If (3a + 5b) : (3a − 5b) = (3c + 5d) : (3c − 5d), prove that a : b = c : d.

Given,

$\dfrac{3a + 5b}{3a - 5b} = \dfrac{3c + 5d}{3c - 5d}$

Applying componendo and dividendo, we get :

$\Rightarrow \dfrac{(3a + 5b) + (3a - 5b)}{(3a + 5b) - (3a - 5b)} = \dfrac{(3c + 5d) + (3c - 5d)}{(3c + 5d) - (3c - 5d)} \\[1em] \Rightarrow \dfrac{3a + 5b + 3a - 5b}{3a + 5b - 3a + 5b} = \dfrac{3c + 5d + 3c - 5d}{3c +5d - 3c + 5d} \\[1em] \Rightarrow \dfrac{6a}{10b} = \dfrac{6c}{10d} \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}\\[1em]$

Hence, proved that a : b = c : d.