Let (R, m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set {l(M/IM)/e(I, M)}(root I=m) is bounded below by 1/d!e((R) over bar) where (R) over bar = R/ Ann(M). Moreover, when (M) over cap is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stuckrad-Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.