AMC10 2021 B

AMC10 2021 B · Q3

AMC10 2021 B · Q3. It mainly tests Systems of equations, Percent.

In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\%$ of the juniors as a class and $10\%$ of the seniors as a class are on the debate team. How many juniors are in the program?

在一个为高二和高三学生开设的课后项目中，辩论队中有来自每个年级的学生数量相等。该项目共有 $28$ 名学生，高二年级学生的 $25\%$ 和高三年级学生的 $10\%$ 在辩论队中。项目中有多少高二学生？

(A) 5 5

(B) 6 6

(D) 11 11

(E) 20 20

Answer

Correct choice: (C)

正确答案：（C）

Solution

Say there are $j$ juniors and $s$ seniors in the program. Converting percentages to fractions, $\frac{j}{4}$ and $\frac{s}{10}$ are on the debate team, and since an equal number of juniors and seniors are on the debate team, $\frac{j}{4} = \frac{s}{10}.$ Cross-multiplying and simplifying we get $5j=2s.$ Additionally, since there are $28$ students in the program, $j+s = 28.$ It is now a matter of solving the system of equations \[5j=2s\]\[j+s=28,\] and the solution is $j = 8, s = 20.$ Since we want the number of juniors, the answer is \[\boxed{(C) \text{ } 8}.\]

设项目中有 $j$ 名高二学生和 $s$ 名高三学生。将百分比转换为分数，$\frac{j}{4}$ 和 $\frac{s}{10}$ 在辩论队中，由于每个年级的辩论队成员数量相等，$\frac{j}{4} = \frac{s}{10}$。交叉相乘并简化得到 $5j=2s$。此外，由于项目中共有 $28$ 名学生，$j+s = 28$。现在求解方程组 \[5j=2s\]\[j+s=28,] 解得 $j = 8, s = 20$。我们需要高二学生数量，即 $\boxed{(C) \text{ } 8}$。

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