Question

How many times in a day do the two hands on a clock coincide?

Answer 1

`22`

Explanation:

In one hour, the minute hand goes through a complete circle, while the hour hand traces `1/12` of a circle.

So the hour hand travels at `1/12` of the speed of the minute hand.

Consider the time after `1` o'clock at which the hands coincide.

At `1` o'clock, the minute hand is still at "12", so requires `1/12` of an hour to catch up with where the hour hand currently is. By that time, the hour hand has moved on `1/12^2` of a revolution. So the minute hand will need to travel that additional distance, etc.

In terms of revolutions of the hour hand, the minute and hour hand will coincide after:

`1/12+1/12^2+1/12^3+... = 1/11`

That is: the hour hand will have travelled round by `1/11` of a revolution by the time the minute hand catches up with it again.

So the times at which the two hands coincide divide the clock into `11`ths. In `24` hours the hour hand goes round twice, resulting in `22` times at which the hands coincide.