Problem 20 · 2004 AMC 8
Medium
Fractions, Decimals & Percents
fraction-from-empty
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
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Answer: D — 27 people.
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Hint 1 of 2
The one concrete number you're given is '6 empty chairs'. Find what fraction of the chairs is empty — that fraction equals 6, which unlocks the total number of chairs. Start from the fact that ties to a real count.
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Hint 2 of 2
The strategy is start from the known quantity and work outward: 3/4 of chairs are filled, so 1/4 are empty → chairs → seated people → (that's 2/3 of everyone) → total people. Each fraction is a stepping stone, not a thing to combine all at once.
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Approach: anchor on the empty chairs, then chain
- 3/4 of the chairs are filled, so 1/4 are empty — and that 1/4 is the 6 empty chairs we're told about. So 1/4 of chairs = 6 ⇒ total chairs = 24.
- Filled chairs = 3/4 × 24 = 18 people seated. But seated people are 2/3 of everyone in the room.
- So 2/3 of the people = 18 ⇒ total people = 18 ÷ (2/3) = 18 × 3/2 = 27.
- Sanity check: 27 people — 18 seated (2/3) and 9 standing (1/3); 9 is indeed a third, so it's consistent. The habit that transfers: grab the lone numerical fact, convert it into a count, then let each fraction pull you to the next quantity.
Mark:
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