  Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely
generated $R$-module of dimension $d$. We prove that the set
$\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}$ is bounded
below by ${1}/{d!e(\overline{R})}$ where $\overline{R}=R/Ann(M)$. Moreover,
when $\widehat{M}$ 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 St\"{u}ckrad--Vogel in the
affirmative. As an application, we obtain results on uniform behavior of the
lengths of Koszul homology modules.
