AMC 8 2020 Problem 24

AMC8 2020 年第 24 题

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A large square region is paved with n^2 gray square tiles, each measuring s inches on a side. A border d inches wide surrounds each tile. The figure below shows the case for n=3 . When n=24 , the 576 gray tiles cover 64% of the area of the large square region. What is the ratio

\displaystyle \frac{d}{s}

for this larger value of n?

一个大正方形区域用

n^2

块灰色方形瓷砖铺成，每块边长为

s

英寸。每块瓷砖四周都有宽

d

英寸的边框。下图为 n=3 的情形。当 n=24 时，576 块灰色瓷砖覆盖了大正方形面积的 64％。那么对于这个较大的

n

值，

\displaystyle \frac{d}{s}

的比值是多少？

A. $\frac6{25}$
B. $\frac14$
C. $\frac9{25}$
D. $\frac7{16}$
E. $\frac9{16}$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 拼出边长列方程大方边长 =24 段砖 $+25$ 段边框 =24s+25d（23 条砖间 + 2 条外缘 = 25 条）。灰占比

\displaystyle \frac{(24s)^2}{(24s+25d)^2}=\frac{64}{100}

，开方

\displaystyle \frac{24s}{24s+25d}=\frac{8}{10}

，得

240s=192s+200d\Rightarrow 48s=200d

，故

\frac{d}{s}=\frac{48}{200}=\frac{6}{25}

Solution 1

Elementary · build the side length, set up an equation The big square's side is 24 tiles plus 25 borders =24s+25d (23 gaps between tiles plus 2 outer margins). The gray fraction is

\displaystyle \frac{(24s)^2}{(24s+25d)^2}=\frac{64}{100}

; taking square roots,

\displaystyle \frac{24s}{24s+25d}=\frac{8}{10}

, so

240s=192s+200d\Rightarrow 48s=200d

and

\frac{d}{s}=\frac{48}{200}=\frac{6}{25}.

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