If $H . M . : G . M . = 4 : 5$ for the two positive numbers, then the ratio of the numbers is

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$3 : 2$

$3 : 4$

$2 : 3$

$4 : 1$

answer is A.

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Detailed Solution

Given: $H . M . : G . M . = 4 : 5$

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

$\begin{array}{l} \frac{\frac{2 a b}{a + b}}{\sqrt{a b}} = \frac{4}{5} \\ \Rightarrow \frac{2 \sqrt{a b}}{a + b} = \frac{4}{5} \\ \Rightarrow \frac{a + b}{2 \sqrt{a b}} = \frac{5}{4} \\ \Rightarrow \frac{a + b + 2 \sqrt{a b}}{a + b - 2 \sqrt{a b}} = \frac{5 + 4}{5 - 4} \\ \Rightarrow {(\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}})}^{2} = \frac{9}{1} \\ \Rightarrow \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} = \frac{3}{1} \Rightarrow \frac{\sqrt{a}}{\sqrt{b}} = \frac{3 + 1}{3 - 1} \Rightarrow \frac{\sqrt{a}}{\sqrt{b}} = \frac{4}{2} \Rightarrow \frac{a}{b} = \frac{4}{1} \end{array}$ $[A p p l y i n g c o m p o n e n d o a n d d i v i d e n d o]$