Quite recently, the first author investigated vanishing coefficients of the arithmetic progressions in several $q$-series expansions. In this paper, we further study the signs of coefficients in two $q$-series expansions and establish some arithmetic relations for several $q$-series expansions by means of Ramanujan's theta functions. We obtain the 5-dissections of these two $q$-series and give combinatorial interpretations for these dissections. Moreover, we obtain four $q$-series identities involving the aforementioned $q$-series, two of which were proved by Kim and Toh via modular forms.