Problem 22 · 2015 Math Kangaroo
Stretch
Number Theory
divisibilitycasework
A train has 12 carriages. In each carriage there is the same number of compartments. Mike is sitting in the 18th compartment behind the engine, this is in the 3rd carriage. Joanna is sitting in the 50th compartment behind the engine, this is in the 7th carriage. How many compartments are in one carriage?
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Answer: B — 8
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Hint 1 of 3
Every carriage holds the same number of compartments, so try the answer choices one at a time.
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Hint 2 of 3
If each carriage has 8 compartments, carriages 1 and 2 use compartments 1–16, so carriage 3 holds 17–24.
Still stuck? Show hint 3 →
Hint 3 of 3
Check that the same guess also lands compartment 50 inside carriage 7.
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Approach: test the answer choices: find how many compartments fit before each carriage
- Try 8 compartments per carriage. Then carriages 1 and 2 fill compartments 1 through 16, so carriage 3 holds compartments 17 through 24 — and Mike's seat 18 lands there. ✓
- With 8 each, carriages 1–6 fill compartments 1 through 48, so carriage 7 holds 49 through 56 — and Joanna's seat 50 lands there. ✓
- Both facts work only with 8, so there are 8 compartments in one carriage.
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