Students in Sports Camp

Let number of boys be $B$ and girls be $G$. $B + G = 65$. Fractions of boys and girls joining camps must be integers: $\frac{1}{5}B$ and $\frac{2}{3}B$ implies $B$ must be a multiple of $\text{lcm}(5, 3) = 15$. $\frac{1}{4}G$ and $\frac{3}{5}G$ implies $G$ must be a multiple of $\text{lcm}(4, 5) = 20$. Since $B + G = 65$ with $B \in \{15, 30, 45, 60\}$ and $G \in \{20, 40, 60\}$: The only combination is $B = 45$ and $G = 20$. Number of girls joining sports camp = $\frac{3}{5} \times 20 = 12$.