$P$--$Q$ "mixed" modular equations of degree 15

Ramanujan in his second notebook recorded total of seven $P$–$Q$ modular equations involving theta–function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta–functions $\phi (q)$ and $\psi (-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta–function and a continued fraction of order 12 are explicitly evaluated.