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Math Help - Clock problem

  1. #1
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    Clock problem

    Interchangeable positions of minute hand and hour hand occur when the original interval between the two hands is 60/13 minute spaces or a multiple of this.
    Could anyone explain this with examples?
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  2. #2
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    Re: Clock problem

    Hey hisajesh.

    Can you show us what you have tried? (Hint: Try and formulate the problem in terms of congruence equations and number theory).
    Thanks from topsquark
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  3. #3
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    Re: Clock problem

    I learnt congruence equation from Congruence Equation -- from Wolfram MathWorld

    But I have no clue what can I do with congruence equation.
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  5. #5
    MHF Contributor ebaines's Avatar
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    Re: Clock problem

    I think the question is stated incorrectly. The hour hand and minute hand line up every 12/11 hours, or in other words after the minute hand has made 12/11 revolutions. The new position of the minute hand is therefore 1/11 further along than on the previous alignmemt, which works out to be 60/11 = 5.4545 minutes.

    The derivation of the 12/11 figure come from this: the position of the minute hand is M= 2 \pi t where t is in hours, and the position of the hour hand is H = 2 \pi \frac t {12}. The hands line up when M is equal to H plus a multiple of  2  \pi:

     2 \pi t  = 2 \pi ( \frac t {12})+ k (2 \pi)

    where k = 1, 2, 3... This occurs when

      t =  \frac {k (2 \pi)} {2 \pi - \frac {2\pi} {12}} = \frac k {1 - \frac 1 {12}} = \frac {12 }{11}k

    Thus every multiple of 12/11 hours the hands are realigned, at which time the hands are 60/11 "minutes space" further on.
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