correct answer is: a) 30 boys

Explanation

The ratio of boys and girls is 13 : 9
Let the number of boys was 13x and the number of girls was 9x
\therefore Total number of students were = \left(13x\ +\ 9x\right)
\rightarrow\ 22x
According to the question,
22x\ =\ 660
\rightarrow\ x\ =\ \frac{660}{22}
\rightarrow\ x\ =\ 30
\therefore The number of boys was 13x or \left(13\ \times\ 30\right) or 390
And, the number of girls was 9x or \left(9\ \times\ 30\right) or 270

Other side,
30 girls joined the school.
Let, ‘b’ is the number of boys left.
According to the question,
The new ratio of boys and girls became 6 : 5
So that,
\left(390\ -\ b\right) ∶ \left(270+\ 30\right)\ =\ 6 ∶ 5
\rightarrow\ \frac{\left(390\ -\ b\right)}{\left(270\ +\ 30\right)}\ =\ \frac{6}{5}
\rightarrow\ 5\left(390\ -\ b\right)\ =\ 6\left(270\ +\ 30\right) _{[cross multiply]}
\rightarrow\ 1950\ -\ 5b\ =\ 1620\ +\ 180
\rightarrow\ 1950\ -\ 5b\ =\ 1800
\rightarrow\ -\ 5b\ =\ 1800\ -\ 1950
\rightarrow\ -\ 5b\ =\ -\ 150
\rightarrow\ b\ =\ \frac{-\ 150}{-\ 5}
\therefore\ b\ =\ 30
\therefore\ 30 boys left the school.
Ans: Total 30 boys left the school.