Routing in wireless ad hoc networks is a challenging issue because it dynamically controls the network topology and determines the network performance. Most of the available protocols are based on single-rate radio networks and they use hop-count as the routing metric. There have been some efforts for multirate radios as well that use transmission-time of a packet as the routing metric. However, neither the hop-count nor the transmission-time may be a sufficient criterion for discovering a high-throughput path in a multirate wireless ad hoc network. Hop-count based routing metrics usually select a low-rate bound path whereas the transmission-time based metrics may select a path with a comparatively large number of hops. The trade-off between transmission time and effective transmission range of a data rate can be another key criterion for finding a high-throughput path in such environments. In this paper, we introduce a novel routing metric based on the efficiency of a data rate that balances the required time and covering distance by a transmission and results in increased throughput. Using the new metric, we propose an on-demand routing protocol for multirate wireless environment, dubbed MR-AODV, to discover high-throughput paths in the network. A key feature of MR-AODV is that it controls the data rate in transmitting both the data and control packets. Rate control during the route discovery phase minimizes the route request (RREQ) avalanche. We use simulations to evaluate the performance of the proposed MR-AODV protocol and results reveal significant improvements in end-to-end throughput and minimization of routing overhead.