Problem 17 · AMC 8 Stretch
Core
Ratios, Rates & Proportions
Arithmetic & Operations
asking-key-questionswork-backward
Two trains are \(200\) miles apart on the same track, heading toward each other. One goes \(60\) mph, the other \(40\) mph. A fly starts on the front of the slower train and flies back and forth between the two trains at \(240\) mph, turning around instantly each time it reaches a train, until the trains crash and squash it. How far does the fly travel in total?
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Answer: 480 miles
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Hint 1 of 3
Adding up the fly's endless shrinking back-and-forth trips is a nightmare. Ask the key question: what single thing, times the fly's speed, gives its total distance?
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Hint 2 of 3
Distance = speed \(\times\) time, and the fly is flying the WHOLE time until the trains crash. So the real question is: how long until they crash?
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Hint 3 of 3
The two trains close the \(200\)-mile gap together at \(60 + 40 = 100\) mph. Find how long that takes, then multiply by the fly's \(240\) mph.
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Approach: Asking the key question — find the time, not the zig-zags
- Don't add up the fly's zig-zags. Ask the key question: how long does the fly fly? Its distance is just speed times time, and it flies until the trains crash.
- The trains approach each other at a combined speed of \(60 + 40 = 100\) mph. To close a \(200\)-mile gap at \(100\) mph takes time \(= \frac{200}{100} = 2\) hours.
- The fly flies for those whole \(2\) hours at \(240\) mph: \(240 \times 2 = 480\) miles.
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