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2016 AMC 10B Problem 11

Problem 11 of 25IntermediateAlgebraGeometryCombinatorics

Carl decided to fence in his rectangular garden. He bought 2020 fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly 44 yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?

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Solution

Let ll be the number of posts on a long side, including corners, and let ss be the number of posts on a short side, including corners. The total number of posts is 2l+2s−4=20,2l+2s-4=20, where the −4-4 avoids double-counting the corners. Since a long side has twice as many posts as a short side, l=2sl=2s. Thus 2(2s)+2s−4=202(2s)+2s-4=20, so s=4s=4 and l=8l=8. There are 33 equal gaps on a short side and 77 equal gaps on a long side. With 44 yards between neighboring posts, the side lengths are 1212 and 2828, so the area is 12⋅28=33612\cdot28=336. Thus, the correct answer is B.
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Tagged: system of equations · perimeter · area · fencepost counting

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