2001 AMC 12 Problems/Problem 10

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Problem

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

$\text{(A) }50\qquad\text{(B) }52 \qquad\text{(C) }54\qquad\text{(D) }56\qquad\text{(E) }58$

Solution

Consider any single tile:

If the side of the small square is $a$ , then the area of the tile is $9a^2$ , with $4a^2$ covered by squares and $5a^2$ by pentagons. Hence exactly $5/9$ of any tile are covered by pentagons, and therefore pentagons cover $5/9$ of the plane. When expressed as a percentage, this is $55.\overline{5}\%$ , and the closest integer to this value is $56$ . $\boxed{\mathrm{D}}$

2001 AMC 12 (Problems • Resources)
Preceded by Problem 9	Followed by Problem 11
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2001 AMC 12 Problems/Problem 10

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