Relation Between AM, GM, and HM
2026-02-28 17:30 Diff
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Last updated on December 8, 2025

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. AM × HM = GM^2. We will learn about the formulas, derivations, and the relationship between these means with the help of FAQs and examples.

What is Arithmetic Mean (AM)?

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The average of a set of numbers is called 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 = x / n (new line)

Where x is the sum of all the given numbers

(new line) 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 = x / n

= 190 / 2

= 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 either a very large or an extremely small number in a set. The geometric mean (GM) can be calculated by using the formula:

GM = (x1 × x2 × x3 × ... × xn)(1/n)

Where, x1, x2, x3 . . . xn are the given numbers 

N is the total number given.

Let's learn more about 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 = \((2 \,\times\, 8 \,\times\, 4)^{\frac{1}{3}} \,=\, 64^{\frac{1}{3}} \,=\, 4 \)

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. In simple terms, it is defined as the reciprocal of the arithmetic mean of reciprocals. The formula for HM is:

HM = \(\frac{n}{\frac{1}{x_1} \,+\, \frac{1}{x_2} \,+\, \cdots \,+\, \frac{1}{x_n}} \)


Here, n is the total number of values.

xn 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.


Reciprocals: 1/60 + 1/40 = 2/120 + 3/120 = 5/120

Dividing 2 by the sum, we get:

= \(\frac{2}{\frac{5}{120}} \,=\, 2 \,\times\, \frac{120}{5} \,=\, \frac{240}{5} \,=\, 48\)

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What is the Relationship Between AM, GM, and HM?

The relationship is: AM ≥ GM ≥ HM


This means that the arithmetic mean is always the greatest, followed by the geometric mean. 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 = √(a × b)

HM = \(\frac{2 \, a \, b}{a + b} \)

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. This can be expressed as follows:


AM × HM = GM2


By expanding this formula, we are able to identify it better.


AM × HM = \(\left(\frac{a \,+\, b}{2}\right) \,\times\, \frac{2 \, a \, b}{a \,+\, b} \)

= ab

= \(\left(\sqrt{a \, b}\right)^2 \,=\, GM^2 \)

Tips and Tricks for Relation between AM GM and HM

The Arithmetic Mean (AM) is the most common average and is usually the highest of the three, while the Geometric Mean (GM) is helpful for things like growth rates and is always less than or equal to the AM. The Harmonic Mean (HM) is great for things like speed or efficiency, where smaller numbers matter more, making it typically lower than both AM and GM.

AM ≥ GM ≥ HM: The Arithmetic Mean (AM) is always greater than or equal to the Geometric Mean (GM), and the Geometric Mean is always greater than or equal to the Harmonic Mean (HM).

When Are They Equal?: AM, GM, and HM are equal only when all the numbers in the set are identical.

AM as the average: The Arithmetic Mean (AM) is simply calculated by adding all the numbers and dividing by how many numbers there are.

  • GM for growth and multiplication: The Geometric Mean (GM) is especially useful when dealing with rates of growth or multiplying numbers, like calculating compound interest or population increases.
  • HM for rates and reciprocals: The Harmonic Mean (HM) is ideal for averaging rates or values where reciprocals are involved, such as speed or efficiency.
  • GM and AM in comparison: The Geometric Mean (GM) is always less than or equal to the Arithmetic Mean (AM), reflecting the difference between additive and multiplicative relationships.

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 ways to avoid them.

Real-Life Applications of Relationship Between AM, GM, HM

AM, GM, and HM are often used in the field of finance. Here are some real-life applications; learning about them makes students realize the importance of AM, GM, and HM. 

  • 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 for multiplicative numbers, such as growth rates and investment returns. In investments and stock markets, it is used to find the profit in a year, the average population growth rate, etc.
  • Average speed: The harmonic mean is used in cases where rates like speed, efficiency, or density are involved. The harmonic mean is used to find the average speed, fuel efficiency, and other rate-based calculations.
  • Health and medicine: In healthcare, doctors rely on the Harmonic Mean (HM) to determine the average dosage of medications, particularly when the effectiveness of treatment depends on factors like how quickly a drug is absorbed. This allows for more personalized treatment plans, ensuring patients receive the optimal amount of medicine for the best outcomes. 
  • Environmental studies: In environmental research, the Geometric Mean (GM) is often used to calculate the average pollution levels over time. Since pollution levels can increase or decrease at varying rates, GM offers a more accurate view of these changes, especially when the data shows significant fluctuations, such as in air or water quality.

Download Worksheets

Problem 1

Find AM, GM, and HM for a = 4, b = 9.

Okay, lets begin

AM = 6.5, GM = 6, HM ≅ 5.54
 

Explanation

AM = (a+b) / 2 = (4+9) / 2 = 13/2 = 6.5

GM = √(a × b) = √(4 × 9) = √36 = 6

HM = \(\frac{2 \, a \, b}{a \,+\, b} \) = \(\frac{2 \,\times\, 4 \,\times\, 9}{4 \,+\, 9} \) = \(\frac{72}{13} \) ≅ 5.54

Well explained 👍

Problem 2

If AM = 18 and GM = 12, find the HM.

Okay, lets begin

HM = 8
 

Explanation

HM = GM2 / AM

= 122 / 18

= 144 / 18

= 8

Well explained 👍

Problem 3

If AM = 30 and HM = 20, find GM.

Okay, lets begin

GM ≅ 24.49
 

Explanation

GM = √(AM × HM)

= √(30 × 20)

= √600 ≅ 24.49

Well explained 👍

Problem 4

If the geometric mean (GM) of two numbers is 15 and the harmonic mean (HM) is 12, find the arithmetic mean (AM).

Okay, lets begin

AM ≅ 16.33
 

Explanation

AM = GM2 / HM

= 152 / 12

= 225 / 12

≅ 18.75

Well explained 👍

Problem 5

If the arithmetic mean (AM) of two numbers is 9 and their harmonic mean (HM) is 81/10, find the geometric mean (GM).

Okay, lets begin

GM ≅ 8.5
 

Explanation

GM = √(AM × HM)

= √(9 × 81/10)

= √(9 × 8.1)

= √72.9

≅ 8.54

Well explained 👍

FAQs on Relationship Between AM, GM, HM

1.What is the relationship between AM, GM, and HM of any two unequal positive numbers?

For any two unequal positive numbers, AM > GM > HM.
 

2.Find the two positive numbers if their arithmetic mean is 34 and their geometric mean is 16.

The two numbers are 64 and 4 because their average is 34 and their geometric mean is 16.

3.What is the formula of GM between any two unequal positive numbers?

The formula for the geometric mean (GM) between any two unequal positive numbers a and b is:

GM = √a × b.

4.What happens when AM is equal to GM?

If the AM and GM are equal, then the two numbers must be exactly equal.
 

5.What is the GM of two positive numbers whose AM and HM are 75 and 48 respectively?

The geometric mean is 60 when the AM and HM are 75 and 48.
 

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.