The formula to calculate the wavelength of $H_{\alpha }$-line in Balmer series is given by

$\frac{hc}{{\lambda }_{\alpha }}=13.6\left[\frac{1}{4}-\frac{1}{9}\right] ...\left(1\right)$

The formula to calculate the wavelength of $H_{\beta }$-line in Balmer series is given by

$\frac{hc}{{\lambda }_{\beta }}=13.6\left[\frac{1}{4}-\frac{1}{16}\right] ...\left(2\right)$

Divide equation (2) by equation (1) 
$\Rightarrow \frac{{\lambda }_{\alpha }}{{\lambda }_{\beta }}=\frac{\frac{1}{4}-\frac{1}{16}}{\frac{1}{4}-\frac{1}{9}}$
$=\frac{36}{64}\times \frac{12}{5}$
$=\frac{27}{20}$