In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If (1/3) of all the ninth graders are paired with (2/5) of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?

AMC 8 2015 Problem 16

AMC8 2015 年第 16 题

比与比例★★★☆☆

In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If

\displaystyle \frac{1}{3}

of all the ninth graders are paired with

\displaystyle \frac{2}{5}

of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?

在一所中学的一项导师项目中，由六年级学生和九年级学生成对组队完成项目。每个九年级学生都不能和超过 1 个六年级学生组成队友。如果

\displaystyle \frac{1}{3}

的所有九年级学生和

\displaystyle \frac{2}{5}

的所有六年级成对，那么已经组成队友的学生占了六年级和九年级总人数的比例是多少？

A. $\displaystyle \frac{2}{15}$
B. $\displaystyle \frac{4}{11}$
C. $\displaystyle \frac{11}{30}$
D. $\displaystyle \frac{3}{8}$
E. $\displaystyle \frac{11}{15}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

2 2 out of every 5 5 sixth graders are paired with a ninth grade buddy, and 2 2 out of every 6 6 ninth graders are paired with a sixth grade buddy. (The ratios are now expressed so that the number of sixth graders matches the number of ninth graders.) So 4 4 out of every 11 11 students are in the mentoring program. The fraction is 4 11 11 4 . OR OR Suppose that 𝑛 n sixth graders are paired with 𝑛 n ninth graders. Then the total number of sixth graders is 5 2 𝑛 2 5 n, the total number of ninth graders is 3 𝑛 3n, and the total number of sixth and ninth graders is 5 2 𝑛 + 3 𝑛 = 11 2 𝑛 2 5 n+3n= 2 11 n. There are 2 𝑛 2n students in the mentoring program, which is 2 𝑛 11 2 𝑛 = 4 11 2 11 n 2n = 11 4 of the total number of students. Answer: 𝐵 B .

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