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This paper discusses the issue of routing packets over an IEEE 802.11 wireless ad hoc network with multiple data rates (1, 2, 5.5, and 11 Mb/s). With the characteristics of modulation schemes, the data rate of a wireless network is inversely proportional with the transmission distance. The conventional shortest path of the minimum-hops approach will be no longer suitable for the contemporary multi-rate/multi-range wireless networks (MR2WN). We propose an efficient delay-oriented multi-rate/multi-range routing protocol (MR2RP) for MR2WN to maximize the channel resource utilization as well as to minimize the network end-to-end transfer delay. By analyzing the medium access delay of the IEEE 802.11 medium access control (MAC) protocol, the proposed MR2RP is capable of predicting the end-to-end transfer delay of a routing path and find the best one. The proposed MR2RP may choose a longer path but with less contention competitors and buffer queuing delay. Simulation results show that MR2RP performs the load balancing and fast routing very well and its call blocking probability is obviously lower than that of conventional minimum-hops approach with fixed transmission rate.