Boys and girls

Answer: 1300

Using letters to represent the 'parts'

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Boys: $3a-80=7b$            $\times2$ gives $6a-160=14b$

Girls: $2a - 20 = 5b$            $\times3$ gives $6a-60=15b$

Subtract: $(6a-60)-(6a-160)=15b-14b$

                                           $100=b$

In June there were $7b + 5b = 12b = 1200$ students

In September there were $1200 + 20 + 80 = 1300$ students

Using a letter to represent the number of students in September

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September: $\frac35x$ boys out of $x$ students

June: $\frac35x-80$ boys is $\frac7{12}$ of $x-100$ students

$$\begin{array}{rl}{\displaystyle \frac{\frac{3}{5}x-80}{x-100}}& ={\displaystyle \frac{7}{12}}\\ \Rightarrow 12\times (\frac{3}{5}x-80)& =7\times (x-100)\\ \Rightarrow 12\times (3x-4000)& =35\times (x-100)\\ \Rightarrow 36x-4800& =35x-3500\\ \Rightarrow x& =1300\end{array}$$

Using letters to represent the numbers of boys and girls in September

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September: $B:G$ is $3:2$, and so $\frac B3=\frac G2\Rightarrow B=\frac{3G}2$

June: $B-80:G-20$ is $7:5$, so $\frac{B-80}7=\frac{G-20}5$

                                            $\Rightarrow 5(B-80)=7(G-20)$

                                            $\Rightarrow 5B-260=7G$

Substitute $B=\frac{3G}2$: $$\begin{array}{rl}5\frac{3G}{2}-260& =7G\\ \Rightarrow 15G-520& =14G\\ \Rightarrow G& =520\end{array}$$

So $B=\frac{3G}2=780$ and in September there were $520 + 780 =1300$ students.