Name: Proportion and Variation
Uploaded: 2019-07-19
Duration: 12 min
Description: In this video, the basic concepts of Proportion and Variation are discussed. The content in this video is helpful to understand variation and its applications.

Question

Write the result after applying componendo and dividendo to

3 a^{2} + 2 b^{2} : 3 a^{2} - 2 b^{2} : : 3 c^{2} + 2 d^{2} : 3 c^{2} - 2 d^{2}

.

Accepted Answer

If two ratios are such that $x : y = p : q or \frac{x}{y} = \frac{p}{q}$ , then $\frac{x + y}{x - y} = \frac{p + q}{p - q}$ . This is called the componendo and dividendo rule for ratios.

Here, given, $3 a^{2} + 2 b^{2} : 3 a^{2} - 2 b^{2} : : 3 c^{2} + 2 d^{2} : 3 c^{2} - 2 d^{2}$ .

Applying componendo and dividendo rule, we obtain, $3 a^{2} + 2 b^{2} : 3 a^{2} - 2 b^{2} : : 3 c^{2} + 2 d^{2} : 3 c^{2} - 2 d^{2}$

$\Rightarrow \frac{3 a^{2} + 2 b^{2}}{3 a^{2} - 2 b^{2}} = \frac{3 c^{2} + 2 d^{2}}{3 c^{2} - 2 d^{2}}$

$\Rightarrow \frac{3 a^{2} + 2 b^{2} + (3 a^{2} - 2 b^{2})}{3 a^{2} + 2 b^{2} - (3 a^{2} - 2 b^{2})} = \frac{3 c^{2} + 2 d^{2} + (3 c^{2} - 2 d^{2})}{3 c^{2} + 2 d^{2} - (3 c^{2} - 2 d^{2})}$

$\Rightarrow \frac{3 a^{2} + 2 b^{2} + 3 a^{2} - 2 b^{2}}{3 a^{2} + 2 b^{2} - 3 a^{2} + 2 b^{2}} = \frac{3 c^{2} + 2 d^{2} + 3 c^{2} - 2 d^{2}}{3 c^{2} + 2 d^{2} - 3 c^{2} + 2 d^{2}}$

$\Rightarrow \frac{6 a^{2}}{4 b^{2}} = \frac{6 c^{2}}{4 d^{2}}$

$\Rightarrow \frac{a^{2}}{b^{2}} = \frac{c^{2}}{d^{2}}$

$\Rightarrow a^{2} : b^{2} : : c^{2} : d^{2}$

Therefore, the required result is $a^{2} : b^{2} : : c^{2} : d^{2}$ .

Write the result after applying componendo and dividendo to $3 a^{2} + 2 b^{2} : 3 a^{2} - 2 b^{2} : : 3 c^{2} + 2 d^{2} : 3 c^{2} - 2 d^{2}$ .

Important Questions on Ratio, Proportion and Variation

Learn and Prepare for any exam you want !

Write the result after applying componendo and dividendo to 3a2+2b2:3a2-2b2::3c2+2d2:3c2-2d2.

Important Questions on Ratio, Proportion and Variation

Learn and Prepare for any exam you want !

Write the result after applying componendo and dividendo to $3 a^{2} + 2 b^{2} : 3 a^{2} - 2 b^{2} : : 3 c^{2} + 2 d^{2} : 3 c^{2} - 2 d^{2}$ .