Example methods, apparatus, systems, and articles of manufacture are disclosed to estimate population reach for different unions based on marginal ratings. An example apparatus includes an association controller to generate a tree association corresponding to a union of a first and second margin of time; and one or more commercial solvers to perform parallel computations on a processor to determine first multipliers corresponding to a total number of panelists exposed to media at the first margin, the second margin, and the union by solving first equations corresponding to the tree association; discard a first multiplier and a second multiplier of the first multipliers corresponding to the first margin and the second margin; perform parallel computation on the processor to determine second multipliers corresponding to a total audience exposed to the media at the first margin and the second margin by solving second equations corresponding to the tree association, the second system of equations including a subset of the first multipliers; and determine an estimate for a population reach of the union based on the second multipliers.