The ratio of wavelength of spectral lines H _ α and H _ β in the...

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

$\dfrac{hc}{{\lambda}_{\alpha}} = 13 . 6\left[\dfrac{1}{4} - \dfrac{1}{9}\right] . . . \left(1\right)$

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

$\dfrac{hc}{{\lambda}_{\beta}} = 13 . 6\left[\dfrac{1}{4} - \dfrac{1}{16}\right] . . . \left(2\right)$

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