Problem 11 · 1995 AJHSME
Hard
Ratios, Rates & Proportions
relative-speedperimeter
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Answer: D — Point D.
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Hint 1 of 2
When two people start together and walk opposite ways around a loop, at the instant they meet their distances ADD UP to the whole loop. That one fact replaces any speed/time algebra.
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Hint 2 of 2
Speeds split the loop in the same ratio as the speeds. Jane is twice as fast, so of the 18 blocks she walks 2 parts and Hector 1 part — Hector walks just 13 of 18 = 6 blocks. Then trace those 6 blocks on the picture.
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Approach: their combined distance equals the loop; split it by the speed ratio
- The key idea: going opposite ways around the rectangle, when they first meet their two distances together equal the full perimeter, 18 blocks. No need to find time or actual speed.
- Speed ratio is 2 : 1 (Jane : Hector), so they split the 18 blocks the same way: Jane walks 12, Hector walks 6.
- Trace Hector's 6 blocks from the start (bottom middle): 3 blocks right to corner E, then 3 blocks up the right side — landing exactly on corner D. (Jane's 12 blocks bring her to D too, confirming the meeting point.)
- Why this transfers: for two travelers closing a loop or a gap, 'distances sum to the whole' plus 'distances split as the speed ratio' solves almost every meeting problem with no equations.
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