Solution to Clock hands meet

How many times do clock hands meet ?

How many times a day do the minute and hour hands of a clock meet?

Solution:

1. 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, N = 22

Thus, the hands of a clock overlap 22 times a day. They overlap at 12:00, ~1:05, ~2:10, ~3:15, ~4:20, ~5:25, ~6:30, ~7:35, ~8:40, ~9:45, ~10:50. Note that there is no ~11:55. This becomes 12:00.