Methods and apparatus to solve for a minimum variance solution for a constrained weight problem in a scalable and memory efficient manner are disclosed. An example method includes storing a constraint matrix and a population matrix into a memory, the constraint matrix representing demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population; determining, by executing an instruction with a processor, a linear system based on a set of constraints including the constraint matrix and the population matrix; determining, by executing an instruction with the processor, a solution to the linear system by determining an inverse of a first matrix of the linear system, the determination of the inverse including breaking down the first matrix into blocks, dimensions of the blocks being no larger than dimensions of the constraint matrix and outputting a set of weights based on the inverse of the linear system, the set of weights satisfying the set of constraints.