¶ 2020 AMC8 Problem 24
Problem:
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
Answer Choices:
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Solution:
When there are gray tiles, each with an area of square inches. Together the tiles cover an area of square inches. Each side of the large square region spans tiles plus $254 borders for a side length of inches, giving a total area of square inches. It is given that the gray tiles cover of the total area. It follows that
which simplifies to
Solving the equation for produces the following result:
Therefore, when the ratio .
Let the side length of each gray tile bes inch. Then the tiles have a total area of square inches. Because of the large square region is covered by gray tiles, the large square region has an area of square inches, and hence its side length is inches. Each side of the large square region tiles plus borders, so and . The desired ratio is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions