Population balances for simultaneous coagulation and breakage (and their analogs, e.g., polymerization and depolymerization) are employed in describing many systems including aerosols, powders and polymers, and many unit operations including reactors, crystallizers, and size reduction/enlargement equipment. The population balance equations for homogeneous coagulation rate, power-law breakage rate and self-similar daughter particles are developed in terms of a similarity transform which at the steady state yields limiting analytical approximations for very large and very small particles. These solutions become basis functions for reconstructing the particle size distribution using a moment methodology.