Question

At what time between 4 and 5 O'clock will the minute hand and the hour hand make an angle of ${30}^o$ with each other?

Answer 1

$\theta =\frac{11}{2}m-30h$,
here $\theta ={30}^o,h=4,m$
$\Rightarrow 11m=60h+2\theta$
$\Rightarrow m=\frac{60h+2\theta }{11}=\frac{60\times 4+2\times 30}{11}$
$=\frac{240+60}{11}=\frac{300}{11}=27\frac{3}{11}$ min past 4.
$\theta =30h-\frac{11}{2}m$
$11m=60h-2\theta \Rightarrow m=\frac{60h-2\theta }{11}$
$\Rightarrow \frac{60\times 4-2\times 30}{11}=\frac{240-60}{11}$
$=\frac{180}{11}=16\frac{4}{11}$ min pas 4 O'clock
Therefore, the angle between the two hands is ${30}^o$ when the time is 4 hours $16\frac{4}{11}$ min and 4 hours $27\frac{3}{11}min$.