Easy Google Interview Question!

Actually, I would like to generalize the formula for finding the angle between the hour hand and minute hand in degrees.

Consider a time representation $h:m$ where $h$ is the no. of complete hours passed and $m$ the exact and precise minutes elapsed. Then:

The exact hours elapsed = $h + \frac{m}{60}$ Starting from $12$ , part of the clock swiped by the hour hand = $\frac{\left(h+\frac{m}{60}\right)}{12}$

Angle (in degrees) between $12$ and the hour hand = $\frac{\left(h+\frac{m}{60}\right)}{12}\cdot 360 = 30h\ +\ \frac{m}{2}$

Similarly, starting from 12, part of the clock swiped by the minute hand = $\frac{m}{60}$

Angle (in degrees) between $12$ and the minute hand = $\frac{m}{60}\cdot 360 = 6m$

Clearly then, the angle between the hour hand and minute hand will be = $|30h\ +\ \frac{m}{2} - 6m| = \boxed{|30h\ -\ \frac{11m}{2}|}$

Note: One must notice that when $h=12$ , one must take $h=0$ since $12$ has been taken as the frame of reference here. (12 is the starting point)

Simple, the hour hand moves $30^\circ$ in $60$ mins. ie, it moves at a rate of $\dfrac {30}{60} =0. 5^\circ /min$

At quarter past three the minute hand is points directly at $3$ while hour hand has moved $0.5\times 15 = 7.5^\circ$

Hence, the angle between hour and minute hand is $(90^\circ + 7.5^\circ) - 90^\circ = \boxed{7.5^\circ}$