This paper presents a new partitioned scheduling approach to schedule fixed-priority periodic real-time tasks on multi-core platforms under the Rate Monotonic Scheduling (RMS) policy. We first develop a novel task set scaling method to scale all tasks with respect of any task’s period such that the ratio of the maximum period over the minimum period is less than 2. Based on our task set scaling method, we present an enhanced utilization bound for single-core systems under RMS policy. Then we exploit our task set scaling method and the enhanced utilization bound for multi-core scheduling, and propose a new partitioned scheduling algorithm to maximize the system utilization while guaranteeing the real-time constraints for all tasks. Our experimental results demonstrate that the proposed algorithm can improve the scheduling performance compared with existing works.