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A sea beacon consists of two rotating bulbs emitting green [#permalink]
09 Aug 2003, 23:17

A sea beacon consists of two rotating bulbs emitting green and white rays. At midnight, the bulbs are directly opposite and start to rotate clockwise. The green bulb makes 8 rotations per hour; the white 7 rph. At what time will the rays coincide with each other?

Solving the problem intuitively, the green bulb gains 1 rotation every 1 hour as in (8-7) rotations. now this means that every one hour, the green bulb gains 360degrees over the white one. now, the initial difference between them is 180degrees which the green bulb has to make up. the rate of gain being 360degrees every hour, it would take 1/2 hours to cover the required 180degrees. to check, in 1/2 hours, the green bulb moves 4 rotations while the white one moves 3.5, after which they both should coincide having started 1/2 a rotation apart.

or another way to solve is that you know that after every hour, the 2 bulbs should return to their original positions as they each complete a whole number of rotations (8 and 7 respectively) and this is true because 8 and 7 have no common factors. the common factors thing is important as say if you had 4 and 6 instead, then after 1/2 hour, the 4 one would complete 2 rotations and the 6 one would complete 3, hence both returning to their original positions. now since the earliest both return to their respective original positions is an hour, they should cover half of the respective decisions and hence be opposite each other in 1/2 an hour.