The authors consider the following delayed ratio-dependent predator-prey system \[ \begin{aligned} x(t) &= x(t)\Biggl[a(t)- b(t) \int^t_{-\infty} K(t- s) x(s)\,ds\Biggr]- e(t) g\Biggl({x(t)\over y(t)}\Biggr),\\ y(t) &= y(t)\Biggl[e(t) g\Biggl({x(t)- \tau(t)\over y(t)- \tau(t)}\Biggr)- d(t)\Biggr],\end{aligned}\tag{\(*\)} \] where \(x(t)\) and \(y(t)\) represent the predator and prey densities, respectively, \(a(t)\), \(b(t)\), \(c(t)\), \(d(t)\), \(e(t)\) and \(\tau(t)\) are positive periodic continuous functions with period \(\omega> 0\), \(\omega\) is a positive real constant. \(K(s): \mathbb{R}^+\to \mathbb{R}^+\) is a measurable, \(\omega\)-periodic, normalized function such that \(\int^{+\infty}_0 K(s)\,ds= 1\). By using the continuation theorem of the coincidence degree theory [see R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)], the authors establish two main theorems on the existence of at least one positive \(\omega\)-periodic solution of system \((*)\) when the functional response function \(g\) is monotonic or nonmonotonic. As corollaries, some applications are listed.