Let the two numbers be \(a\) and \(b\). Then GM = \(\sqrt{ab}\) and HM = \(\frac{2ab}{a+b}\).

\(\frac{GM}{HM} = \frac{a+b}{2\sqrt{ab}} = \frac{5}{4} \Rightarrow \frac{a+b}{\sqrt{ab}} = \frac{5}{2}\)

Put \(t = \sqrt{a/b}\): \(t + \frac{1}{t} = \frac{5}{2} \Rightarrow 2t^2 - 5t + 2 = 0 \Rightarrow t = 2\) or \(t = \frac{1}{2}\).

So \(a/b = 4\) or \(\frac{1}{4}\). The numbers are in the ratio 1 : 4.

Answer: (A) 1 : 4