At what time, exactly, do the two hands of a clock overlap for the first time after 12:00 noon?
Let d = the distance between numbers (on the arc). It’s clear that the hands will cross at a few minutes after 1:00, let’s say at d + x.
Now, the minute hand (the big one) has gone the same fraction of the whole circumference, namely 12d, that the hour hand has gone the distance between 1 and 2.
That is
| d + x | = | x |
| 12d | d |
d(d + x) = 12dx
d2 + dx = 12dx
d2 – 11dx = 0
d(d – 11x) = 0
x = d/11.
So the hands cross at exactly 1/11 of an hour after 1:00.
But 1/11 of 3600 seconds is 327 3/11 seconds, or 5 minutes, 27 3/11 seconds.
So, the hands cross at 1:05:27 3/11.
(Now, what about the next time after that? and the next? And, uh-oh? Does this suggest another way to go at the problem?)