Problem 20 · 2016 Math Kangaroo
Stretch
Algebra & Patterns
substitutionsum-constraint
Luigi owns a few square tables and some chairs for his little restaurant. If he sets out his tables individually with 4 chairs each, he is 6 chairs short. If he always puts two tables together to make a bigger table with 6 chairs, he has 4 chairs left over. How many tables does Luigi have?
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Answer: B — 10
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Hint 1 of 3
Pairing tables uses 6 chairs for every 2 tables, which is 3 chairs per table; singling uses 4 chairs per table.
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Hint 2 of 3
So switching from paired to single needs 1 more chair for each table.
Still stuck? Show hint 3 →
Hint 3 of 3
Compare how the chair situation swings from '4 left over' to '6 short'.
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Approach: see how 1 extra chair per table swings the count
- Paired up, the tables use 6 chairs per 2 tables, which is 3 chairs per table; set out singly they use 4 chairs per table.
- So going from paired to single costs 1 extra chair for each table.
- The situation swings from 4 chairs left over to 6 chairs short, a change of \(4 + 6 = 10\) chairs, and that swing is exactly 1 chair per table, so there are 10 tables, choice (B).
With algebra
Chairs \(=4T-6\) (singly) and \(=3T+4\) (paired); setting them equal, \(4T-6=3T+4\) gives \(T=10\).
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