Problem 5 · AMC 8 Stretch
Stretch
Ratios, Rates & Proportions
Logic & Word Problems
logical-reasoningvisual-representationpattern-recognition
Anton and Ben start running toward each other from the two ends of a long straight path (Anton from end A, Ben from end B). Each runs at his own steady speed. They first meet 800 m from Ben's end. They keep going, reach the far ends, instantly turn around, and meet a second time 400 m from Anton's end. How long is the path (in meters)?
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Answer: 2000 m
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Hint 1 of 4
Don't track each runner separately at first — track the TOTAL distance the two of them run together.
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Hint 2 of 4
At the FIRST meeting, the two of them together have covered the path exactly once (their two pieces fill it).
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Hint 3 of 4
At the SECOND meeting, together they have covered the path exactly three times. (Draw it: each one went to the far end and partway back.)
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Approach: Combined distance triples between the two meetings
- Because both run steadily, their distances grow in the same proportion. At the first meeting they have together run one full path length; at the second meeting they have together run three path lengths (each finishes the path and comes partway back).
- So the combined distance tripled, and since both run steadily, each runner's own distance also triples.
- Ben ran 800 m to the first meeting (it was 800 m from his end B), so by the second meeting Ben has run \(3 \times 800 = 2400\) m. The second meeting is 400 m from Anton's end A, meaning Ben ran the whole path and came back 400 m, so path \(= 2400 - 400 = 2000\) m.
- The path is 2000 m long. (Check Anton: he ran \(2000 - 800 = 1200\) m to the first meeting, then \(3 \times 1200 = 3600\) m by the second, which is one path plus 1600 m back, leaving him \(2000 - 1600 = 400\) m from end A. It fits.)
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