At what time between 2 and 3 will the hour and minute hands of a clock

At 2:00, the hour hand is at the 2 o’clock mark (10 minute divisions or 60 degrees from 12), and the minute hand is at the 12 o’clock mark (0 degrees). The angle between them is 60 degrees, with the hour hand ahead of the minute hand.

The minute hand moves at 6 degrees per minute, and the hour hand moves at 0.5 degrees per minute. The relative speed of the minute hand with respect to the hour hand is 6 – 0.5 = 5.5 degrees per minute.

Let ‘t’ be the number of minutes past 2 when the hands are 72 degrees apart.
At time ‘t’, the angle of the minute hand from 12 is 6t degrees.
At time ‘t’, the angle of the hour hand from 12 is (60 + 0.5t) degrees.

We want the absolute difference between these angles to be 72 degrees:
|6t – (60 + 0.5t)| = 72
|5.5t – 60| = 72

Two possible cases:
1) 5.5t – 60 = 72
5.5t = 132
t = 132 / 5.5 = 1320 / 55 = 264 / 11 = 24 minutes.
This corresponds to the minute hand being ahead of the hour hand by 72 degrees.

2) 5.5t – 60 = -72
5.5t = -12
t = -12 / 5.5. This is a negative time, which is not applicable for the time between 2 and 3. This situation (hour hand 72 degrees ahead of minute hand) occurs before 2:00.

Therefore, the only time between 2 and 3 when the hands are 12 minute divisions (72 degrees) apart is exactly 24 minutes past 2.