Question

At what time, between five o'clock and six o'clock,the minute and hour hands make an angle of 34${}^o$ with each other?

Answer 1

The angle between the minute hand and the hour hand at five o'clock is 150${}^o$.
The angle between the hands becomes 34${}^o$ when the angle changes by 116${}^o$ and 184${}^o$ i.e., (150${}^o$ - 34${}^o$) and (150${}^o$ +34${}^o$).
The angle changes by 5.5${}^o$ in 1 minute.
The angle changes by 116${}^o$ in $\frac{1}{5.5}\times 116=21\frac{1}{11}$ minutes.
The angle changes by 184${}^o$ in $\frac{1}{5.5}\times 184=33\frac{5}{11}$ minutes.
Therefore, the angle betweenthe two hands is 34${}^o$ when the time is 5 hours $21\frac{1}{11}$ minutes, and again at 5 hours $33\frac{5}{11}$ minutes.