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Ratio and Proportion — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsRatio and Proportion4 Marks
Question
If $\frac{a-2 b-3 c+4 d}{a+2 b-3 c-4 d}=\frac{a-2 b+3 c-4 d}{a+2 b+3 c+4 d}$ show that 2ad = 3bc
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Answer

$\frac{a-2 b-3 c+4 d}{a+2 b-3 c-4 d}=\frac{a-2 b+3 c-4 d}{a+2 b+3 c+4 d}$
Applying componendo and dividendo
$\Rightarrow \frac{a-2 b-3 c+4 d+a+2 b-3 c-4 d}{a-2 b-3 c+4 d-a-2 b+3 c+4 d}=\frac{a-2 b+3 c-4 d+a+2 b+3 c+4 d}{a-2 b+3 c-4 d-a-2 b-3 c-4 d}$
$\begin{aligned} & \Rightarrow \frac{2 a-6 c}{-4 b+8 d}=\frac{2 a+6 c}{-4 b-8 d} \\ & \Rightarrow \frac{2(a-3 c)}{-4(b-2 d)}=\frac{2(a+3 c)}{4(b+2 d)} \\ & \Rightarrow \frac{a-3 c}{a+3 c}=\frac{b-2 d}{b+2 d}\end{aligned}$
Applying componendo and dividendo
$\begin{aligned} & \Rightarrow \frac{a-3 c+a+3 c}{a-3 c-a-3 c}=\frac{b-2 d+b+2 d}{b-2 d-b-2 d} \\ & \Rightarrow \frac{2 a}{-6 c}=\frac{2 b}{-4 d} \\ & \Rightarrow-4 \mathrm{da}=-6 \mathrm{cb} \\ & \Rightarrow 2 \mathrm{ad}=3 \mathrm{bc}\end{aligned}$

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