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If the hands start off together at midnight, say, then the minute hand must catch up with the hour hand 11 times before they both end up on top of each other again at noon.

Hence between one and two o'clock the time when the hands are in the same place is 60/11 = 5.4545... minutes past the hour, between two and three o'clock 2*60/11 = 10.9090... minutes past the hour and so on.

So the time you want is 10.91 minutes past 2 o'clock which, to the nearest second, is 2:10:54.

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Q: What time between 2 and 3 o' clock do the minute hand and the hour hand overlap each other?
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