Problem 4 · 2003 AMC 8
Easy
Algebra & Patterns
substitution
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted 7 children and 19 wheels. How many tricycles were there?
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Answer: C — 5 tricycles.
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Hint 1 of 2
Don't set up two equations — pretend every child is on a bicycle first and watch what's missing.
Still stuck? Show hint 2 →
Hint 2 of 2
Each tricycle is just a bicycle with one extra wheel, so the leftover wheels count the tricycles directly.
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Approach: assume all bicycles, then the leftover wheels count the tricycles
- Suppose all 7 children rode bicycles. That would be 7 × 2 = 14 wheels — but only counts as a starting guess.
- We actually see 19 wheels, so 19 − 14 = 5 wheels are unaccounted for. A tricycle is just a bicycle with one extra wheel, so each leftover wheel marks one tricycle: 5 tricycles.
- You'll see this again: the "assume the cheapest option, then spend the surplus" trick cracks chickens-and-rabbits, coins, and stamp problems without any algebra.
Another way — solve the system:
- With b bicycles and t tricycles: b + t = 7 and 2b + 3t = 19.
- Subtract twice the first from the second: t = 19 − 14 = 5.
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