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Relation Between AM GM HM

Relation between AM GM HM is useful to better understand arithmetic mean(AM), geometric mean(GM), harmonic mean(HM). The product of arithmetic mean and harmonic mean is equal to the square of the geometric mean. AM × HM = GM².

Among the three means, the arithmetic mean is greater than the geometric mean, and the geometric mean is greater than the harmonic mean.

AM > GM > HM

Let us learn more about the relation between AM GM HM, the formulas, derivations of AM GM HM, with the help of examples, FAQs.

1.	What Is The Relation Between AM GM HM?
2.	Formula For Relation Between AM GM HM
3.	Examples On Relation Between AM GM HM
4.	Practice Questions
5.	FAQs On Relation Between AM GM HM

What Is The Relation Between AM GM HM?

The relation between AM GM HM can be understood from the statement that the value of AM is greater than the value of GM and HM. For the same given set of data points, the arithmetic mean is greater than geometric mean, and the geometric mean is greater than the harmonic mean. This relation between AM, GM HM can be presented as the following expression.

AM > GM > HM

Here arithmetic mean is written in short form as AM, the geometric mean is written in short form as GM, and harmonic mean is written in short form as HM.

For understanding this, let us first understand how to find AM, GM, HM. For any two numbers a, b the formula for the arithmetic mean(AM), geometric mean(GM), and harmonic mean(HP) is as follows. The arithmetic mean is also called the average of the given numbers, and for two numbers a, b, the arithmetic mean is equal to the sum of the two numbers, divided by 2.

AM = \(\dfrac{a + b}{2}\)

The geometric mean of two numbers is equal to the square roots of the product of the two numbers a, b. Further, if there are n number of data, then their geometric mean is equal to the nth root of the product of the n numbers.

GM = \(\sqrt {ab}\)

The harmonic mean of two numbers 1/a. 1/b is equal to the inverse of their arithmetic mean. The arithmetic mean of these two numbers 1/a, 1/b is equal to (a + b)/2ab, and the inverse of this results in the harmonic mean of the two numbers.

HM = \(\dfrac{2ab}{a + b}\)

Formula For Relation Between AM GM HM

The formula for the relation between AM, GM, HM is the product of arithmetic mean and harmonic mean is equal to the square of the geometric mean. This can be presented here in the form of this expression.

AM × HM = GM²

Let us try to understand this formula clearly, but deriving this formula.

AM × HM = \(\dfrac{a + b}{2}\) ×\(\dfrac{2ab}{a + b}\) = ab

ab = \(\sqrt{(ab)^2}\) = \((\sqrt{ab})^2\) = GM²

Thus the square of the geometric mean is equal to the product of the arithmetic mean and harmonic mean.

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Examples on Relation Between AM GM HM

Example 1: For the two data points 10, 12, find the relationship between AM, GM, HM, and arrange them in ascending order

Solution:

The given data points are 10, 12

Arithmetic Mean (AM) = (10 + 12)/2 = 22/2 = 11

Geometric Mean(GM) = \(\sqrt {10 × 12}\) = \(\sqrt 120\) = \(2\sqrt 30 \) = 2 ×5.47 = 10.94

Harmonic Mean (HM) = \(\dfrac{2(10)(12)}{10 + 12} \) = \(\dfrac{120}{11}\) = 10.9

From the above values we have 11 > 10.94 > 10.9

AM > GM > HM

Therefore, we have AM > GM > HM.
Example 2: Find the value of geometric mean GM, if the arithmetic mean AM is 7, and harmonic mean HM is 48/7.

Solution:

Arithmetic Mean (AM) = 7

Harmonic Mean(HM) = 48/7

Geometric Mean(GM) = ?

The relation between AM, GM, HM can be computed using the following formula.

AM × HM = GM²

7 × 48/7 = GM²

GM² = 48

GM = \(\sqrt {48}\)

GM = \(4\sqrt 3\)

Therefore the geometric mean of the two numbers is \(4\sqrt 3\).

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Practice Questions on Relation Between AM GM HM

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FAQs on Relation Between AM GM HM

What Is The Relation Between AM, GM HM?

The relation between AM GM HM can be represented by the formula AM × HM = GM². Here the product of the arithmetic mean(AM) and harmonic mean(HM) is equal to the square of the geometric mean(GM).

What Is The Inequality Relation Between AM GM HM?

The relation between AM GM HM can be understood from the values of each of these means. The arithmetic mean(AM) is greater than the geometric mean(GM), and the geometric mean(GM) is greater than harmonic mean(HM). This inequality can be represented as an expression AM > GM > HM.

How To Find The Relation Between AM, GM, HM?

The relation between AM GM HM can be found by first taking the product of the arithmetic mean(AM) and harmonic mean(HM). AM × HM = \(\dfrac{a + b}{2}\) ×\(\dfrac{2ab}{a + b}\) = ab = \(\sqrt{(ab)^2}\) = \((\sqrt{ab})^2\) = GM². Thus we have AM × HM = GM².

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