AMC10 2010 B

AMC10 2010 B · Q15

AMC10 2010 B · Q15. It mainly tests Linear inequalities, Arithmetic misc.

On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and −1 point for an incorrect answer. Jesse’s total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?

在一场 50 题选择题数学竞赛中，正确得 4 分，空题 0 分，错误 −1 分。Jesse 总分为 99。Jesse 最多能正确答多少题？

(A) 25 25

(B) 27 27

(D) 31 31

(E) 33 33

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): If Jesse answered $R$ questions correctly and $W$ questions incorrectly, then $R+W\le 50$, and Jesse’s score is $99=4R-W\ge 4R-(50-R)=5R-50$. Thus $5R\le 149$, and because $R$ is an integer, $R\le 29$. Jesse could achieve a score of 99 by answering 29 questions correctly and 17 incorrectly, leaving 4 answers blank.

答案（C）：如果 Jesse 正确回答了 $R$ 题、错误回答了 $W$ 题，则 $R+W\le 50$，并且 Jesse 的得分为 $99=4R-W\ge 4R-(50-R)=5R-50$。因此 $5R\le 149$，且由于 $R$ 是整数，$R\le 29$。Jesse 可以通过答对 29 题、答错 17 题得到 99 分，并留下 4 题空白。

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