Answer
The fraction of the rope that can hang over the edge of the table is $\frac{\mu_s}{\mu_s + 1}$.
Work Step by Step
Let $x$ be the fraction of the rope that hangs over the table. If the rope does not slide, then the weight of the hanging section is equal in magnitude to the force of static friction. Let's find $x$ when the force of static friction is at its maximum.
$x~mg = (1-x)~mg~\mu_s$
$x = (1-x)~\mu_s$
$x + x~\mu_s = \mu_s$
$x = \frac{\mu_s}{\mu_s + 1}$
The fraction of the rope that can hang over the edge of the table is $\frac{\mu_s}{\mu_s + 1}$.
