The ratio of wavelength of spectral lines ${\mathrm{H}}_{\mathrm{\alpha }}\text{ and }{\mathrm{H}}_{\mathrm{\beta }}$ in the Balmer series is x/20. The value of x is ……

$\begin{array}{l}\frac{1}{\mathrm{\lambda }}=\mathrm{R}(\mathrm{z}-\mathrm{b}{)}^2\frac{1}{{\mathrm{n}}_1^2}-\frac{1}{{\mathrm{n}}_2^2}\\ \Rightarrow \frac{1}{{\mathrm{\lambda }}_1}=\mathrm{R}(\mathrm{z}-\mathrm{b}{)}^2\left[\frac{1}{2^2}-\frac{1}{3^2}\right]\\ \Rightarrow \frac{1}{{\mathrm{\lambda }}_1}=\mathrm{R}[\mathrm{z}-\mathrm{b}{]}^2\left[\frac{5}{36}\right] .......... (1)\\ \mathrm{and}, \frac{1}{{\mathrm{\lambda }}_2}=\mathrm{R}[\mathrm{z}-\mathrm{b}{]}^2\left[\frac{3}{16}\right] ......... (2)\\ \Rightarrow \frac{\left(2\right)}{\left(1\right)}=\frac{{\mathrm{\lambda }}_1}{{\mathrm{\lambda }}_2}=\frac{3}{16}\times \frac{36}{5}=\frac{27}{20}\end{array}$