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 ${\displaystyle \frac{a+b}{2}}=9$ . Also, we know that HM is a harmonic mean and we are given that HM is 4, so we get ${\displaystyle \frac{2ab}{a+b}}=4$ .
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:
 $AM={\displaystyle \frac{a+b}{2}}$
Since AM of a & b is given as 9.
Therefore, we can write:
 ${\displaystyle \frac{a+b}{2}}=9$
 $a+b=18......(1)$

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

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:
 $\begin{array}{rl}& GM=\sqrt{36}\\ & GM=6\end{array}$

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:
 $\begin{array}{rl}& {\left(GM\right)}^2=\left(9\right)\left(4\right)\\ & =36\end{array}$
 $\Rightarrow GM=6$ $\Rightarrow GM=6$
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. $HM\le GM\le AM$