When the Clock Hands Meet

Measure angles in laps and time in hours. The minute hand turns at $1$ lap per hour and the hour hand at $\frac{1}{12}$ lap per hour. The minute hand therefore gains on the hour hand at

The hands start together. They next coincide when the minute hand has gained exactly one full lap. That takes

Equivalently, in degrees, $360t=30t+360$, which gives $t=\frac{12}{11}$.

This is $1$ hour and $5\frac{5}{11}$ minutes, so the meeting is at about 1:05:27.

Generalisation. The minute hand gains a lap every $\frac{12}{11}$ hours. The hands therefore coincide at times $\frac{12k}{11}$ hours after noon, for $k=0,1,2,\dots$. That is $11$ meetings in every $12$ hours, not $12$.