Two trains 100 km apart move toward each other at 50 km/h each. A fly starts on the front of one train, flies to the other at 75 km/h, turns instantly on arrival, and keeps shuttling until the trains meet. How far does the fly travel in total?

Two trains 100 km apart move toward each other at 50 km/h each. A fly starts on the front of one train, flies to the other at 75 km/h, turns instantly on arrival, and keeps shuttling until the trains meet. How far does the fly travel in total?

Approach: Find how long the trains take to meet before thinking about any individual leg of the fly's path, and note the fly's speed never changes.

75. The fly travels 75 km. The trains close at a relative speed of 50 + 50 = 100 km/h across a gap of 100 km, so they meet after exactly 1 hour. Getting the total time first is the whole solution: the fly is airborne for that entire hour at a constant 75 km/h, so it covers 75 km. Summing the individual legs gives the same number, since each leg is a fixed fraction of the one before and the resulting geometric series converges to 75. That route is slower and easy to get wrong. The fly makes infinitely many turns inside the hour, and the total distance stays finite because the legs shrink geometrically.

Follow-up: How many turns has the fly made by the halfway point in time, and where is it at that moment?

Key concepts: relative speed, total time first, geometric series.