When the Hands Overlap

What is the first time after 3 PM when the hour and minute hands of a clock are exactly on top of each other?

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#Set the two angles equal

Measure each hand in degrees clockwise from 12, using $m$ for the minutes past 3. The minute hand sweeps $6^\circ$ per minute, so it sits at $6m$ . The hour hand begins at the 3, which is $90^\circ$ , and creeps on at $0.5^\circ$ per minute, so it sits at $90 + 0.5m$ . They coincide when

6m = 90 + 0.5m. \tag*{(1)}

#Solve for the minute

Starting at 3:00 the hour hand leads by 90 degrees, and the minute hand closes that gap at 5.5 degrees a minute. They meet after 180 over 11 minutes, so the first overlap past 3pm lands at 3:16 and 4/11 minutes, near 3:16:22.

Collecting terms leaves the minute hand gaining $5.5^\circ$ each minute on a $90^\circ$ lead,

5.5\,m = 90 \;\Longrightarrow\; m = \frac{180}{11} = 16\tfrac{4}{11}. \tag*{(2)}

So the hands first align at 3

and

\tfrac{4}{11}

of a minute. Since

\tfrac{4}{11}

of a minute is

21\tfrac{9}{11}

seconds, the exact moment is 3:16

and

\tfrac{9}{11}

, near 3:16

. The same

\tfrac{12}{11}

hour spacing separates every consecutive overlap around the dial.