In DS-CDMA systems with Successive Interference Cancellation (SIC), the numbers of possible rate modes and decoding orders grow exponentially with the number of users. How to allocate rates in order to maximize the system utility is an NP problem, which involves the joint optimization of rate allocation and decoding order adjustment. This paper proposes two suboptimal greedy algorithms for addressing this NP problem. The main idea of the algorithms is that the rate mode is greedily updated step by step, then the decoding order is adjusted for the updated rate mode in every step and the feasibility of the corresponding rate mode is quickly examined according to the adjusted decoding order. The proposed algorithms have very low computational complexity and can be used to implement the quick and efficient allocation of rates. Simulation experiments validate this analysis and demonstrate the effectiveness of the algorithms.