AMC10 2019 B

AMC10 2019 B · Q3

AMC10 2019 B · Q3. It mainly tests Percent, Averages (mean).

In a high school with 500 students, 40% of the seniors play a musical instrument, while 30% of the non-seniors do not play a musical instrument. In all, 46.8% of the students do not play a musical instrument. How many non-seniors play a musical instrument?

一所高中共有 500 名学生，40% 的高三学生会演奏乐器，而 30% 的非高三学生不会演奏乐器。总共有 46.8% 的学生不会演奏乐器。有多少非高三学生会演奏乐器？

(A) 66 66

(B) 154 154

(D) 220 220

(E) 266 266

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Let $n$ be the number of non-seniors. Because there are 500 students in all, there are $500-n$ seniors. Because 40% of the seniors play a musical instrument, 60% of the seniors do not play a musical instrument. This leads to the equation $$ 0.60(500-n)+0.30n=0.468\cdot 500. $$ The equation simplifies to $0.30n=66$. Solving this equation for $n$ gives $n=220$. Because $100\%-30\%=70\%$ of the non-seniors play a musical instrument, there are $0.70\cdot 220=154$ non-seniors who play a musical instrument.

答案（B）：设 $n$ 为非高年级学生（non-seniors）的人数。因为一共有 500 名学生，所以高年级学生（seniors）人数为 $500-n$。因为高年级学生中有 40% 会演奏乐器，所以有 60% 的高年级学生不会演奏乐器。于是得到方程 $$ 0.60(500-n)+0.30n=0.468\cdot 500. $$ 该方程化简为 $0.30n=66$。解得 $n=220$。因为非高年级学生中有 $100\%-30\%=70\%$ 会演奏乐器，所以会演奏乐器的非高年级学生人数为 $0.70\cdot 220=154$。

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