In Otunta and Ikhile [1996], the authors in a frame work for studying the stability and convergence of a class of variable order one-step rational schemes based on rational interpolants of Luke et al [1975], Fatunla [1982] and Lambert [1974], left open the solution of the resultant order conditions of some rational nonlinear methods. In this paper, we show that the order conditions reduce to the solution of linear systems of equations for which the Gaussian elimination method comes in handy. In fact, this has been exploited herein to construct a new class of rational methods for stiff and singular initial value problems in ordinary differential equations.