Starting from the time shown, at what time will the two hands next lie exactly on top of each other?

The clock shows 4:05. At 4:00 the minute hand is $120^{\circ }$ behind and gains $5.5^{\circ }$ a minute, so they meet $\frac{120}{5.5}=21\frac{9}{11}$ min past 4, which is after 4:05. (20 min ignores the hour hand's motion.)