Problem 6 · 2011 AMC 8
Easy
Counting & Probability
complementary-counting
In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle?
Show answer
Answer: D — 306.
Show hints
Hint 1 of 2
Flip the question: instead of "car owners without a motorcycle," ask "who has no motorcycle at all?" Since everyone owns at least one vehicle, anyone without a motorcycle must own a car — so those two groups are the same people.
Still stuck? Show hint 2 →
Hint 2 of 2
The 331 car-owners is a decoy here — complementary counting (total minus the unwanted group) sidesteps it entirely.
Show solution
Approach: complementary counting — car-only = everyone who lacks a motorcycle
- Every adult owns at least one vehicle, so an adult without a motorcycle owns a car and nothing else — exactly the "car but no motorcycle" group we want.
- Count the non-motorcycle owners: 351 − 45 = 306.
- Why this transfers: when every item is in "A or B or both," the "A only" group is just everyone minus the B group — counting the complement beats wrestling with the overlap.
Another way — inclusion-exclusion as a check:
- Both = cars + motorcycles − total = 331 + 45 − 351 = 25 own both.
- Car owners without a motorcycle = 331 − 25 = 306, matching above.
Mark:
· log in to save