Problem 30 · 2019 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraint
A train has 18 carriages. There are 700 passengers on the train. In every five successive carriages there are exactly 199 passengers in total. How many passengers are in the two middle carriages of the train altogether?
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Answer: D — 96
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Hint 1 of 2
Every five consecutive carriages hold the same total, which forces a repeating pattern.
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Hint 2 of 2
Carriage counts repeat with period 5, so carriage i equals carriage i+5.
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Approach: use the period-5 pattern from equal 5-sums
- Since each window of 5 carriages sums to 199, sliding by one shows carriage i and carriage i+5 carry the same number.
- So the 18 carriages repeat with period 5; carriages 1-15 give 3 × 199 = 597, leaving carriages 16,17,18 to total 700 − 597 = 103.
- Those equal carriages 1,2,3, so carriages 4+5 = 199 − 103 = 96; the two middle carriages 9,10 equal carriages 4,5, giving 96.
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