When are the hands opposite between 4 and 5 o'clock?

Question

Solve in your head. At what exact time between 4:00 and 5:00 do the hour and minute hands point in exactly opposite directions (180 degrees apart)? Aim: under 30 seconds.

Accepted Answer

At 4:00 the hour hand is at 4 	imes 30^\circ = 120^\circ, the minute hand at 0^\circ: the hour hand leads by 120^\circ. The relative speed of the minute hand is 5.5^\circ/min. For the hands to be opposite in this hour, the minute hand must move around so the separation becomes 180^\circ. Starting 120^\circ behind, the minute hand needs to gain enough relative angle to make the gap 180^\circ. The clean way: the minute hand must gain 120^\circ to reach overlap, then continue until it leads by 180^\circ, for a total relative gain of 120 + 180 = 300^\circ: t = \frac{300}{5.5} = \frac{600}{11} = …

When are the hands opposite between 4 and 5 o'clock?

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