Problem 16 · 1994 AJHSME
Medium
Geometry & Measurement
scaling
The perimeter of one square is 3 times the perimeter of another square. The area of the larger square is how many times the area of the smaller square?
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Answer: E — 9.
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Hint 1 of 2
Perimeter is just 4 sides added up, so it grows in lockstep with the side. If the perimeter tripled, the side tripled too — the '4' and the units cancel out of the ratio.
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Hint 2 of 2
Here's the trap the answer choices set: area does NOT triple. Length is one-dimensional, area is two-dimensional, so the area ratio is the side ratio MULTIPLIED BY ITSELF.
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Approach: area scales as the square of the side ratio
- Perimeter = 4 × side, so a 3× perimeter means a 3× side. The big square's side is 3 times the small one's.
- Area = side × side. Triple a side and you triple it in BOTH directions, so area becomes 3 × 3 = 9 times as big.
- Picture it: lay the small square inside the big one and you fit a 3-by-3 grid of them — 9 copies. That's the whole reason answer '3' is a trap. The rule (scale length by k → area by k², volume by k³) shows up everywhere from maps to model rockets.
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