If $\dfrac{4m + 3n}{4m - 3n} = \dfrac{7}{4}$, use properties of proportion to find :

(i) m : n

(ii) $\dfrac{2m^2 - 11n^2}{2m^2 + 11n^2}$

(i) Given,

$\dfrac{4m + 3n}{4m - 3n} = \dfrac{7}{4}$

Applying componendo and dividendo:

$\dfrac{4m + 3n + 4m - 3n}{4m + 3n - (4m - 3n)} = \dfrac{7 + 4}{7 - 4} \\[1em] \Rightarrow \dfrac{8m}{6n} = \dfrac{11}{3} \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{11 \times 6}{3 \times 8} \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{11}{4}.$

Hence, m : n = 11 : 4.

(ii) We know,

$\phantom{\Rightarrow} \dfrac{m}{n} = \dfrac{11}{4} \\[1em] \Rightarrow \dfrac{m^2}{n^2} = \dfrac{121}{16} \\[1em] \Rightarrow \dfrac{2m^2}{11n^2} = \dfrac{2 \times 121}{11 \times 16} \\[1em] \Rightarrow \dfrac{2m^2}{11n^2} = \dfrac{242}{176}$

Applying componendo and dividendo:

$\Rightarrow \dfrac{2m^2 + 11n^2}{2m^2 - 11n^2} = \dfrac{242 + 176}{242 - 176} \\[1em] \Rightarrow \dfrac{2m^2 + 11n^2}{2m^2 - 11n^2} = \dfrac{418}{66} = \dfrac{19}{3}$

Applying invertendo:

$\Rightarrow \dfrac{2m^2 - 11n^2}{2m^2 + 11n^2} = \dfrac{3}{19}$

Hence, $\dfrac{2m^2 - 11n^2}{2m^2 + 11n^2} = \dfrac{3}{19}.$