The thermodynamic properties of a ternary magnetic system are studied by considering a random-site Ising model in the context of the Bogoliubov inequality for the free energy. Only nearest-neighbor sites are considered in order to account for the different interactions in the model. The critical temperature, magnetization, and other quantities have been obtained in an approximation which reproduces the exact results at least in one dimension. The calculations are used to interpret the experimental data of the mean hyperfine field obtained by M\"ossbauer spectroscopy and the critical temperature obtained by calorimetry measurements of the antiferromagnetic ternary Fe-Mn-Al alloys in the fcc-disordered phase.