The double-differential cross sections for the Cd-116(p, n) and Sn-116(n, p) reactions were measured at 300 MeV for studying the nuclear matrix element of the Cd-116 beta beta decay. A multipole decomposition technique was applied to the spectra to extract the Gamow-Teller (GT) component. The integrated GT strengths up to an excitation energy of 30 MeV in the intermediate nucleus In-116 are 40 +/- 6 and 11 +/- 1 in the (p, n) and (n, p) spectra, respectively, including the component from the isovector spin monopole (IVSM). The spectra are compared with those obtained from a theoretical calculation with a quasiparticle random phase approximation, where the contribution from the IVSM component as well as the interference effect between the GT and IVSM components are fully taken into account.