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
Solution 1
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 .
Solution 2
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 576 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 . ~jmansuri
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.