Problem 18 · 2022 Math Kangaroo
Hard
Logic & Word Problems
work-backward
The bus stops in the villages A, B, C and D lie along a road in this order, and neighbouring villages are 10 km apart. There are 10 children in village A, 20 in B, 30 in C and 40 in D. Every child takes the bus to school. A new school will be built where the total number of kilometres travelled by all the children is as small as possible. Where will the new school be built?
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Answer: D — in C
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Hint 1 of 3
The best spot has about as many children on one side of the school as on the other side.
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Hint 2 of 3
Start at one end and add up children until you reach more than half of all of them.
Still stuck? Show hint 3 →
Hint 3 of 3
The village where you pass the halfway count is the best place for the school.
Show solution
Approach: walk from one end and stop where you pass half the children
- There are 10 + 20 + 30 + 40 = 100 children in all, so half of them is 50.
- Counting from A: A has 10, then A and B have 30, then A, B and C have 60 - we pass 50 right at C.
- Since just as many children sit on each side once we reach C, building the school in C makes the total travel smallest.
- So the answer is D.
Check by trying neighbours
Moving the school 10 km from C toward D saves 40 children 10 km each (400 km) but costs the other 60 children 10 km each (600 km), a net loss; moving it toward B is worse too, so C truly is best.
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