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14. Hands of a Clock

When are the hands of a clock in a straight line?

Clearly 6 o'clock and 12 o'clock are possible answers. They are not the only ones, though, but the others are not so easy to find...
 

Scroll down the page for the answer.


 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Many will suggest a quarter past nine, twenty past ten, and so on, as answers but, of course, while they might be close, they're not exact...

...so the hour hand and minute hand turn through angles xh and xm respectively in time t minutes.

the minute hand takes 60 minutes to turn 2 pi radians

so  xm = 2 pi t / 60

so  xm = pi t / 30

the hour hand takes 12 * 60 ( = 720 ) minutes to turn 2 pi radians

so  xh = 2 pi t / 720

so  xh = pi t / 360

now for the hands to be in a straight line, then the angle between them must be 0 or 180 degrees, ie a multiple of pi

so  xm - xh = n pi

where n = 0, 1, 2, 3...

so  (pi t / 30) - (pi t / 360) = n pi

so  pi t( 1 / 30 - 1 / 360 ) = n pi

so  t( 12 / 360 - 1 / 360 ) = n

so  t( 11 / 360 ) = n

so  t = 360n / 11

Taking different values for n and starting with n = 0 you should verify that n = 0 corresponds to 12 o'clock, n = 1 gives 32 and 8/11 minutes (which is shortly after 12 30), n = 2 gives 65 and 5/11 minutes (which is shortly after 1 05) and so on.
 

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