# 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.**

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.

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