Problem 25 · 2022 Math Kangaroo
Stretch
Logic & Word Problems
caseworksum-constraint
Four villages A, B, C and D lie (not necessarily in this order) along a straight road. A and C are 75 km apart, B and D are 45 km apart, and B and C are 20 km apart. Which of these distances cannot be the distance from A to D?
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Answer: C — 80 km
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Hint 1 of 2
Place the points on a line using the known gaps AC = 75, BD = 45, BC = 20; AD depends on the left-right order.
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Hint 2 of 2
Try the possible orderings of A, B, C, D consistent with the gaps and list the AD distances that can occur.
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Approach: place the points with signed positions on the line
- Put C at 0. Then B is at +20 or -20 (since BC = 20), A is at +75 or -75 (since AC = 75), and D sits 45 from B.
- Trying all the consistent combinations, A to D comes out as 10, 50, 100 or 140 km.
- AD = 80 km never appears, so the distance that cannot occur is C.
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