2020 AMC 8 Problems/Problem 24
A large square region is paved with gray square tiles, each measuring inches on a side. A border inches wide surrounds each tile. The figure below shows the case for . When , the gray tiles cover of the area of the large square region. What is the ratio for this larger value of
WLOG, let . Then, the total area of the squares of side is , of the area of the large square, which would be , making the side of the large square . Then, borders have a total length of . Since if is the value we're asked to find, the answer is .
When , we see that the total height of the large square is . Similarly, when , the total height of the large square is . The total area of the gray tiles is and the area of the large white square is . We are given that the ratio of the gray area to the area of the large square is . Thus, our equation becomes . Square rooting both sides, we get . Cross multiplying, we get . Combining like terms, we get , which implies that .
The area of the shaded region is . The total area of the square is because for each side of the square there one extra row/column of the border. Our equation is . Taking the square root of both sides gives . Cross multiplying and rearranging gives .
WLOG . For large enough , we can approximate the percentage of the area covered by the gray tiles by subdividing most of the region into congruent squares, as shown: Each red square has side length , so by solving , we obtain . The actual fraction of area covered by the gray tiles is slightly less than , which implies . Hence (and ) is less than , and the only choice that satisfies this is . ~scrabbler94
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