101 LearnersLast updated on July 5th, 2025
Relation Between AM, GM, and HM
The relationship between AM, GM, and HM helps us better understand the three mean types: Arithmetic Mean, Geometric Mean, and Harmonic Mean. The product of the Arithmetic Mean (AM) and Harmonic Mean (HM) is equal to the square of the Geometric Mean (GM): AM × HM = GM2. We will learn about the formulas, derivations, and the relationship between these means with the help of FAQ’s and examples.
What is the Arithmetic Mean (AM)
The average of a set of numbers is called the Arithmetic Mean (AM). The arithmetic mean can be found by dividing the sum of all given numbers by the total number of values. It helps us identify the center value of a data set, and it’s called the mean. The formula for AM is:
AM = xn
Here x is the sum of all the given numbers
n is the total number.
The arithmetic mean is used to find the average in daily life, such as finding the average marks in exams, the average temperature, etc.
Example: Find the average of 90 and 100.
To find the average, first we need to add the numbers.
Adding 90 and 100, we get 190.
The total number given is 2.
AM = xn
= 1902
= 95
What is Geometric mean (GM)
The Geometric Mean (GM) is a unique type of average that involves multiplying all the given numbers and then taking the nth root. Where n is the total number of values. This is useful while calculating population growth, interest rates, etc. It helps in reducing the effect of very large or very small numbers in a set. The Geometric Mean (GM) can be calculated by using the formula:
GM = nX1X2X3. . . Xn
Where X1X2X3. . . Xn are the given numbers
N is the total number given.
Let's learn the geometric mean by using the simple example given below,
Find the geometric mean for 2, 8, and 4.
Multiplying 2, 8, and 4, we get 64.
The total number is 3
GM = 3284
GM = 364 = 4
What is Harmonic Mean (HM)
The Harmonic mean is a special type of average used for dealing with rates, speed, etc. In the Harmonic mean, first we take the reciprocal of each number, find their average, and then flip the result back. In simple terms, it is defined as the reciprocal of the arithmetic mean of reciprocals. The formula for HM is:
HM = nin1xi
Here, n is the total number of values.
xi is an individual value.
For example: Find the HM using the given values 60 and 40.
First, find the reciprocals of the numbers, find the sum, and then use the formula.
160 + 140 = 2120 + 3120 = 5120
HM = 25120 = 2 × 1205 = 2405 = 48
What is the relationship between AM, GM, and HM?
The relationship between Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) shows that:
AMGMHM
This means that the Arithmetic Mean is always the largest, followed by the Geometric Mean, and the Harmonic Mean is the smallest.
To understand this, we need to understand the following. For any two numbers a and b the formula for arithmetic mean, geometric mean, and harmonic mean are as follows. The arithmetic mean is the average of two numbers. The geometric mean is equal to the square root of the product of two numbers. The harmonic mean of two numbers is calculated by taking the reciprocal of the arithmetic mean of their reciprocals.
AM = a+b2
GM = ab
HM = 2aba+b
The formula for the relation between AM, GM, and HM is that the product of the arithmetic and harmonic mean is equal to the square of the geometric mean.
AM × HM = GM2
By deriving this formula, we are able to identify it better.
AM × HM = a+b2 × 2aba+b
= ab
Here ab can be derived as ab2 = (ab)2 = GM2
Real Life Applications of Relationship Between AM, GM, HM
AM, GM, and HM have real-life applications like fields such as averaging, finance, and speed calculations. Here are some real life applications, which make students learn easily and apply the means effectively in real life.
- Finding Average: Arithmetic means are used to find the average easily. It can be used to find the average of marks in exams, calculate the average income, daily temperature, etc.
- Investments and Stock Market Returns: The geometric mean is useful when numbers are multiplicative, such as growth rates and returns. In investments and stock markets, it is used to find the profit in a year, the average population growth rate, etc.
- Average Speed of a Journey: The harmonic mean is used in cases where rates like speed, efficiency, or density are involved. The harmonic mean is used to find average speed, fuel efficiency, and other rate-based calculations.
Common Mistakes and How to Avoid Them in the Relationship Between AM, GM, and HM
Mistakes are common when dealing with the relation between AM, GM, and HM. Here are some mistakes and the ways to avoid them which can help us to prevent those mistakes.
Mistake 1
Considering AM, GM, and HM are always equal
We assume that all three means are equal. They are equal only if all the numbers are the same. Otherwise, AM GMHM. For 4 and 4, AM = GM = HM = 4, but for 2 and 8, they are different.
Mistake 2
Forgetting the order
Students might change the order. Always remember that the order is AM GMHM. Remember that AM is the largest, GM is in the middle, and HM is the smallest. For example, writing GM AMHM, instead of the correct order.
Mistake 3
Forgetting to take reciprocals when using HM.
Always remember to take reciprocals of the arithmetic mean when using the HM. Not taking the reciprocals leads to a mistake. For example, for 2 and 6, HM = 3 only if you use 1/2 and 1/6 first.
Mistake 4
Using the wrong roots in GM calculations
The root value depends on the value of the numbers. For example, if there are 2 numbers, it's square root; for 3, the root value is 3 (cube root); for 4 numbers, the root value is 4, and so on.
Mistake 5
Using GM for simple averages
The Arithmetic Mean is best for simple averages, not GM. Understand the use of all three means and apply them to the problems. For calculating marks 6, 7, and 8, the AM = 7 is the best average to use; if we use GM instead of AM, the calculation becomes difficult.
Solved Examples of the Relationship Between AM, GM, HM
Problem 1
Find AM, GM, and HM for a = 4, b = 9.
AM = 6.5, GM = 6, HM ≅ 5.54
Explanation
AM = a+b2 = 4+92 = 132 = 6.5
GM = ab = 4 × 9 = 36 = 6
HM = 2aba+b = 2 × 4×94+9 = 7213 ≅ 5.54
Problem 2
If AM = 18 and GM = 12, find HM.
HM = 8
Explanation
HM = GM2AM = 12218 = 14418 = 8
Problem 3
If AM = 30 and HM = 20, find GM.
GM ≅ 24.49
Explanation
GM = AM × HM = 30 × 20 = 600 ≅ 24.49
Problem 4
If the geometric mean (GM) of two numbers is 15 and the harmonic mean (HM) is 12, find the arithmetic mean (AM).
AM ≅ 16.33
Explanation
AM = GM2HM = 14212 = 19612 ≅ 16.33
Problem 5
If the arithmetic mean (AM) of two numbers is 9 and their harmonic mean (HM) is 8110, find the geometric mean (GM).
GM ≅ 8.5
Explanation
GM = AM × HM = 9 × 8110 = 72910 = 2710 ≅ 8.5
FAQs on Relationship Between AM, GM, HM
1.What is the relationship between AM, GM, and HM of any two unequal positive numbers?
2.What is the answer to the am between two positive numbers is 34 and their GM is 16?
3.What is the formula of GM between any two unequal positive numbers?
4.What happens when AM is equal to GM?
5.What is the GM of two positive numbers whose am and hm are 75 and 48?
6.How can children in Vietnam use numbers in everyday life to understand Relation Between AM, GM, and HM?
7.What are some fun ways kids in Vietnam can practice Relation Between AM, GM, and HM with numbers?
8.What role do numbers and Relation Between AM, GM, and HM play in helping children in Vietnam develop problem-solving skills?
9.How can families in Vietnam create number-rich environments to improve Relation Between AM, GM, and HM skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
