It was some time between 7 and 8 PM that Sam looked at his watch. Some time later, between 10 and 11 PM, he looked again, and coincidentally, the hour hand and minute hand were exactly switched from the first time he looked. What were the two times?
Suppose the two times are x minutes after 7 pm and y minutes after 10 pm.
At each time, equating the fraction of a revolution covered by alternate hands since they were last at the top gives:
7/12 + x/720 = y/60 and 10/12 + y/720 = x/60 or, more simply:
420 + x = 12y and 600 + y = 12x
Solving simultaneously gives: x = 7620/143 and y = 5640/143.
So the two times are: 41/143 of a minute after 7:53 pm and 63/143 of a minute after 10:39 pm.

Posted by Harry
on 20090425 17:29:56 