In this paper, we propose a fast adaptive power allocation method for an orthogonal frequency division multiple access (OFDMA) system that employs an adaptive modulation and coding (AMC) scheme. The proposed scheme aims to reduce the calculation complexity of greedy adaptive power allocation (APA), which is known as the optimal algorithm for maximizing the utility argument of power. Unlike greedy APA, which starts power allocation from "0", the proposed algorithm initially allocates a certain level of power determined by the water-filling scheme. We theoretically demonstrate that the proposed algorithm has almost the same capability of maximizing the utility argument as the greedy APA while reducing the number of operations by 2M, where M is the number of AMC levels.