If $\dfrac{7m + 2n}{7m - 2n} = \dfrac{5}{3}$, use properties of proportion to find :

(i) m : n

(ii) $\dfrac{m^2 + n^2}{m^2 - n^2}$

(i) Given,

$\Rightarrow \dfrac{7m + 2n}{7m - 2n} = \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{7m + 2n + 7m - 2n}{7m + 2n - (7m - 2n)} = \dfrac{5 + 3}{5 - 3} \\[1em] \Rightarrow \dfrac{14m}{4n} = \dfrac{8}{2} \\[1em] \Rightarrow \dfrac{7m}{2n} = \dfrac{4}{1} \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{8}{7}$

Hence, m : n = 8 : 7.

(ii) We know that,

$\Rightarrow \dfrac{m}{n} = \dfrac{8}{7} \\[1em] \Rightarrow \dfrac{m^2}{n^2} = \dfrac{64}{49}$

Applying componendo and dividendo,

$\Rightarrow \dfrac{m^2 + n^2}{m^2 - n^2} = \dfrac{64 + 49}{64 - 49} \\[1em] \Rightarrow \dfrac{m^2 + n^2}{m^2 - n^2} = \dfrac{113}{15} = 7\dfrac{8}{15}.$

Hence, $\dfrac{m^2 + n^2}{m^2 - n^2} = 7\dfrac{8}{15}$.