AMC12 2018 A

AMC12 2018 A · Q2

AMC12 2018 A · Q2. It mainly tests Word problems (algebra), Money / coins.

While exploring a cave, Carl comes across a collection of 5-pound rocks worth \$14 each, 4-pound rocks worth \$11 each, and 1-pound rocks worth \$2 each. There are at least 20 of each size. He can carry at most 18 pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

卡尔在探洞时发现了一些5磅重的石头每块价值14美元、4磅重的石头每块价值11美元、1磅重的石头每块价值2美元。每种石头至少有20个。他最多能携带18磅。问他能带出洞的最大价值（美元）是多少？

(A) 48 48

(B) 49 49

(D) 51 51

(E) 52 52

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The 5-pound rocks have a value of \$14 ÷ 5 = \$2.80 per pound; the 4-pound rocks have a value of \$11 ÷ 4 = \$2.75 per pound; the 1-pound rocks have a value of \$2 per pound. It is not to Carl’s advantage to take 1-pound rocks when he can take the larger rocks. Therefore the only issue is how many of the more valuable 5-pound rocks to take, including as many 4-pound rocks as possible in each case. The viable choices are displayed in the following table. \[ \begin{array}{cccc} \text{5-pound rocks} & \text{4-pound rocks} & \text{1-pound rocks} & \text{value} \\ (\$14\ \text{each}) & (\$11\ \text{each}) & (\$2\ \text{each}) & \\ 3 & 0 & 3 & \$48 \\ 2 & 2 & 0 & \$50 \\ 1 & 3 & 1 & \$49 \\ 0 & 4 & 2 & \$48 \\ \end{array} \] The maximum possible value is \$50.

答案（C）：5磅石头的价值为 \$14 ÷ 5 = \$2.80 每磅；4磅石头的价值为 \$11 ÷ 4 = \$2.75 每磅；1磅石头的价值为每磅 \$2。当卡尔可以拿更大块的石头时，拿1磅石头并不划算。因此，唯一的问题是要拿多少块更有价值的5磅石头，并且在每种情况下尽可能多地拿4磅石头。可行的选择如下表所示。 \[ \begin{array}{cccc} \text{5磅石头} & \text{4磅石头} & \text{1磅石头} & \text{价值} \\ (\$14\ \text{每块}) & (\$11\ \text{每块}) & (\$2\ \text{每块}) & \\ 3 & 0 & 3 & \$48 \\ 2 & 2 & 0 & \$50 \\ 1 & 3 & 1 & \$49 \\ 0 & 4 & 2 & \$48 \\ \end{array} \] 最大可能价值是 \$50。

Topics