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 $\frac{d}{5}$ for this larger value of $n$?

A. $\frac{6}{25}$

B. $\frac{1}{4}$

C. $\frac{9}{25}$

D. $\frac{7}{16}$

E. $\frac{9}{16}$

1. 2020 AMC 8 官方解析

When $n = 24$, there are $24^2 = 576$ gray tiles, each with an area of $s^2$ square inches. Together the tiles cover an area of $24^2 s^2$ square inches. Each side of the large square region spans 24 tiles plus 25 borders for a side length of $24s + 25d$ inches, giving a total area of $(24s + 25d)^2$ square inches. It is given that the gray tiles cover $64\%$ of the total area. It follows that