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Mathematics

If (ma + nb) : (mc + nd) = (ma − nb) : (mc − nd), prove that a : b = c : d.

Ratio Proportion

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Answer

Given,

(ma + nb) : (mc + nd) = (ma − nb) : (mc − nd)

⇒ma+nbmc+nd=ma−nbmc−nd\Rightarrow \dfrac{ma + nb}{mc + nd} = \dfrac{ma - nb}{mc - nd}

Apply Alternendo,

ma+nbma−nb=mc+ndmc−nd\dfrac{ma + nb}{ma - nb} = \dfrac{mc + nd}{mc - nd}

Applying componendo and dividendo:

⇒(ma+nb)+(ma−nb)(ma+nb)−(ma−nb)=(mc+nd)+(mc−nd)(mc+nd)−(mc−nd)⇒ma+nb+ma−nbma+nb−ma+nb=mc+nd+mc−ndmc+nd−mc+nd⇒2ma2nb=2mc2nd⇒ab=cd.\Rightarrow \dfrac{(ma + nb) + (ma - nb)}{(ma + nb) - (ma - nb)} = \dfrac{(mc + nd) + (mc - nd)}{(mc + nd) - (mc - nd)} \\[1em] \Rightarrow \dfrac{ma + nb + ma - nb}{ma + nb - ma + nb} = \dfrac{mc + nd + mc - nd}{mc + nd - mc + nd} \\[1em] \Rightarrow \dfrac{2ma}{2nb} = \dfrac{2mc}{2nd} \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}.

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

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