
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
162.6k+ views
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.
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.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More
