AMC 8 2011 Problem 14

AMC8 2011 年第 14 题

比与比例★★★☆☆

There are 270 students at Colfax Middle School, where the ratio of boys to girls is

5 : 4

. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is

4 : 5

. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?

Colfax 中学有 270 个学生，男生和女生的比例是 5:4。Winthrop 中学有 180 个学生，男生和女生的比例是 4:5。这两个学校举行了一次舞会，并且两个学校的学生都参加了这次舞会。那么舞会上女生所占总学生数的比例是多少？

A. $\dfrac7{18}$
B. $\dfrac7{15}$
C. $\dfrac{22}{45}$
D. $\dfrac12$
E. $\dfrac{23}{45}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

At Colfax Middle School, there are

\frac49×270 = 120

girls. At Winthrop Middle School, there are

\frac59×180 = 100

girls. The ratio of girls to the total number of students is

\displaystyle \frac{120+100}{270+180} = \frac{220}{450} = \frac{22}{45}

. Since we know there are 270 kids at Colfax Middle School and we know the ratio of boys to girls is

5:4

, we can use guess and check to find the number of girls at the dance (and vice versa). 5:4=150:120 and 4:5=80:100. Now that the ratios sum to 270 and 180, we know that in Colfax there are 120 girls and in Winthrop, there are 100 girls. 120+100=220 total girls and 270+180=450 total people.

\displaystyle \frac{220}{450} = \frac{22}{45}

. We know that since 270 is greater than 180 and how the ratios are opposites, we know that there will be more boys than girls, so the fraction of students at the dance that are girls should be less than

\displaystyle \frac{1}{2}

. We also know that the amount of girls and the amount of boys are close, so the fraction of students at the dance that are girls should be really close to

\displaystyle \frac{1}{2}

. The only answer choice that/best satisfies these conditions is

\displaystyle \frac{22}{45}

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