Let $c$ denote the size of the class and $g$ denote the number of girls. We want $0.45c< g< 0.5c\Leftrightarrow 0.45< \frac{g}{c}< 0.5$.

If we try $g=1$ there is no such $c$, since if $c=1$ or $c=2$, $\frac{g}{c}$ is too large, and if $c> 2$, $\frac{g}{c}$ is too small.

Similarly for $g=2$, $3$ or $4$. However for $g=5$ setting $c=11$ leads to $\frac{g}{c}=\frac{5}{11}=\frac{500}{11}$% $=(\frac{440}{11}+\frac{60}{11})$% $=(40+\frac{55}{11}+\frac{5}{11})$% $=45\frac{5}{11}$%.

*This problem is taken from the UKMT Mathematical Challenges.*