AMC8 2022

AMC8 2022 · Q19

AMC8 2022 · Q19. It mainly tests Averages (mean), Logic puzzles.

Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points? (Note that the median test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)

拉莫斯先生给他的20名学生出了一次测试。下方的点图显示了测试分数的分布。后来拉莫斯先生发现有一道题评分错误。他重新评分，给一些学生加了5分，这使得中位数测试分数增加到$85$。至少有多少名学生获得了额外分数？（注意，如果将20个测试分数按升序排列，中位数等于中间两个分数的平均值。）

(A) 2 2

(B) 3 3

(D) 5 5

(E) 6 6

Answer

Correct choice: (C)

正确答案：（C）

Solution

We notice that $13$ students have scores under $85$ currently, and only $5$ have scores over $85$. We find the median of these two numbers, getting: \[13-5=8\] \[\frac{8}{2}=4\] \[13-4=9\] Thus, we realize that $4$ students must have their score increased by $5$. So, the correct answer is $\boxed{\textbf{(C)}4}$.

我们注意到目前有$13$名学生的分数低于$85$，只有$5$名学生的分数高于$85$。我们计算这两个数的差： \[13-5=8\] \[\frac{8}{2}=4\] \[13-4=9\] 因此，必须有$4$名学生的分数增加$5$分。正确答案是$\boxed{\textbf{(C)}4}$。

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