A method to obtain (approximate) analytical expressions for the radial distribution functions and structure factors in a multi-component mixture of additive hard spheres is introduced. In this method, only contact values of the radial distribution function and the isothermal compressibility are required and thermodynamic consistency is achieved. The approach is simpler than but yields equivalent results to the Generalized Mean Spherical Approximation. Calculations are presented for a binary and a ternary mixture at high density in which the Boublik-Mansoori-Carnahan-Starling-Leland equation of state is used. The results are compared with the Percus-Yevick approximation and the most recent simulation data.