Problem 15 · 2014 Math Kangaroo
Medium
Algebra & Patterns
sum-constraintcasework
A restaurant has 16 tables, each with 3, 4, or 6 chairs. The tables with 3 or 4 chairs seat 36 guests in total. The restaurant seats 72 guests altogether. How many tables have three chairs?
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Answer: A — 4
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Hint 1 of 2
First find how many seats the 6-chair tables provide, then how many such tables there are.
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Hint 2 of 2
Once you know the number of 3-or-4-chair tables and their 36 seats, set up two simple equations for the 3s and 4s.
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Approach: peel off the 6-chair tables, then nudge 4-chair tables down to 3
- The 3- and 4-chair tables seat 36 guests, so the 6-chair tables seat 72 − 36 = 36, which is 36 ÷ 6 = 6 tables of six.
- That leaves 16 − 6 = 10 tables with 3 or 4 chairs, and they seat 36 guests in total.
- Pretend all 10 had 4 chairs: that would be 40 seats, which is 4 too many.
- Each time you change a 4-chair table into a 3-chair table you lose one seat, so you need 4 of those changes.
- There are 4 tables with three chairs.
The same idea with a quick equation
If t tables have 3 chairs, then 3t + 4(10 − t) = 36, so 40 − t = 36 and t = 4.
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