smc

Let $n$ be the number of minutes after 6:00. Let $h(n)=180+\frac{n}{2}$ be the angle, in degrees, of the hour hand (with $0^{\circ }$ at the top and increasing in the clockwise direction); similarly, let $m(n)=6n$ be the angle of the minute hand. We want $|h(n)-m(n)|=110$. This is equivalent to $$180-\frac{11n}{2}=\pm 110$$ $$-\frac{11n}{2}\in \{-70,-290\}$$ $$\frac{11n}{2}\in \{70,290\}$$ $$11n\in \{140,580\}$$ $$n\in \{\frac{140}{11},\frac{580}{11}\}$$ The difference between the two values of $n$ is $\frac{440}{11}=\boxed{40}$.