We have examined the accuracy of a modified first-order perturbation theory of nonideal binary mixtures and van der Waals one-fluid theory, which uses an accurate equation of state for pure fluids. Perturbation theory consists of the first-order perturbation theory of high temperature approximation and the random phase approximation. Inclusion of random phase approximation enhances the applicability of perturbation theory to much wider ranges of temperatures and densities. In this part of the paper, results are reported for pressure, residual chemical potentials at finite concentration and in infinite dilution, total and excess properties of several fluid mixtures under highly nonideal conditions. In these mixtures, size and energy parameters of like as well as unlike components differ significantly. Comparisons of theoretical predictions with accurate simulation results show, in general, a very good performance of the perturbation theory in varying nonideal conditions. Van der Waals one-fluid theory is successful in predicting equation of state, residual chemical potentials at finite concentration and in infinite dilution in moderately nonideal mixtures only. It becomes less reliable in describing residual chemical potentials in infinite dilution in highly nonideal mixtures, and becomes unreliable in predicting excess properties of even moderately nonideal mixtures. In general, the present form of perturbation theory offers a significant improvement over the previous forms based on high temperature approximation, and is essentially more accurate than the van der Waals one-fluid theory.