Problem 12 · 2021 Math Kangaroo
Medium
Logic & Word Problems
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The map shows three bus stations at points A, B and C. A tour from station A to the Zoo and the Port and back to A is 10 km long. A tour from station B to the Park and the Zoo and back to B is 12 km long. A tour from station C to the Port and the Park and back to C is 13 km long. Also, a tour from the Zoo to the Park and the Port and back to the Zoo is 15 km long. How long is the shortest tour from A to B to C and back to A?
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Answer: B — 20 km
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Hint 1 of 2
Each given tour is a there-and-back loop, so it is twice some pair of legs.
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Hint 2 of 2
Add up the right combination of half-loops to build the A-B-C-A circuit.
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Approach: route through the inner points and use the loop totals
- Send the trip A→Zoo→B→Park→C→Port→A. Then A→Zoo and Port→A are the two legs of the 10-loop minus its Zoo–Port leg, and similarly for the other two stations.
- Adding up, the six legs total (10 − Zoo–Port) + (12 − Park–Zoo) + (13 − Port–Park) = 35 − (Zoo–Port + Park–Zoo + Port–Park).
- But Zoo–Port + Park–Zoo + Port–Park is exactly the 15-km Zoo–Park–Port loop, so the tour is 35 − 15 = 20 km.
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