Continuous-mixture rate expressions are formulated for particle breakage and molecular fragmentation when two fragments result from each fission event. The proposed general stoichiometric coefficient for binary fragmentation reduces to the limit cases of totally random and proportioned fragmentation terms. Combinations of the new binary fragmentation expressions describe ternary and other multiple product events resulting in more than two fragments. New formulations for the stoichiometric coefficients and the rate constants allow final products of the fragmentation process to have a nonzero size. Moment methods and other procedures provide solutions for the frequency distribution functions of the population balance equations that apply to batch and continuous-flow cases. A superposition of fragmentation events yields a moment solution for an infinite sequence of scissions and the general stoichiometric coefficient.