2020 AMC 8 Problem 24

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}

The large side length is

24s+25d

Compare shaded area

24^2s^2

with total area

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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.

Problem 24 in other years