Question

At what time, between four o'clock and five o'clock, are the hands 2 minutes space apart?

Answer 1

The hands are two minutes space apart is the other way of telling that the angle between the hands is 12${}^o$ (one minute space apart is equal to 6${}^o$).
At four o'clock, the angle between the hands is 120${}^o$. When this angle changes by $({120}^o-{12}^o)={108}^o$ and $({120}^o+{12}^o)={132}^o$, the angle between the hands reduces to 12${}^o$.
The angle between the hands changes by 5.5${}^o$ in 1 minute.
The angle changes by ${108}^o$ in $\frac{1}{5.5}\times 108=19\frac{7}{11}$ minutes.
The angle changes by ${132}^o$ in $\frac{1}{5.5}\times 132=24$ minutes.
Thus, the hands are 2 minutes space apart, once at $19\frac{7}{11}$ minutes past 4 and again at 24 minutes past
4 [i.e., approximately at 4 hours, 19 minutes, 38 seconds and at 4 hours, 24 minutes].