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Problem 24 (2020 AMC 8)

AMC 8Hard

From module Composite Figures and Shaded Areas

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Understand the scale
Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
AreaComposite FiguresRatios

Problem

A large square region is paved with n2n^2 gray square tiles, each measuring ss inches on a side. A border dd inches wide surrounds each tile. The figure below shows the case for n=3.n = 3. When n=24n = 24, the 576576 gray tiles cover 64%64\% of the area of the large square region. What is the ratio ds\frac{d}{s} for this larger value of nn? [See figure on AoPS]

(A) 625(B) 14(C) 925(D) 716(E) 916\textbf{(A)}\ \frac{6}{25} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{9}{25} \qquad \textbf{(D)}\ \frac{7}{16} \qquad \textbf{(E)}\ \frac{9}{16}

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