Dear Aspirant,

We know , 1 day = 24 hours.

The Minute hand covers 360 degree in 60 , hence, speed of minute hand is 6 degree per minute.

The Hour hand covers 360 degree in 12 hours , So

Speed of hour hand = 30 degree per hour

= 1/2 degree per minute.

Now, we calculate the Angular relative speed between minute and hour hand = 6–1/2 = 11/2 degree per minute.

When both minute hand and hour hand overlap (taking reference of 12:00 AM, as at this point both hands are at same position but will move at different speed), they will infact overlap at the point when the relative speed between them covers the angular distance of 360 degree.

Now, Time = angular distance/angular relative speed

= 360/ (11/2)

= 720/11 minutes.

Hence, in every 720/11 minute, the two hands overlap each other.

So, the number of times the two hands overlap will actually be equal to number of same minutes in 24 hour duration

= 24*60 minutes / (720/11)

=  24 *60/ (720/11)

= 24 * 60 * 11/720

= 22 times.

So, the hands of a clock will overlap 22 times in 24 hour duration.

Hence, they shall overlap 12 times in 12 hour duration.

Hope it helps. Thank you.