Question

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ 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?

$\ 256$

$\ 336$

$\ 384$

$\ 448$

$\ 512$

Solution:

Let $l$ be the number of posts on a long side, including corners, and let $s$ be the number of posts on a short side, including corners. The total number of posts is $2l+2s-4=20,$ where the $-4$ avoids double-counting the corners.

Since a long side has twice as many posts as a short side, $l=2s$ . Thus $2(2s)+2s-4=20$ , so $s=4$ and $l=8$ .

There are $3$ equal gaps on a short side and $7$ equal gaps on a long side. With $4$ yards between neighboring posts, the side lengths are $12$ and $28$ , so the area is $12\cdot28=336$ .

Thus, the correct answer is B.

2016 AMC 10B Problem 11

Solution:

Problem 11 in Other Years