Question

how many times a day do the hour & minutes hand on a clock line up exactly with each other

Answer 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. Thus, the hands of a clock overlap 22 times a day.

Answer 2

24 hours 84600 sec a day