Convergence rates of Markov chains have been widely studied in recent years. In particular, quantitative bounds on convergence rates have been studied in various forms by Meyn and Tweedie [Ann. Appl. Probab. 4 (1994) 981.1101], Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558.566], Roberts and Tweedie [Stochastic Process. Appl. 80 (1999) 211.229], Jones and Hobert [Statist. Sci. 16 (2001) 312.334] and Fort [Ph.D. thesis (2001) Univ. Paris VI]. In this paper, we extend a result of Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558.566] that concerns quantitative convergence rates for time-homogeneous Markov chains. Our extension allows us to consider f-total variation distance (instead of total variation) and time-inhomogeneous Markov chains. We apply our results to simulated annealing.