Question

At what time, between four o'clock and five o'clock, the two hands of the clock overlap?

Answer 1

At four o'clock, the hour hand will be at 4 and the minute hand will be at 12. So, the angle between the two hands at four o'clock is 120${}^o$.
The hands overlap when the angle between them reduces to zero.
The angle changes by 5.5${}^o$ in 1 minute.
So, time taken to change the angle by ${120}^o=\frac{1}{55}\times 120=21\frac{9}{11}$ minutes.
Therefore, the hands overlap at $21\frac{9}{11}$ minutes past 4.