Question

If the ratio of $H.M$ 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. $noneofthese$

Answer 1

The correct option is D

$noneofthese$

Find the ratio of the numbers :

Let $a$ and $b$ be the numbers,

Then $HM=\frac{2ab}{(a+b)}$

$GM=\sqrt{\left(ab\right)}$

Given $\frac{HM}{GM}=\frac{12}{13}$

$\Rightarrow$ $\raisebox{1ex}{$\frac{\left[2ab\right]}{(a+b)}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{\left(ab\right)}$}\right.=12/13$

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

$\Rightarrow$ $\frac{\sqrt{\left(ab\right)}}{(a+b)}=\frac{6}{13}$

$\Rightarrow$ $\frac{(a+b)}{\sqrt{\left(ab\right)}}=\frac{13}{6}$

$\Rightarrow$$\sqrt{\left(\frac{a}{b}\right)}+\sqrt{\left(\frac{b}{a}\right)}=\frac{13}{6}$

Let $\sqrt{\frac{a}{b}}=p,p+\frac{1}{p}=\frac{13}{6}$

$\Rightarrow$ $6p^2-13p+6=0$

Solving using quadratic formula

We get $p=\frac{2}{3}or\frac{3}{2}$

$\Rightarrow$ $p^2=\frac{4}{9}or\frac{9}{4}$

So, $\frac{a}{b}=\frac{4}{9}\mathrm{or}\frac{9}{4}$

Hence, the option (D) is correct.