Question

If the ratio of two numbers is $9:1$ then find the ratio of geometric and harmonic mean between the numbers.
A. $1:9$
B. $5:3$
C. $3:5$
D. $2:5$

Answer 1

Hint: Using the basic formula for Harmonic Mean and Geometric mean we will solve the given problem and then we will be calculating the ratio between the two.

Formula Used:
Geometric Mean - $\sqrt {\left( {ab} \right)} $
Harmonic Mean - $\dfrac{{2ab}}{{\left( {a + b} \right)}}$ where $a,b$ are variables.

Complete step by step solution:
Let us Assume the numbers are $a$ and $b$
According to the ratio given in the question –
$\dfrac{a}{b} = \dfrac{9}{1}$
Simplifying , we will get –
$a = 9b$ ---- (i)
Now Finding out the ratio of Geometric Mean : Harmonic Mean
According to the formula
Geometric Mean - $\sqrt {\left( {ab} \right)} $ and Harmonic Mean - $\dfrac{{2ab}}{{\left( {a + b} \right)}}$
$\dfrac{{GM}}{{HM}} = \dfrac{{\sqrt {ab} \left( {a + b} \right)}}{{2ab}}$
Substituting the value of $a$ from equation (i) we will get –
$a = 9b$
$ \Rightarrow \dfrac{{\sqrt {\left( {9b} \right)b} \left( {9b + b} \right)}}{{2\left( {9b} \right)b}}$
$ \Rightarrow \dfrac{{3b\left( {10b} \right)}}{{18{b^2}}}$
$ \Rightarrow \dfrac{{10b}}{{6b}}$
$ \Rightarrow \dfrac{5}{3}$
Hence the Ratio we have calculated is $5:3$

Option ‘B’ is correct

Additional information
Geometric mean: By using the product of the values of the numbers, the geometric mean is a mean or average that represents the central tendency or typical value of a set of numbers.
Harmonic mean: One of various averages, and more specifically one of the Pythagorean means, is the harmonic mean. In some circumstances, when the average rate is desired, it is appropriate.

Note: Here we need to know about the geometric mean and harmonic mean. From the ratio of the number, first calculate the one number in terms of the other number. Then calculate the geometric mean and harmonic mean of the given number. At the end find the ratio of geometric mean and harmonic mean of the given number.