturbation theory (TPT). Two simplifying modifications of these mixing rules are proposed for the TPT mixture model. The accuracy of these simplified rules is examined and found to give accurate representation of the theoretically derived Helmholtz energy over a wide range of densities and temperatures.