Question

At what time between 4 and 5 O'clock will the minute hand and the hour hand coincide with each other?

Answer 1

When the two hands coincide with each other the angle between them is $0^o$.
$\theta =\frac{11}{2}m-30h$. Here $\theta =0^o,h=4\frac{11}{2}m=30h$
$\Rightarrow 11m=60h\Rightarrow m=\frac{60h}{11}$
$m=\frac{60\times 4}{11}=\frac{240}{11}=21\frac{9}{11}$ min past 4.
The two hands of the clock coincide at 4 hours $21\frac{9}{11}min$.