Question

Between $1$ o'clock and $2$ o'clock, find the time when the minute hand and an hour hand are at angle of ${90}^{\circ }$ for the first time.

Answer 1

When the hands are at right angle, they are $15$ minutes apart. So the hands are at right angle for the first time when the minute hand will gain $(5+15)$ minutes i.e. $20$ minutes and it can do it in $20\times \frac{12}{11}$ minutes i.e. in $21\frac{9}{11}$ minutes. The time will be $21\frac{9}{11}$ minutes past 1 o'clock.