At what time immediately prior to six o'clock are the hands of the clock exactly opposite to each other?
Give the exact time in hours, minutes and seconds.

I propose a completely arithmetic solution.

Since the hour hand takes 60 minutes to move through the equivalent of 5 minutes for the minutes hand, we can say that the hour hand catches up with the minutes hand at the rate of 1/12.
The same phenomenon (hands opposing, or at 90 degrees, etc.) repeats itself approximately every hour, increased by the rate of 12/11 (the hour hand catches up!):
So number of minutes before 6 o'clock before the hands were opposing:
60*12/11=720/11=65min 27+3/11 sec.,
or 4:54:32+8/11
It happens again at 7:05:27+3/11, etc.

For example, when will the two hands be perfectly superimposed after 12:00?
Answer is 13:05:27+3/11, 14:10:54+6/11, etc.

Posted by P C on 2004-02-04 22:17:36