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2023 AMC 8

Problem 13

Problem 13 · 2023 AMC 8 Medium
Fractions, Decimals & Percents proportionfraction-to-decimal
Figure for AMC 8 2023 Problem 13
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Answer: D — 48 miles.
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Hint 1 of 2
Forget miles for a moment — locate each station as a fraction of the whole route. The watch-out: stations sit between start and finish, so 2 stations make 3 equal gaps (not 2).
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Hint 2 of 2
2 repair stations → route split into thirds, so the 1st repair is at 1/3. 7 water stations → eighths, so the 3rd water is at 3/8. The gap between those positions is the given 2 miles.
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Approach: convert station positions to fractions of the route
  1. Pin each station as a fraction of the route. The key (and the trap): n stations spaced between start and finish cut it into n+1 equal pieces. So 2 repair stations → thirds (1st repair at L/3) and 7 water stations → eighths (3rd water at 3L/8).
  2. The 3rd water is 2 miles past the 1st repair, so 3L/8 − L/3 = 2.
  3. Combine over 24: (9L − 8L)/24 = L/24 = 2, so L = 48 miles. Worth keeping: ‘k things evenly spaced between two ends’ always means k+1 gaps — the classic fencepost catch.
Another way — spacing variables (MAA):
  1. Let w be the gap between water stations and r the gap between repair stations. The whole route is 8w = 3r.
  2. The 3rd water (at 3w) is 2 past the 1st repair (at r): 3w = r + 2. Sub r = 3w − 2 into 8w = 3r: 8w = 9w − 6, so w = 6.
  3. Race length = 8w = 48 miles.
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