AMC12 2016 B

AMC12 2016 B · Q9

AMC12 2016 B · Q9. It mainly tests Fractions, Basic counting (rules of product/sum).

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?

卡尔决定把他的长方形花园围起来。他买了 20 根栅栏桩，在四个角各放了一根，其余的均匀地沿着花园边缘放置，使相邻两根桩之间的距离恰好为 4 码。花园的长边（包含两端角上的桩）所包含的桩数是短边（包含两端角上的桩）桩数的两倍。卡尔的花园面积是多少（单位：平方码）？

(A) 256 256

(B) 336 336

(D) 448 448

(E) 512 512

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Let $x$ be the number of posts along the shorter side; then there are $2x$ posts along the longer side. When counting the number of posts on all the sides of the garden, each corner post is counted twice, so $2x + 2(2x) = 20 + 4$. Solving this equation gives $x = 4$. Thus the dimensions of the rectangle are $(4 - 1)\cdot 4 = 12$ yards by $(8 - 1)\cdot 4 = 28$ yards. The requested area is given by the product of these dimensions, $12\cdot 28 = 336$ square yards.

答案（B）：设较短边上的柱子数为 $x$，则较长边上有 $2x$ 根柱子。统计花园四边柱子总数时，每个角上的柱子会被重复计算两次，因此有 $2x + 2(2x) = 20 + 4$。解此方程得 $x = 4$。因此矩形的尺寸为：$(4 - 1)\cdot 4 = 12$ 码和 $(8 - 1)\cdot 4 = 28$ 码。所求面积为这两个尺寸的乘积：$12\cdot 28 = 336$ 平方码。

Topics