Problem 24 · 2020 AMC 8
Hard
Geometry & Measurement
area-fractionratio
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Answer: A — d/s = 6/25.
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Hint 1 of 2
No real measurements are given, only ratios — so set the big square's side to 1 and work in fractions. The single most useful move: 64% is a perfect square, 64% = (4/5)2, so areas turn into lengths cleanly.
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Hint 2 of 2
Gray covers (4/5)2 of the area split among 242 tiles, so each tile's side is (4/5)/24 = 1/30. Then the side of the big square = 24 tiles + 25 borders, which solves for d.
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Approach: normalize the big square to side 1; turn the % into a length
- Only ratios matter, so let the large square have side 1. The gray fraction 64% = (4/5)2, and it's shared by 576 = 242 equal tiles, so each tile has area (4/5)2 / 242 = (1/30)2.
- Take the square root to get the tile's side: s = 1/30.
- Lay out one side: 24 tile widths and 25 border widths (a border on each side of every tile) fill the unit length: 24·(1/30) + 25d = 1 ⇒ 25d = 1 − 4/5 = 1/5 ⇒ d = 1/125.
- d/s = (1/125) ÷ (1/30) = 30/125 = 6/25.
- Why this transfers: when a problem gives only proportions, fix the overall size to 1 so everything becomes a fraction. And a percentage that's a perfect square (64% → 4/5) is a strong hint to bridge from area to length by taking a square root.
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