Name: Proportion
Uploaded: 2019-07-19
Duration: 5 min 17 s
Description: A T-Rex looks strange because it has disproportionately sized arms. But, what are proportions? To understand their significance in mathematics, here are some...

Question

Find the third proportional to

$5 + 2 \sqrt{3}, 37 + 20 \sqrt{3}$

Accepted Answer

Given, Two numbers $5 + 2 \sqrt{3}, 37 + 20 \sqrt{3}$ and $37 + 20 \sqrt{3}$

Let the third proportional to $5 + 2 \sqrt{3}$ and $15$ be $x$ .

If $x, y$ and $z$ are in continued proportion, then $z$ is called third proportional.

Therefore,

$5 + 2 \sqrt{3} : 37 + 20 \sqrt{3} : : 37 + 20 \sqrt{3} : x \Rightarrow \frac{5 + 2 \sqrt{3}}{37 + 20 \sqrt{3}} = \frac{37 + 20 \sqrt{3}}{x} \Rightarrow x = \frac{{(37 + 20 \sqrt{3})}^{2}}{5 + 2 \sqrt{3}} \Rightarrow x = \frac{{(37)}^{2} + {(20 \sqrt{3})}^{2} + 2 \times 37 \times 20 \sqrt{3}}{5 + 2 \sqrt{3}} \Rightarrow x = \frac{1369 + 1200 + 1480 \sqrt{3}}{5 + 2 \sqrt{3}}$

Rationalize the denominator

$\Rightarrow x = \frac{2569 + 1480 \sqrt{3}}{(5 + 2 \sqrt{3})} \times \frac{(5 - 2 \sqrt{3})}{(5 - 2 \sqrt{3})}$

$\Rightarrow x = \frac{(2569 + 1480 \sqrt{3}) (5 - 2 \sqrt{3})}{{(5)}^{2} - {(2 \sqrt{3})}^{2}} \Rightarrow x = \frac{2569 (5 - 2 \sqrt{3}) + 1480 \sqrt{3} (5 - 2 \sqrt{3})}{25 - 12} \Rightarrow x = \frac{12845 - 5138 \sqrt{3} + 7400 \sqrt{3} - 8880}{13} \Rightarrow x = \frac{3965 + 2262 \sqrt{3}}{13} \Rightarrow x = \frac{13 (305 + 174 \sqrt{3})}{13} \Rightarrow x = (305 + 174 \sqrt{3})$

Hence, the required third proportional is $305 + 174 \sqrt{3}$ .

Find the third proportional to

$5 + 2 \sqrt{3}, 37 + 20 \sqrt{3}$

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !

Find the third proportional to 5+23, 37+203

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !

Find the third proportional to

$5 + 2 \sqrt{3}, 37 + 20 \sqrt{3}$