Question

Find the geometric mean of 20 and 45.
(a) 30
(b) 60
(c) 50
(d) 40

Answer 1

To find the geometric mean (GM) of 20 and 45, we have to use the formula for GM which is given by $GM=\sqrt[n]{{{x}_{1}}{{x}_{2}}...{{x}_{n}}}$ , where ${{x}_{1}},{{x}_{2}},..,{{x}_{n}}$ are the observations. We have to take the square root of the product of 20 and 45 to their GM.

Complete step by step answer:
We have to find the geometric mean (GM) of 20 and 45. Let us recollect what geometric mean is. We know that the Geometric Mean of a series containing n observations is the nth root of the product of the values. Let ${{x}_{1}},{{x}_{2}},..,{{x}_{n}}$ be the observations, then the G.M is given by
$GM=\sqrt[n]{{{x}_{1}}{{x}_{2}}...{{x}_{n}}}$
Or we can write the GM as
$GM={{\left( {{x}_{1}}{{x}_{2}}...{{x}_{n}} \right)}^{\dfrac{1}{n}}}$
We are given two numbers 20 and 45. Therefore, $n=2$ . We can find the geometric mean of 20 and 45 by taking the square root of the product of 20 and 45.
$\begin{aligned} & \Rightarrow GM=\sqrt{20\times 45} \\\ & \Rightarrow GM=\sqrt{900} \\\ \end{aligned}$
We have to take the square root of 900 which will be 30. Therefore, the above equation can be written as
$\Rightarrow GM=30$

So, the correct answer is “Option a”.

Note: Students have a chance of making mistake by writing the formula for GM as $GM=\sqrt[n]{{{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}}$ . They should not get confused with GM and AM (Arithmetic Mean). Arithmetic mean or simplify, mean can be found by adding all the numbers of a data set and dividing the sum by the number of data points in a set. Geometric Mean is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values.