Problem 25 · 2012 Math Kangaroo
Hard
Fractions, Decimals & Percents
ratiosum-constraint
Tango is being danced in pairs, a man with a woman. No more than 50 people attend a dance evening. At a certain moment 34 of the men were dancing with 45 of the women. How many people were dancing at this moment?
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Answer: B — 24
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Hint 1 of 2
Dancing men and dancing women are equal in number (they pair up).
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Hint 2 of 2
Set (3/4) of the men equal to (4/5) of the women and use the at-most-50 limit.
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Approach: equal dancing partners
- Pairs mean dancing men = dancing women: (3/4)M = (4/5)W, so 15M = 16W, giving M = 16k, W = 15k.
- Total people 31k ≤ 50 forces k = 1, so M = 16, W = 15.
- Dancers = 2 × (3/4 × 16) = 2 × 12 = 24.
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