At 6 A.M. or P.M. (it doesn’t matter which), a clock’s minute hand and hour hand point in opposite directions, forming a straight line across the clockface. When is the next time this will happen?
This will next occur at 55⁄11 minutes after seven o’clock, or approximately 7:05:27 A.M. or P.M. There are several ways to set up the problem, but we’ll work with angles.
At six o’clock, the minute hand points at the 12 on the clockface, which we’ll label as zero degree, or 0°. The minute hand completes a full 360° rotation every 60 minutes, which is a rate of 6° per minute. So after m minutes pass, the minute hand will be at angle 6m°.
At six o’clock, the hour hand points at the 6 on the clockface, so it begins at 180°. The hour hand does a full 360° rotation every 12 hours, or 720 minutes, which is a rate of 0.5° per minute (360° / 720 minutes = 0.5° per minute). So after m minutes pass, the hour hand will be at angle (180 + 0.5m)°.
For the hands to point in opposite directions in a straight line again, the difference between their angles must be 180°. The minute hand moves faster than the hour hand, so this will happen after the minute hand passes over the hour hand and then exceeds its angle by exactly 180°. So we set up our equation as (angle of minute hand after m minutes) – (angle of hour hand after m minutes) = 180° and solve for m:
6m°– (180° + 0.5m°) = 180°
This simplifies to 5.5m° = 360°, or m = 655/11 minutes, or one hour, five minutes and about 27 seconds later. By adding this amount of time to six o’clock, you get the answer of approximately 7:05:27.
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