In this paper we study the distributed average consensus problem in multi-agent systems with dynamically-changing directed communication links that are subject to quantized information flow. We present and analyze a distributed averaging algorithm which operates exclusively with quantized values (i.e., the information stored, processed and exchanged between neighboring agents is subject to deterministic uniform quantization) and relies on event-driven updates (e.g., to reduce energy con-sumption, communication bandwidth, network congestion, and/or processor usage). We characterize the properties of the proposed distributed algorithm over dynamic directed communication topologies subject to some connectivity conditions and we show that its execution allows each agent to reach, in finite time, a fixed state that is equal (within one quantization level) to the average of the initial states. The main idea of the proposed algorithm is that each agent (i) models its initial state as two quantized fractions which have numerators equal to the agent's initial state and denominators equal to one, and (ii) transmits one fraction randomly while it keeps the other stored. Then, every time an agent receives one or more fractions, it averages their numerators with the numerator of the fraction it stored, and then transmits them to randomly selected out-neighbors. Finally, we provide examples to illustrate the operation, performance, and potential advantages of the proposed algorithm. We compare against various quantized average consensus algorithms and show that our algorithm's convergence speed is among the fastest in the current literature.