Question
If $a : b :: c : d$, prove that $\frac{5 a+11 b}{5 c+11 d}=\frac{5 a-11 b}{5 c-11 d}$
✓
Answer
$\begin{aligned}
& \because a : b :: c : d \\
& \therefore \frac{a}{b}=\frac{c}{d} \\
& \Rightarrow \frac{5 a}{11 b}=\frac{5 c}{11 d} \ldots\left(\text { Multiplying by } \frac{5}{11}\right)
\end{aligned}$
Applying componendo and dividendo,
$
\frac{5 a+11 b}{5 a-11 d}=\frac{5 c+11 d}{5 c-11 d}
$
$\Rightarrow \frac{5 a+11 b}{5 c+11 b}=\frac{5 a-11 b}{5 c-11 d} \ldots$ (Applying alternendo).
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