At what time immediately prior to six o'clock are the hands of the clock exactly opposite to each other?
Give the exact time in hours, minutes and seconds.
(In reply to
problem solution by K Sengupta)
Between 5 and 6 o'clock, we observe that if the hour hand stood still, then at precisely 5*5+30=55 minutes past 5 o' clock the hands of the clock would be exactly opposite to each other.
When both hands are active, the minute hand gains precisely 1 minute mark over the hour hand in exactly 12/11 minutes.
Therefore, it gains 55 minute mark over the hour hand in (55*12)/11 =60 minutes past 5 o'clock or at precisely 6 o'clock. This violates the given conditions.
Between 4 and 5o'clock, we observe that for the clock hands to be opposite each other, the minute hand must gain 4*5+30=50 minute marks over the hour hand. This will happen at (50*12)/11 = 54+6/11 minutes past 4.
Now, 6/11 minutes = 360/11 seconds = 32+8/11 seconds ~ 32.7273 seconds.
Consequently, immediately prior to 6 o'clock, the clock hands are opposite to each other at approximately 54 minutes 32.7273 seconds past 4.
COMMENT: MY earlier result made 14 years ago was 54 minutes 32.74 seconds past 4 o' clock which is slightly incorrect.
Edited on January 3, 2022, 9:37 pm
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