The ratio of the longest wavelength in the Lyman series to the longest wavelength in the Balmer series of hydrogen is

Longest wavelength in a series corresponds to the smallest energy jump: $n=2 \to 1$ for Lyman, $n=3 \to 2$ for Balmer. Using $\tfrac{1}{\lambda} = R\left(\tfrac{1}{n_1^2} - \tfrac{1}{n_2^2}\right)$: Lyman max gives $\tfrac{1}{\lambda_L} = R(1 - \tfrac{1}{4}) = \tfrac{3R}{4}$. Balmer max gives $\tfrac{1}{\lambda_B} = R(\tfrac{1}{4} - \tfrac{1}{9}) = \tfrac{5R}{36}$. Ratio $\tfrac{\lambda_L}{\lambda_B} = \tfrac{5R/36}{3R/4} = \tfrac{5}{36} \cdot \tfrac{4}{3} = \tfrac{20}{108} = \tfrac{5}{27}$.