Problem 18 · 1987 AJHSME
Hard
Fractions, Decimals & Percents
compose-fractions
Half the people in a room left. One third of those remaining started to dance. There were then 12 people who were not dancing. The original number of people in the room was what?
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Answer: C — 36.
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Hint 1 of 2
Follow the 12 non-dancers backward. What single fraction of the ORIGINAL crowd are they?
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Hint 2 of 2
Half the people stay, and of those 1⁄3 dance — so 2⁄3 of the stayers don't. Chain the fractions: 1⁄2 of the room, then 2⁄3 of that.
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Approach: compose the fractions back to the original
- Half stay, so the stayers are 1⁄2 of the original. Of the stayers, 1⁄3 dance, leaving 2⁄3 not dancing. Chain them: the non-dancers are 2⁄3 × 1⁄2 = 1⁄3 of the original room.
- That third equals 12, so the whole room is N = 3 × 12 = 36.
- Why this transfers: 'a fraction of a fraction' multiplies — collapsing the two steps into one fraction of the start lets you solve in a single division instead of tracking three separate counts.
Another way — walk forward and check:
- Start with 36: half leave → 18 remain. A third of 18 dance → 6 dancers, so 18 − 6 = 12 not dancing.
- Matches the given 12, confirming 36.
Mark:
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