Question

It the ratio of H.M and G.M of two
quantities is 12: 13 then show that
the ration of the numbers is 9:4​

Answer 1

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Solution

Answer 2

Answer:

Step-by-step explanation:

Given :

Ratio of H.M & G.M of two numbers is 12 : 13,

Prove that,

Ratio of the two numbers is 9 : 4.

Solution :

let the two numbers be a & b,.

then,

H.M of a & b is given by,

$H.M(a,b) = \frac{2ab}{a+b}$.

G.M of a & b is given by,

$G.M(a,b) = \sqrt{ab}$

so, it is given that,

$\frac{(\frac{2ab}{a+b})}{\sqrt{ab}} = \frac{12}{13}$

To Prove :

$\frac{a}{b} = \frac{9}{4}$

Proof :

$\frac{(\frac{2ab}{a+b})}{\sqrt{ab}} = \frac{12}{13}$

⇒ $\frac{2ab}{a+b} \times \frac{1}{\sqrt{ab}} = \frac{12}{13}$

⇒ $\frac{2\sqrt{a}{b}}{a+b} = \frac{12}{13}$

Adding 1 both the sides,

⇒ $\frac{2\sqrt{ab}}{a+b} + 1 = \frac{12}{13} + 1$

⇒ $\frac{a + b + 2\sqrt{ab}}{a+b} = \frac{12 + 13}{13}$

⇒ $\frac{(\sqrt{a} + \sqrt{b})^2}{a+b} = \frac{25}{13}$

By taking square root both the sides,

⇒ $\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a+b}} = \frac{5}{\sqrt{13}}$

So,

⇒ $\sqrt{a} + \sqrt{b} = 5x$ ; (x ∈ R)

⇒ $\sqrt{a+b} = \sqrt{13x}$ (or) $a+b=13x$

These equations are true when a = 9x & b = 4x or vice-versa

hence,

$\frac{a}{b}=\frac{9}{4}$

Hence,proved.