2016 AMC 10B Problem 11

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

Answer: B

Concepts:system of equations perimeter area fencepost counting

Difficulty rating: 1280

Small hint:

Let

s

and

l

be posts on the short and long sides, including corners

Big hint:

Use

l=2s

and

2l+2s-4=20

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.