Question

if the harmonic mean between two numbers is to their geometric mean as 20 to 29, prove that the numbers are in the ratio of 4:25

Answer 1

Let a and b be the numbers.
Then, harmonic mean $H=\frac{2}{\left({\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}\right)}=\frac{2ab}{a+b}$
Geometric mean, $G=\sqrt{ab}$

Then, the ratio of the harmonic mean to geometric mean will be

$$\begin{gathered} H:G=\frac{H}{G}=\frac{20}{29} \\ \Rightarrow \frac{{\displaystyle \frac{2ab}{a+b}}}{\sqrt{ab}}=\frac{20}{29} \\ \Rightarrow \frac{\sqrt{ab}}{a+b}=\frac{10}{29} \\ \Rightarrow 10a+10b-29\sqrt{ab}=0 \end{gathered}$$

divide whole equation by $\sqrt{ab}$

Let $\sqrt{\frac{a}{b}}=k$ then

$$\begin{gathered} 10k^2-29k+10=0 \\ \Rightarrow 10k^2-25k-4k+10=0 \\ \Rightarrow \left(2k-5\right)\left(5k-2\right)=0 \\ \Rightarrow k=\frac{5}{2},\frac{2}{5} \\ \Rightarrow \sqrt{\frac{a}{b}}=\frac{5}{2},\frac{2}{5} \\ \Rightarrow \frac{a}{b}=\frac{25}{4},\frac{4}{25} \end{gathered}$$

Thus, the numbers are in the ratio 4:25.