A bee between two trains
Two trains are 100 km apart on the same track and head towards each other, one at 50 km/h and the other at 70 km/h. A bee starts at the front of the first train and flies at 80 km/h to the second train, turns around instantly, flies back to the first, and keeps going back and forth until the trains meet.
How many kilometres does the bee fly in total?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Do not follow the bee. How long does it fly?
The trains close the gap at km/h.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The zigzag is a distraction. The bee flies at a constant speed for as long as the trains take to meet, so the distance is speed times time.
Step 1: the time. The gap of 100 km closes at km/h, so the trains meet after
Step 2: the distance.
The long way. Summing the legs one by one also works. The first leg lasts hour, after which the trains are only 20 km apart; each round trip shrinks the gap by the same factor, so the legs form geometric series whose total is again km. The story goes that John von Neumann answered at once; asked whether he knew the trick, he said he had summed the series.