2007 AMC 12B 第 5 题

先试着解答 2007 AMC 12B 第 5 题,然后核对你的答案与精心整理的解答,解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2007 AMC 12B 解答,或核对答案。

所有题目均经美国数学协会(MAA)官方合法授权使用。

5.

20072007 年 AMC 1212 的计分方式为:每答对一题得 66 分,答错得 00 分,空题得 1.51.5 分。Sarah 看过 2525 道题后,决定尝试前 2222 题,最后 33 题留空。为了至少得到 100100 分,她在前 2222 题中至少要答对多少题?
The 20072007 AMC 1212 contests will be scored by awarding 66 points for each correct response, 00 points for each incorrect response, and 1.51.5 points for each problem left unanswered. After looking over the 2525 problems, Sarah has decided to attempt the first 2222 and leave the last 33 unanswered. How many of the first 2222 problems must she solve correctly in order to score at least 100100 points?
1313
1414
1515
1616
1717
答案:D
知识点:不等式
小提示:
33 道空题已经贡献 3⋅1.53\cdot1.5 分。
The 33 unanswered problems already contribute 3⋅1.53\cdot1.5 points
大提示:
解 6x≥100−4.56x\ge100-4.5,再把答对题数 xx 向上取整。
Solve 6x≥100−4.56x\ge100-4.5 and round the number correct xx up to a whole number
解答:
三道空题给 3⋅1.5=4.53\cdot1.5=4.5 分。
所以 Sarah 需要从答对题中得到至少 100−4.5=95.5100-4.5=95.5 分。因为 15<95.56<16, 15\lt\dfrac{95.5}{6}\lt16\text{,} 她至少要答对 1616 题,此时得分为 16⋅6+4.5=100.516\cdot6+4.5=100.5。
所以正确答案是 D。
The three unanswered problems give 3⋅1.5=4.53\cdot1.5=4.5 points.
So Sarah needs at least 100−4.5=95.5100-4.5=95.5 points from correct answers. Since 15<95.56<16, 15\lt\dfrac{95.5}{6}\lt16, she must solve at least 1616 problems correctly, which would give her 16⋅6+4.5=100.516\cdot6+4.5=100.5 points.
Thus, the correct answer is D.
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