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A novel analytical approximated solution for the gain of broadband Raman amplifiers with multiple counter-pump lasers is derived taking into account pump-pump interactions. Validation against experimental data available in the literature with two and three pumps is performed. In order to test the limits of the approximated analytical solution, signal gain is numerically obtained by simulation for Raman amplifiers built with different types of fibers and covering bandwidths up to 100 nm. The results show that our analytical solution presents a good agreement with both numerical and experimental results in most of the practical situations. This allows the reduction of the computation time in several orders of magnitude when optimizing such amplifiers.