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The time is past 2 o'clock in 10 minutes. The minute hand will be as much as ahead of the hour hand as it is behind it. What time is it?

The Answer is 2:05.91

I am having trouble interpreting " Minute Hand will be as much ahead of the hour Hand as it is Behind it ". Does it mean the hour hand somehow bisects the two times of the minute hand? I am more of a visual learner... but i dont know why clock problems for me are so hard to visualize. maybe there is an easier way?

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  • $\begingroup$ Is this the exact wording of the problem? $\endgroup$ – OnceUponACrinoid Jun 8 '15 at 2:42
  • $\begingroup$ @OnceUponACrinoid yes $\endgroup$ – james Jun 8 '15 at 4:32
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Here is how I would interpret the question: "The time is past 2 o'clock. In 10 minutes, the minute hand will be as much ahead of the hour hand as it is behind it [right now]." It's not quite right to say that the hour hand bisects the two positions of the minute hand, since the hour hand is also moving, but you're on the right track. Here is what we can infer:

  • Right now the minute hand is behind the hour hand (in particular, it is probably between 2:00 and 2:10).
  • In ten minutes the minute hand will be in front of the hour hand.
  • The distance right now between the minute hand and the hour hand is the same as the distance between them will be in ten minutes.

These points make 2:05.91 seem like a reasonable answer, although I'll admit that I'm getting a slightly different answer when I work the problem myself.

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I would suggest that the interpretation given by Annie Carter is correct, and would solve the problem as follows. Suppose that the time now is $m$ minutes past $2$. For convenience we measure angles in units of full circles. Then the angles between "12" on the clock, the centre of the clock, and the minute and hour hands are $$\angle TOM=\frac{m}{60}\ ,\quad \angle TOH=\frac{2+\frac{m}{60}}{12}\ .$$ In ten minutes time it will be $$\angle TOM=\frac{m+10}{60}\ ,\quad \angle TOH=\frac{2+\frac{m+10}{60}}{12} \ .$$ So we have $$\frac{m+10}{60}-\frac{2+\frac{m+10}{60}}{12} =\frac{2+\frac{m}{60}}{12}-\frac{m}{60}\ .$$ Probably the best way to solve this is to write it as $$\frac{2m+10}{60}=\frac{4+\frac{2m+10}{60}}{12}\ ;$$ you then easily get $2m+10=\frac{240}{11}$ and so $m=\frac{65}{11}=5.91$.

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