Question
If (ma + nb): b :: (mc + nd) : d, prove that a, b, c, d are in proportion.

Answer


$
\begin{aligned}
& ( ma + nb ): b ::( mc + nd ): d \\
& \Rightarrow \frac{ ma + nb }{ b }=\frac{ mc + nd }{ d } \\
& \Rightarrow mad + nbd = mbc + nbd \\
& \Rightarrow mad = mbc \\
& \Rightarrow ad = bc \\
& \Rightarrow \frac{a}{b}=\frac{c}{d}
\end{aligned}
$
Hence a : b :: c : d.

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