Accurate numerical methods are needed to determine phase equilibria in fluid mixtures for industrial applications, especially when mixtures exhibit critical points. In this paper we test the reliability and consistency of predictions of three major approaches for computing vapor-liquid-phase equilibria in binary fluid mixtures based on the Soave-Redlich-Kwong equation of state and van der Waals one-fluid mixing rules. These approaches are fugacity method, Gibbs minimization method and area method. To examine the accuracy of the three methods, calculated results for the vapor-liquid-phase equilibria in two selected binary mixtures, namely, carbon dioxide/butane and carbon dioxide/propane, are compared with experimental data over a range of high temperatures (278-398 IK) and elevated pressures (0.3-7 MPa). It has been found that the area method, which provides both necessary and sufficient conditions for Gibbs energy minimization of the entire system, is more reliable than either the fugacity method or Gibbs minimization method in determining the vapor-liquid-phase equilibria, especially close to the critical loci of the mixtures. The Gibbs minimization performs better than the fugacity method in locating phase equilibria in the vicinity of the critical locus of the mixtures. Far away from the critical loci of the mixtures all the three methods are found to be equally reliable. The area-method-based calculations illustrate that the SRK EOS/VDW MR model can predict reasonably well the vapor-liquid-phase equilibria in these binary mixtures. However, the SRK EOS/VDW MR model, which contains only a single binary interaction parameter, is unable to describe accurately the pressure dependence of the vapor- and the liquid-phase compositions.