Question

Using properties of proportion, solve for $x$
$\frac{\sqrt{x+5}+\sqrt{x-16}}{\sqrt{x+5}-\sqrt{x-16}}=\frac{7}{3}$

Answer 1

Step:solve for $x$
If $\frac{a}{b}=\frac{c}{d}$ then $\frac{a+b}{a-b}=\frac{c+d}{c-d}$

Given, $\frac{\sqrt{x+5}+\sqrt{x-16}}{\sqrt{x+5}-\sqrt{x-16}}=\frac{7}{3}$

$\frac{\sqrt{x+5}+\sqrt{x-16}+\sqrt{x+5}-\sqrt{x-16}}{\sqrt{x+5}+\sqrt{x-16}-\sqrt{x+5}+\sqrt{x-16}}=\frac{7+3}{7-3}$

[By using componendo and dividendo]
$\Rightarrow \frac{2\sqrt{x+5}}{2\sqrt{x-16}}=\frac{10}{4}$

$\Rightarrow \frac{\sqrt{x+5}}{\sqrt{x-16}}=\frac{5}{2}$

Squaring both sides, we get

$\Rightarrow \frac{x+5}{x-16}=\frac{25}{4}$

By doing the cross multiplication, we get

$\Rightarrow 4(x+5)=25(x-16)$
$\Rightarrow 4x+20=25x-400$
$\Rightarrow 20+400=25x-4x$
$\Rightarrow 420=21x$
$\Rightarrow x=\frac{420}{21}=20$

Hence, the value of $x$ is 20.

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