We study the effective interactions, structure, and the isothermal compressibility of a binary mixture interacting with pairwise additive pair potentials. By integrating out the degrees of freedom of species 2 in the partition sum we first show that a binary mixture can be mapped formally onto an effective one-component system with an effective Hamiltonian consisting of a structure-independent term, which contributes to the total pressure and chemical potential of the system, but does not affect the phase behavior, and a structure-dependent potential of mean force, which contains pair-, triplet-, and higher-body interactions. We then show that the 1-1 structure factor and pair correlation function, and the total isothermal compressibility of the mixture are equal to those of the effective one-component system, provided the mapping is exact. We illustrate and confirm these results by calculating the structure factors and pair correlation functions of the binary Asakura-Oosawa model, which is a simple model for colloid-polymer mixtures, and those of the corresponding one-component system for a size ratio such that the mapping onto an effective one-component Hamiltonian with a strictly pairwise potential of mean force is exact. The distinction between the osmotic and total compressibility of the mixture is emphasized.