2015 AMC 8 第 16 题

Question

在一个中学导师项目中，一些六年级学生与一名九年级学生配成伙伴。没有九年级学生被分配超过一名六年级伙伴。如果所有九年级学生的

\dfrac{1}{3}

与所有六年级学生的

\dfrac{2}{5}

配成伙伴，那么六年级和九年级学生总数中有伙伴的比例是多少？

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

\dfrac{1}{3}

of all the ninth graders are paired with

\dfrac{2}{5}

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

\dfrac{2}{15}

\dfrac{4}{11}

\dfrac{11}{30}

\dfrac{3}{8}

\dfrac{11}{15}

答案：B

知识点：比与比例分数

难度评级：1300

小提示：

设六年级和九年级学生人数分别为

s

和

n

Let

s

and

n

be the numbers of sixth and ninth graders.

大提示：

配对的六年级学生数与配对的九年级学生数相同

The paired sixth graders and paired ninth graders are the same number of pairs.

视频讲解：

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文字解答：

设六年级学生有

s

人，九年级学生有

n

人。配对的九年级学生数等于配对的六年级学生数，所以

\dfrac13 n=\dfrac25 s\text{。}

得

5n=6s

，所以

n:s=6:5

。学生总数对应

11

份。

配对的九年级学生占全体学生的

\dfrac13\cdot\dfrac6{11}=\dfrac2{11}

，配对的六年级学生占全体学生的

\dfrac25\cdot\dfrac5{11}=\dfrac2{11}

。合计有

\dfrac4{11}

的学生有伙伴。

所以正确答案是 B。

Let there be

s

sixth graders and

n

ninth graders. The number of paired ninth graders equals the number of paired sixth graders, so

\dfrac13 n=\dfrac25 s.

This gives

5n=6s,

so

n:s=6:5.

The total number of students is therefore proportional to

11

parts.

The paired ninth graders make up

\dfrac13\cdot\dfrac6{11}=\dfrac2{11}

of all students, and the paired sixth graders make up

\dfrac25\cdot\dfrac5{11}=\dfrac2{11}

of all students. Altogether,

\dfrac4{11}

of the students have a buddy.

Thus, B is the correct answer.