Question

Let the H.M and G.M of two positive numbers be in the ratio $4:5$, then largest number divided by smallest number of the two numbers ?

Answer 1

Let the two numbers be $a,b$
Given that $\frac{2ab}{(a+b)\left(\sqrt{ab}\right)}=\frac{4}{5}$
$\Rightarrow 5\sqrt{ab}=2(a+b)$
$\Rightarrow 25ab=4a^2+8ab+4b^2$
$\Rightarrow 4a^2-17ab+4b^2=0$
i.e. $4a^2-16ab-ab+4b^2=0$
i.e. $(4a-b)(a-4b)=0$
i.e. $\frac{a}{b}=0.25$ or $4$
Since we need larger number divided by smaller number, the answer becomes $4$.