Question

一场选择题考试共有 $20$ 题。计分规则是：每答对一题得 $+5$ 分，每答错一题得 $-2$ 分，未答题得 $0$ 分。John 的考试成绩是 $48$ 分。他最多可能答对了多少题？

A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48.$ What is the maximum number of questions he could have answered correctly?

$9$

$10$

$11$

$12$

$16$

答案：D

知识点：一次方程最优化奇偶性

难度评级：1150

解答：

设 $c$ 为答对题数， $w$ 为答错题数，则 $5c - 2w = 48$ 。于是 $2w = 5c - 48$ ，所以 $c$ 必须为偶数。

试取 $c = 14$ ，得到 $w = 11$ ，但 $c + w = 25 \gt 20$ ，超过了题目总数；任何更大的偶数 $c$ 都会需要回答更多题。试取 $c = 12$ ，得到 $w = 6$ ，且 $c + w = 18 \le 20$ ，符合条件。因此最大值是 $12$ 。

所以正确答案是 D。

Let $c$ be the number correct and $w$ the number wrong, so $5c - 2w = 48.$ Then $2w = 5c - 48,$ which requires $c$ to be even.

Trying $c = 14$ gives $w = 11,$ but $c + w = 25 \gt 20,$ too many; every larger even $c$ would require still more answered questions. Trying $c = 12$ gives $w = 6$ and $c + w = 18 \le 20,$ which works. So the maximum is $12.$

Thus, the correct answer is D .

1987 AMC 8 第 24 题

答案：D

难度评级：1150

解答：

其他年份的第 24 题