If p + r = mq and $\dfrac{1}{q} + \dfrac{1}{s} = \dfrac{m}{r};$ then prove that : p : q = r : s.

Given,

$\Rightarrow \dfrac{1}{q} + \dfrac{1}{s} = \dfrac{m}{r} \\[1em] \Rightarrow \dfrac{s + q}{qs} = \dfrac{m}{r} \\[1em] \Rightarrow \dfrac{s + q}{s} = \dfrac{mq}{r} \\[1em] \Rightarrow \dfrac{s + q}{s} = \dfrac{p + r}{r} \text{ (As mq = p + r)} \\[1em] \Rightarrow 1 + \dfrac{q}{s} = \dfrac{p}{r} + 1 \\[1em] \Rightarrow \dfrac{q}{s} = \dfrac{p}{r} \\[1em] \Rightarrow \dfrac{p}{q} = \dfrac{r}{s}.$

Hence, proved that p : q = r : s.