Question

The arithmetic mean of two numbers is $\frac{13}{2}$ and their geometric mean is $6$. Find their harmonic mean.

Answer 1

We know that the arithmetic mean between the two numbers $a$ and $b$ is $AM=\frac{a+b}{2}$ and the geometric mean is $GM=\sqrt{ab}$.

Here, it is given that the arithmetic mean of $a$ and $b$ is $\frac{13}{2}$, therefore,

$AM=\frac{a+b}{2}\Rightarrow \frac{13}{2}=\frac{a+b}{2}\Rightarrow a+b=13...\left(1\right)$

Also, it is given that the geometric mean of $a$ and $b$ is $6$, therefore,

$GM=\sqrt{ab}\Rightarrow 6=\sqrt{ab}\Rightarrow ab=6^2\Rightarrow ab=36...\left(2\right)$

Now, we also know that the harmonic mean between the two numbers $a$ and $b$ is $HM=\frac{2ab}{a+b}$, thus using equations 1 and 2 we have,

$HM=\frac{2ab}{a+b}=\frac{2\times 36}{13}=\frac{72}{13}$

Hence, the harmonic mean is$\frac{72}{13}$.