Question

At what time, between four o'clock and five o'clock, are the hands of the clock at right angle?

Answer 1

At four o'clock the angle between the minute and the hour hand is 120${}^o$. This angle reduces at the rate of 5.5${}^o$ per minute till it reaches zero and thereafter the angle increases at the same rate. The two hands are at right angles when the angle between them changes by 30${}^o$ and 210${}^o$ (i.e., 120${}^o$ - 90${}^o$ and 120${}^o$ + 90${}^o$).
The angle changes by 5.5${}^o$ in 1 minute.
So, the angle changes by 30${}^o$ in $\frac{1}{5.5}\times {30}^o=5\frac{5}{11}$ minutes.
The angle changes by 210${}^o$ in $\frac{1}{5.5}\times {210}^o=38\frac{2}{11}$ minutes.
Thus, the hands are at right angle at $5\frac{5}{11}$ past 4 and again at $38\frac{2}{11}$ minutes past 4.