When do the hands overlap after 3 o'clock?

Question

Solve in your head. At what exact time after 3:00 do the hour and minute hands coincide? Aim: under 30 seconds.

Accepted Answer

Treat it as a pursuit problem. At 3:00 the gap is 90^\circ. The minute hand moves at 6^\circ/min, the hour hand at 0.5^\circ/min, so the gap closes at 5.5^\circ/min: t = \frac{90}{5.5} = \frac{180}{11} = 16	frac{4}{11} 	ext{ minutes.} So the hands coincide at 3:16:21	frac{9}{11} (since 	frac{4}{11} min = 	frac{240}{11} \approx 21.8 s). Takeaway. Everything about clock hands reduces to the relative speed 5.5^\circ/min. Corollary worth caching: overlaps happen every \frac{360}{5.5} = \frac{720}{11} \approx 65	frac{5}{11} minutes, which is why the hands coincide only 11 times in 12 hours, …

When do the hands overlap after 3 o'clock?

Your answer

More Mental Math questions