Question

If the retio ofH.M. and G.M. of two quantities is 12 : 13, then the ratio of the numbers is

• A. 1:2
• B. 2:3
• C. 3:4
• D. None of these

Answer 1

The correct option is D None of these

Let a and b be the two numbers

Given,

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

G.M.=$\sqrt{ab}$

As per question

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

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

$\frac{\sqrt{ab}}{a+b}=\frac{6}{13}$

$\frac{a+b}{\sqrt{ab}}=\frac{13}{6}$

$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\frac{13}{6}$

let $\sqrt{\frac{a}{b}}=x$

$x+\frac{1}{x}=\frac{13}{6}⟹6x^2+6=13x⟹6x^2-13x+6=0⟹(3x-2)(2x-3)=0$

$x=\frac{2}{3},\frac{3}{2}$

$⟹x^2=\frac{4}{9},\frac{9}{4}$

Therefore the ratio of two numbers are $\frac{a}{b}=\frac{4}{9},\frac{9}{4}$