When are the clock hands first exactly opposite?

Question

Solve in your head. At what exact time after 12:00 are the hour and minute hands first pointing in exactly opposite directions (180 degrees apart)? Aim: under 30 seconds.

Accepted Answer

At 12:00 both hands sit at 0^\circ, so the gap starts at 0. The minute hand pulls away from the hour hand at the relative speed 6 - 0.5 = 5.5^\circ/min. You want that gap to reach 180^\circ: t = \frac{180}{5.5} = \frac{360}{11} = 32	frac{8}{11} 	ext{ minutes.} Convert the fraction: 	frac{8}{11} min = 	frac{8}{11}	imes 60 = 	frac{480}{11} \approx 43.6 s. So the hands are first opposite at 12:32:43	frac{7}{11}. Sanity check: opposite alignments, like overlaps, recur every 	frac{720}{11} \approx 65	frac{5}{11} minutes, so they happen 11 times in 12 hours. The first is a bit past the hal…

When are the clock hands first exactly opposite?

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