In this work, we study the existence and uniqueness of weak solutions to the continuous coagulation and multiple fragmentation equations for large classes of kernels. A brief mathematical survey on the well-posedness of the equations under binary fragmentation is given. Later, a uniqueness theorem is proved for mass conserving solutions to the continuous coagulation and binary fragmentation equation under strong fragmentation. Furthermore, we develop the convergence analysis of sectional methods for solving the non-linear pure coagulation equation. Here we examine the most popular of all sectional methods the fixed pivot technique. Finally, we demonstrate practical significance of the mathematical results by performing a few numerical experiments. The fixed pivot technique gives a consistent over prediction of the solution for the large size particles when applied on coarse grids. To overcome this problem, the cell average technique was used which preserves all advantages of the fixed pivot technique and improves the numerical results.