Given a graph $G$, the matching number of $G$, written $α'(G)$, is the maximum size of a matching in $G$, and the fractional matching number of $G$, written $α'_f(G)$, is the maximum size of a fractional matching of $G$. In this paper, we prove that if $G$ is an $n$-vertex connected graph that is neither $K_1$ nor $K_3$, then $α'_f(G)-α'(G) \le \frac{n-2}6$ and $\frac{α'_f(G)}{α'(G)} \le \frac{3n}{2n+2}$. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.

8 pages, 2 figures