2011 AMC 8 Problems/Problem 14

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?

$\text{(A) } \dfrac7{18} \qquad\text{(B) } \dfrac7{15} \qquad\text{(C) } \dfrac{22}{45} \qquad\text{(D) } \dfrac12 \qquad\text{(E) } \dfrac{23}{45}$

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2011 AMC 8 Problems/Problem 14