Hygroscopic growth of mixed aerosol particles and activation of cloud condensation nuclei (CCN) are considered using Köhler theory without the assumption of a dilute solution and accounting for the effect of insoluble fraction. New analytical expressions are derived for the equilibrium wet radius of the wet aerosol and for the critical radii and supersaturations for CCN activation for both volume-distributed soluble fraction and a soluble shell on the surface of an insoluble core (e.g., mineral dust particle). These expressions generalize the known equations of the Köhler theory, and the accuracy and applicability of the classical expressions are clarified. On the basis of these new expressions, a general but simple method is derived for calculation of the wet size spectrum and the CCN activity spectrum from the dry aerosol size spectrum. The method is applicable for any arbitrary shape of the dry aerosol spectra. Some applications for evaluation of aerosol extinction and homogeneous ice nucleation in a polydisperse aerosol are briefly considered. The method described here can be used in cloud and climate models, in particular, for evaluation of the aerosol direct and indirect effects.