The ratio of the longest to shortest wavelengths in Brackett series of hydrogen spectra is

$\frac{25}{9}$

For Bracket series

$\frac{1}{\lambda_{\max }}=R\left[\frac{1}{4^2}-\frac{1}{5^2}\right]=\frac{9}{25 \times 16} R$

and $\frac{1}{\lambda_{\min }}=R\left[\frac{1}{4^2}-\frac{1}{\infty^2}\right]=\frac{R}{16} \Rightarrow \frac{\lambda_{\max }}{\lambda_{\min }}=\frac{25}{9}$