At precisely what time between one and two o'clock is the minute hand exactly over the hour hand?

Question

And please support your answers! Thank you!

Answer 1

You could solve this by figuring out the rate the hour hand turns and the minute hand turns, etc. But it is much easier to think laterally.

The hour and minute hands of a clock line up perfectly at 12 noon, then again at regular intervals after that. But those intervals are not an even hour. It's a little more.

Because the hands line up *eleven* times in a 12 hour period, the interval between alignment is 12/11.

If you convert this to a time, it is exactly:
12/11 = 1.0909... hours
= 1 hour 5.4545... minutes
= 1 hour 5 minutes 27.2727 seconds
= 1 hour 5 minutes 27 3/11 seconds.

So the time when the hour and minute hands align perfectly will be at: 1:05:27 3/11

First alignment: 1:05:27 3/11
Second alignment: 2:10:54 6/11
Third alignment: 3:16:21 9/11
...
Tenth alignment: 10:54:32 8/11
Eleventh alignment: 12:00:00

The time between one and two o'clock is:
1 hour 5 minutes 27 3/11 seconds.

Answer 2

The minute hand moves 6 degrees a minute.
The hour hand moves 1/2 degree a minute.

Since the minute and hour hands are 30 degrees apart (1/12 of the circle) at the start, it takes 30/(6-1/2) minutes after 1:00 for them to match up.

Simply to 30/5.5 or 60/11. Therefore, the hands will be exactly on top of each other 5 5/11 minutes after one

Basically the time will be 1:05:45.45, rounded to the nearest hundredth of a second.

Just for the record, at 1:06 the hands will be 3 degrees apart.

Answer 3

for sure yet 12:20:40 or 8:00:20 are not the solutions because the hour hand also strikes a touch because the minutes bypass. for example for 12:20:40 the hour hand will be between 12 and a million. oh yeah and there can't be endless solutions on your question there is really 12 solutions.

Answer 4

between 1.05 and 1.06 the hour hand should be at 5.5minutes as it moves one minute every 12 minutes the minute hand covers