<PRE><B>Question 188391</B>
Let us start at 12:00 when the two hands are together.


The hour hand will pass through A degrees.


The minute hand will pass through 720 degrees (2 hours) + (180 + A) degrees to be opposite the hour hand.


Since the minute hand travels 12 times as fast as the hour hand,


12A = 720 + 180 + A


11A = 900


A = 81 +  9/11


Which is 21 +  9/11 into the third hour since the first two hours take up 60 degrees.


The hour hand will have passed through 21 +  9/11 degrees since 2 o’clock, it is twice this amount in minutes past 2 o’clock since the hour hand takes 60 minutes to pass through 30 degrees (2 minutes per degree).


Therefore the time is 43 + 7/11 minutes past 2 o’clock.


We can verify this by calculating the proportions of the hour the hands have moved through in degrees with respect to an hour.


i.e. (21 + 9/11)/30 = (180 + 60 + 21 + 9/11)/360 = 8/11


180 is to be opposite, 60 is for the two hours and 21 +  9/11 is A.


Time is 2:43:38 + 2/11 of a second.
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