After 12:00, when are the hands of the clock at 180 degree to each other for the first time?  

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We need to find what the time is when the hands of the clock are at 180 degree to each other for the first time after 12:00. Let this be after x minutes. Now we know that the minute hand moves (360/60)*x degrees in x minutes. The hour hand moves (360/12*60)*x degrees in a minute.  Now when the two hands are at 180 degree to each other it means that (360/60)*x - (360/12*60)*x = 180

canceling 180

=> 2x/60 - 2x/12*60 = 1

=> x (1/ 30 – 1/360) = 1

=> x = (1/ 30 – 1/360) ^-1

=> x = 32.727 min

Therefore the two hands are at 180 degree to each other at 32.72 minutes past 12:00.

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