# 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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Every hour the minute hand of the clock moves one full circle or 360 degree. During the same period the hour hand moves by 1/12th of a complete circle, or 30 degrees.

Thus every 1 hour after 12:00 the net angle between the hour hand and minute hand increases by:

360 - 30 = 330 degrees

Therefore:

The time taken for the angle between hour and minute hand to increase from by 180 degrees from 0 to 180 degrees is given by:

Time for the angle to increase by 180 degrees = 180*(1/330)

= 6/11 hours.

= (6/11)*60 = 32.7272 minutes = 32 minutes and 44 seconds

Therefore:

Time after 12:00 when hands of clock are at 180 degrees to each others = 12:32:44

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We know that the hour hand rotates one full round  in 12 hours or 12*60 =  720 minutes.  So the hour hand  covers 360 degrees in  720 minutes. Therefore the hour hand moves with a speed of 1/2 degrre per minute.

The minute hand moves  360 degree every hour. So the speed of the minute hand per minute is 360/60 = 5 degree per minute.

Therefore  difference  in the angle of hour hand and minute hand per minute =  speed  of minute hand/minute - speed of hour hand per minute = 5-1/2 deg/min = 4.5 degree per minute.

Therefore  it takes 180 deree/ 4.5 = 40 mintes for the angle to be  180 degree for the first time between the hour hand and the minute hand.