AMC 8 2011 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? $\textbf{(A) } \dfrac7{18} \textbf{(B) } \dfrac7{15} \textbf{(C) } \dfrac{22}{45} \textbf{(D) } \dfrac12 \textbf{(E) } \dfrac{23}{45}$