Question:
At $2\!:\!48$ what is the degree measure of the smaller angle formed by the hour and minute hands of a $12$-hour clock?

Answer:
We consider a hand at the 12 to be $0^\circ$. Now we convert the hour and minute hands to a degree measure from $0^\circ$ to $360^\circ$. If we divide $360^\circ$ evenly among 60 minutes, we get that each minute, the minute hand moves $\frac{360^\circ}{60}=6^\circ$. So if the minute hand is at 48 minutes, it is at $48\cdot6^\circ=288^\circ$.

The hour hand is a little trickier. If we divide $360^\circ$ evenly among 12 hours, we get that each hour, the hour hand moves $\frac{360^\circ}{12}=30^\circ$. Note that the hour hand is not at the 2 since it gradually moves toward the 3 throughout the hour. From the 2 toward the 3, the hour hand has moved $\frac{48}{60}=\frac{4}{5}$ of the way. So the degree measure of the hour hand is $2\frac{4}{5}\cdot30^\circ=84^\circ$.

To find the smaller angle formed by the two hands, we can find the larger angle $288^\circ-84^\circ=204^\circ$ and subtract from $360^\circ$ to get $\boxed{156^\circ}$. Or we know that $84^\circ$ is coterminal with (ends at the same place as) $84^\circ+360^\circ=444^\circ$. Now we can subtract $444^\circ-288^\circ=\boxed{156^\circ}$ to find the smaller angle.