Problem 13 · 2022 Math Kangaroo
Hard
Ratios, Rates & Proportions
arithmetic-sequenceunit-rate
The gap between two shelves in Monika’s kitchen is 36 cm. A stack of 8 identical glasses is 42 cm high, and a stack of 2 such glasses is 18 cm high. What is the largest number of glasses in one stack that still fits between the shelves?
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Answer: D — 6
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Hint 1 of 2
A stack's height grows by the same amount for each extra glass; find that per-glass increase.
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Hint 2 of 2
Get the height of one glass, then find the largest count whose stack height stays at or below 36 cm.
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Approach: linear height model for the stack
- From 2 glasses (18 cm) to 8 glasses (42 cm), 6 extra glasses add 24 cm, so each extra glass adds 4 cm.
- One glass is 18 - 4 = 14 cm, and n glasses reach 14 + 4(n-1) = 10 + 4n cm.
- Need 10 + 4n <= 36, so n <= 6.5; the biggest stack is 6 glasses, the answer is D.
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