Question

If AM of two numbers is 9 and their HM is 4, then their GM is

Answer 1

Hint: Let us assume two numbers, a and b. We know that AM is arithmetic mean and is given as 9 in the question, therefore we get $ \dfrac{a+b}{2}=9 $ . Also, we know that HM is a harmonic mean and we are given that HM is 4, so we get $ \dfrac{2ab}{a+b}=4 $ .
Now, we know that GM is the geometric mean and it is given as $ GM=\sqrt{ab} $ . So, to find GM we need to find a value of $ ab $ and then substitute in $ GM=\sqrt{ab} $ to get the value of GM. For getting a value of $ ab $ , evaluate the equations of AM and GM.

Complete step-by-step answer:
Consider two numbers ‘a’ and ‘b’.
The arithmetic mean of a & b is defined as:
 $ AM=\dfrac{a+b}{2} $
Since AM of a & b is given as 9.
Therefore, we can write:
 $ \dfrac{a+b}{2}=9 $
 $ a+b=18......(1) $

Now, the Harmonic mean of a & b is defined as:
 $ HM=\dfrac{2ab}{a+b} $
Since HM of a & b is given as 4.
Therefore, we can write:
  $ \dfrac{2ab}{a+b}=4 $
 $ \begin{align}
  & \Rightarrow ab=\dfrac{4\left( a+b \right)}{2} \\
 & \Rightarrow ab=\dfrac{4\times 18}{2} \\
 & \Rightarrow ab=36......(2) \\
\end{align} $

To calculate the geometric mean of a & b that is defined as:
  $ GM=\sqrt{ab}......(3) $
 Substitute the value of $ ab $ from equation (2) in equation (3), we get:
 $ \begin{align}
  & GM=\sqrt{36} \\
 & GM=6 \\
\end{align} $

Note: We can also find geometric mean of two number by another method stated below:
The geometric mean of any two positive numbers can be given as $ {{\left( GM \right)}^{2}}=\left( AM \right)\left( HM \right) $
For the above question:
 $ \begin{align}
  & {{\left( GM \right)}^{2}}=\left( 9 \right)\left( 4 \right) \\
 & =36
\end{align} $
 $ \Rightarrow GM=6 $ $ \Rightarrow GM=6 $
Also, to check whether the answer is correct, always remember GM lies between harmonic mean (HM) and arithmetic mean (AM) of two numbers.
i.e. $ HM\le GM\le AM $