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

Given: $H.M. : G.M. =4:5$

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

$\begin{array}{l}\frac{\frac{2ab}{a+b}}{\sqrt{ab}}=\frac{4}{5}\\ \Rightarrow \frac{2\sqrt{ab}}{a+b}=\frac{4}{5}\\ \Rightarrow \frac{a+b}{2\sqrt{ab}}=\frac{5}{4}\\ \Rightarrow \frac{a+b+2\sqrt{ab}}{a+b-2\sqrt{ab}}=\frac{5+4}{5-4}\\ \Rightarrow {\left(\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\right)}^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}$$\left[Applying componendo and dividendo \right]$