Question

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

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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.

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. $HM\le GM\le AM$