How many times do the hour and minute hands meet on an analog clock every day? What are those times?

#### Solution

22 times.

Starting at midnight, it would be roughly 12:00, 1:05:27, 2:10:55, 3:16:22, 4:21:49, 5:27:16, 6:32:44, 7:38:11, 8:43:38, 9:49:05, 10:54:33, and then the same 11 times starting at noon.

The easiest way to solve this:

- Every hour, the hour hand moves 1/12 of the way around the clock. Every hour, the minute hand moves the entire way around the clock.
- If we put this in terms of how much the minute hand moves
*relative*to the hour hand every hour, this means minute hand effectively moves 1 – 1/12 = 11/12 of the way towards the hour hand each hour.- In other words, it would take 12/11 of an hour (roughly 65.455 minutes or roughly 1 hour, 5 minutes, and 27 seconds) for the minute hand to catch up to the hour hand again.

- Since there are 24 hours in a day, that would be 24 / (12/11) = 22 times the hands meet each day.