Overlaps

Question:

Determine how many times do the minute and hour hands of a clock overlap in a day?

.

.

C

A

P

T

A

I

N

I

N

T

E

R

V

I

E

W

.

.

Solution: 22 times, not 24 times.

In T hours, the minute hand completes T laps. In the same amount of time, the hour hand completes T/12 laps.
The first time the minute and hour hands overlap, the minute hand would have completed 1 lap more than the hour hand. So we have T = T/12 + 1. This implies that the first overlap happens after T = 12/11 hours (~1:05 am). Similarly, the second time they overlap, the minute hand would have completed two more laps than the hour hand. So for N overlaps, we have T = T/12 + N.
Since we have 24 hours in a day, we can solve the above equation for N:
24 = 24/12 + N
24 = 2 + N
N = 22
The hands will overlap at 12:00, 1:05, 2:10, 3:15, 4:20, 5:25, 6:30, 7:35, 8:40, 9:45, and 10:50. Do consider the fact that there will be no 11:55. It becomes 12:00.

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