How many times in a day, the hands of a clock are together?

Concept

The minute hand moves at 6 degrees per minute and the hour hand at 0.5 degrees per minute, so the minute hand gains on the hour hand at a relative rate of 5.5 degrees per minute. The hands coincide again only once the minute hand has gained a full 360 degrees on the hour hand, which takes 360 / 5.5 = 65 5/11 minutes.

Applying it here

1. Starting from 12:00, the next coincidence is about 65 5/11 minutes later, near 1:05, then near 2:11, 3:16, 4:22, 5:27, 6:33, 7:38, 8:44, 9:49, and 10:55.

2. The coincidence that would next be due lands exactly at 12:00 again, the start of the next 12-hour stretch, so it is not a separate, extra alignment within the same stretch.

3. That gives 11 distinct alignments in every 12-hour stretch, from one 12:00 to the next.

4. A day has two such 12-hour stretches (AM and PM), so the total for 24 hours is 11 × 2 = 22.

Cross-check

Using the general rule directly: in 24 hours (1440 minutes) the hands coincide 1440 / (720/11) = 22 times, since a coincidence recurs every 720/11 minutes. This matches the count obtained by listing the alignments one by one above.