Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist 1,2, a series of remarkable work 1,3–6 provided approximation algorithms for a 23-MMS allocation in which each agent receives a bundle worth at least 23 times her maximin share. More recently, Ghodsi et al. 7 showed the existence of a 34-MMS allocation and a PTAS to find a (34−ϵ)-MMS allocation for an ϵ>0. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain. In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 34-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial time algorithm to find a 34-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a (34+112n)-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that 34 was the best factor known for n>4.