Problem 20 · 2021 Math Kangaroo
Medium
Logic & Word Problems
Algebra & Patterns
substitution
Three villages are connected by paths as shown. From Downend to Uphill, the detour via Middleton is 1 km longer than the direct path. From Downend to Middleton, the detour via Uphill is 5 km longer than the direct path. From Uphill to Middleton, the detour via Downend is 7 km longer than the direct path. How long is the shortest of the three direct paths between the villages?
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Answer: C — 3 km
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Hint 1 of 2
Name the three direct distances and turn each 'detour is k longer' fact into an equation.
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Hint 2 of 2
Add all three equations to get the total of the distances quickly.
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Approach: set up and add the detour equations
- Let the direct paths be DU = a, DM = b, UM = c. The detours give b+c = a+1, a+c = b+5, a+b = c+7.
- Adding all three: 2(a+b+c) = (a+b+c) + 13, so a+b+c = 13.
- Then a = 6, b = 4, c = 3; the shortest direct path is 3 km.
- So the answer is C.
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