We consider a new qualitative approach for treating theoretically the solvation of single-atomic ionic solutes in binary mixtures of polar and nonpolar aprotic solvents. It is based on the implicit continuum electrostatic model of the solvent mixture involving distance-dependent dielectric permittivity epsilon(R) (where R is the distance from the ion) and local concentrations C(1)(R) and C(2)(R) of the solvent ingredients. For a given R, the condition for local thermodynamic equilibrium provides the transcendental equation for explicitly establishing the permittivity and concentration profiles. Computations performed with real Cl(-) and model Cl(+) ions as solutes in benzene/DMSO mixtures are compared with the molecular dynamics simulations of the same systems. A significant discrepancy of molecular and continuum results is revealed for the concentration profiles in the close vicinity of the ion boundary, although the general trends are similar. The continuum methodology cannot account for the formation of rigid solvent structures around ions, which is most significant for the case of Cl(+). Such defect, however, proves to become of less importance in calculations of the solvation free energy, which are quite satisfactory for Cl(-) ion. Free energy calculations for Cl(+) are less successful in the range of low DMSO concentration.