In this study, for the first time we demonstrated that both random and chain-end scissions of polymers can be simulated on a unified Fixed Pivot (FP) framework through an elegant implementation of a discrete-continuous meshing strategy. Achieved using only a fraction of computational expense in solving the full set of exact equations, the FP solutions benchmarked very well against the exact solutions for a polymer with a broad size distribution typical of natural polymers at different degrees of up to approximate to O(10(5)). This is attained despite the use of an efficient computational technique to obtain the exact solutions. Moreover, new observations revealed an additional strength of the current meshing strategy, in that the number of the discrete partitions can be adjusted to improve the accuracy of the solution while retaining the total number of equations to be solved. The FP technique, which in the past was reported to over-predict in cases of pure aggregation, also exhibits marginal over-prediction for pure random scission. The source of this behaviour is further uncovered, leading to a revised guideline on the choice of the number of discrete pivots.