AMC8 2020

AMC8 2020 · Q24

AMC8 2020 · Q24. It mainly tests Fractions, Area & perimeter.

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 $\frac{d}{s}$ for this larger value of $n?$

一个大正方形区域铺有 $n^2$ 块边长 $s$ 英寸的灰色正方形瓷砖。每块瓷砖周围有一条宽度 $d$ 英寸的边框。下图显示 $n=3$ 的情况。当 $n=24$ 时，$576$ 块灰色瓷砖覆盖了大正方形区域的 $64\%$ 面积。对于这个较大的 $n$ 值，$\frac{d}{s}$ 的比值为多少？

(A) \frac{6}{25} \frac{6}{25}

(B) \frac{1}{4} \frac{1}{4}

(D) \frac{7}{16} \frac{7}{16}

(E) \frac{9}{16} \frac{9}{16}

Answer

Correct choice: (A)

正确答案：（A）

Solution

The area of the shaded region is $(24s)^2$. To find the area of the large square, we note that there is a $d$-inch border between each of the $23$ pairs of consecutive squares, as well as from between the first/last squares and the large square, for a total of $23+2 = 25$ times the length of the border, i.e. $25d$. Adding this to the total length of the consecutive squares, which is $24s$, the side length of the large square is $(24s+25d)$, yielding the equation $\frac{(24s)^2}{(24s+25d)^2}=\frac{64}{100}$. Taking the square root of both sides (and using the fact that lengths are non-negative) gives $\frac{24s}{24s+25d}=\frac{8}{10} = \frac{4}{5}$, and cross-multiplying now gives $120s = 96s + 100d \Rightarrow 24s = 100d \Rightarrow \frac{d}{s} = \frac{24}{100} = \boxed{\textbf{(A) }\frac{6}{25}}$. Note: Once we obtain $\tfrac{24s}{24s+25d} = \tfrac{4}{5},$ to ease computation, we may take the reciprocal of both sides to yield $\tfrac{24s+25d}{24s} = 1 + \tfrac{25d}{24s} = \tfrac{5}{4},$ so $\tfrac{25d}{24s} = \tfrac{1}{4}.$ Multiplying both sides by $\tfrac{24}{25}$ yields the same answer as before.

灰色区域面积是 $(24s)^2$。大正方形的边长：相邻 $23$ 对方格之间有 $d$ 英寸边框，加上首尾与大正方形之间的，总共 $23+2 = 25$ 次边框长度，即 $25d$。加上 $24s$，大正方形边长为 $(24s+25d)$，得方程 $\frac{(24s)^2}{(24s+25d)^2}=\frac{64}{100}$。两边取平方根（利用长度非负）得 $\frac{24s}{24s+25d}=\frac{8}{10} = \frac{4}{5}$，交叉相乘得 $120s = 96s + 100d \Rightarrow 24s = 100d \Rightarrow \frac{d}{s} = \frac{24}{100} = \boxed{\textbf{(A) }\frac{6}{25}}$。注意：得到 $\tfrac{24s}{24s+25d} = \tfrac{4}{5}$ 后，取倒数得 $\tfrac{24s+25d}{24s} = 1 + \tfrac{25d}{24s} = \tfrac{5}{4}$，所以 $\tfrac{25d}{24s} = \tfrac{1}{4}$。两边乘 $\tfrac{24}{25}$ 得相同结果。

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