Question

A large square region is paved with $n^2$ shaded 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 $576$ shaded tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n?$

$\dfrac{6}{25}$

$\dfrac{1}{4}$

$\dfrac{9}{25}$

$\dfrac{7}{16}$

$\dfrac{9}{16}$

Video solution:

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Written solution:

For $n=24,$ the shaded tile area is $24^2s^2.$ Each side of the large square consists of $24$ tiles and $25$ borders, so its side length is $24s+25d.$

The shaded tiles cover $64\%=\dfrac{16}{25}$ of the large square, so $\dfrac{24^2s^2}{(24s+25d)^2}=\dfrac{16}{25}.$ Taking positive square roots gives $\dfrac{24s}{24s+25d}=\dfrac{4}{5}.$

Thus $120s=96s+100d,$ so $24s=100d$ and $\dfrac{d}{s}=\dfrac{6}{25}.$

Thus, the correct answer is A.

2020 AMC 8 Problem 24

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Problem 24 in Other Years