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Relation between AM, GM and HM helps us to understand a progression or mathematical series better. AM, GM and HM stand for Arithmetic mean (AM), Geometric mean (GM), and Harmonic mean (HM), respectively. They are all a means of mathematical series used in sequence and series which may be defined as a collection of objects in a sequential manner. We have three popular sequences in mathematics, namely, Arithmetic Sequence, Geometric Sequence and Harmonic Sequence. Each of them has a different significance and hence has different applications.
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Key Terms: Arithmetic Mean, Geometric Mean, Harmonic Mean, Sequence, Series, Average
What is AM?
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Arithmetic Mean, also referred to as Arithmetic average, is used in statistics which is basically the average or mean of a series of numbers. It is the sum of all the numbers in the series divided by the count of numbers in the series. The formula for arithmetic mean is given below,
Arithmetic Mean Formula
where, n = count of numbers in the series and x1 + x2 + x3………xn represents the terms in the sequence.
Let us consider a series of numbers a, b, c, d, e, f and g.
Here, n = 7, so the arithmetic mean of the series would be,
(a + b + c + d + e + f + g) / 7
Read More: Arithmetic Sequence
What is GM?
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Geometric mean unlike arithmetic mean is calculated by taking the product of all the numbers in the series rather than sum as in arithmetic mean. A geometric mean is most commonly used where the data is compounded for example in banks, stock markets, etc.
Geometric mean is calculated by taking the nth root of the products of all the numbers in the series. The formula for geometric mean is given below,
Geometric mean formula
where, n = count of numbers in the series and x1.x2.x3………xn represents the terms in the sequence.
For example, let us consider a series with terms a, b, c and d.
Here, n = 4, so the geometric mean of the series would be,
(a.b.c.d)1/n
Read More: Geometric Progression
What is HM?
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Harmonic mean is another type of average where the reciprocals of the numbers in the series is considered. You can think of harmonic mean as the reciprocal of the mean of reciprocal of the numbers in the series. The formula of the harmonic mean is given below,
Harmonic mean formula
where, n = number of observations or count of numbers in the series series and 1/x1 + 1/x2 + 1/x3………1/xn represents the terms in the sequence.
Relation Between AM, GM and HM
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For a given set of numbers, the arithmetic mean is always greater than the GM and HM of the series.
AM > GM > HM
The relationship between AM, GM, and HM is given by the formula,
AM × HM = GM2
Derivation
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To understand this we must first understand what arithmetic progression, geometric progression and harmonic progression are.
- Arithmetic progression is a sequence of numbers in which the difference between every two consecutive numbers is equal.
For example, 3, 6, 9, 12, 15, 18… In this series or arithmetic progression, the difference between two consecutive numbers is 3.
- Geometric progression is a sequence where each succeeding term is formed by multiplying the corresponding preceding term with a particular fixed number.
For example, 3, 6, 12, 24, 48…
- Harmonic progression is a series formed by the reciprocal of numbers in an arithmetic progression.
For example, 1/2, 1/4, 1/6
Now, let us consider an arithmetic progression (AP) with a, b, c as observations. According to the definition of AP, we can understand that in this progression, b is the AM (mean) of the series. The common difference of this Arithmetic Progression is;
b – a = c – b
a + c = 2b
Hence,
a + c = 2AM…………..(1)
Secondly, let a, b, c be a Geometric Progression, where, b is the GM. Then, the common ratio of this GP is;
b/a = c/b
ac = c2
ac = GM2……………(2)
Lastly is the case of harmonic progression, a, b, c, where the reciprocals of each term will form an arithmetic progression, such as 1/a, 1/b, 1/c is an AP. and b is the HM
Now the common difference of the above AP is;
1/b – 1/a = 1/c – 1/b
2/b = 1/c + 1/a
2/b = (a + c)/ac
2/HM = (a + b)/ab ………….(3)
Substituting eq. 1 and eq.2 in eq. 3 we get;
2/HM = 2AM/GM2
GM2 = AM x HM
Things to Remember
- AM or arithmetic mean is the simple average or mean of a set of numbers. It is the sum of all the numbers divided by the count of numbers in the series.
- GM or geometric mean is used where compounding is done. It is the nth root of the product of all the numbers in the series.
- HM or harmonic mean is the reciprocal of the mean of reciprocal of the numbers in the sequence.
- AM of a given series is always greater than GM and HM. And GM is greater than HM, but lesser than AM.
- The relation between AM, GM, and HM is given by the formula, GM2 = AM x HM
- If any two of AM, GM, or HM are given, one can easily determine the third mean by using the formula for the relationship between AM, GM, and HM.
Solved Questions
Ques. It is given that four times the geometric mean of two numbers ‘a’ and ‘b’ is equal to the arithmetic mean, such that a > b > 0. Calculate the value of (a + b)/(a - b). (5 marks)
Ans. Arithmetic mean of the two numbers, AM=(a+b)/2
Geometric mean of the two numbers, GM = √ab
As given in the question, Arithmetic mean = 4 times the geometric mean
AM = 4 GM
Or, (a+b)/2 = 4 √ ab
Or, a + b = 8 √ab
Or, (a + b)2 = 64 ab ……………(1)
Also, we can write (a + b)2 - (a - b)2 = 4 ab
Or, (a - b)2 = (a + b)2 - 4 ab
Or, (a - b)2 = 64 ab - 4 ab (From eq. 1)
Or, (a - b)2 = 60 ab
Or, a - b = √60ab
Hence, (a+b) / (a−b) = √(64ab / 60ab) = √(16/15) = 1.0327
Ques. If AM and GM of two positive numbers a and b are 16 and 8. Find the value of the harmonic mean of the two numbers. (3 marks)
Ans. AM = a+b/2 = 16
GM = √ab = 8
And as we know from the formula for relation between AM, GM, and HM that,
GM2 = AM x HM
Or, 82 = 16 x HM
Or, HM = 64/16
Or, HM = 4
Therefore, the harmonic mean of a and b is 4.
Ques. A series has two positive numbers a and b. The AM of a and b is 10 and GM is 8. Find the value of a and b. (3 marks)
Ans. AM = a+b/2 = 10
Or, a+b = 20 ………(1)
GM = √ab = 8
Or, ab = 64 ………..(2)
Putting the values of a and b from(1) and (2) in (a – b)2 = (a + b)2 – 4ab, we get
(a – b)2 = 400 – 256 = 144
Or, a – b = ± 12 …………(3)
From solving (1) and (3), we get
a = 4 and b = 16
Ques. Find out the AM, GM and HM of a series 2, 5, 8, 9.(3 marks)
Ans. Arithmetic mean, AM = sum of all the numbers / n
AM = 2+5+8+9 / 4 = 6
Geometric mean, GM = n√product of all the numbers
GM = (2 x 5 x 8 x 9) 1/4 = 5.18
Harmonic mean, HM = GM2 / AM
HM = 26.83/6 = 4.47
Ques. Over ten years, the average monthly wage in a particular municipality increased from 2,500 to 5000. What is the average annual increase using the geometric mean? (3 marks)
Ans. i) Geometric mean calculation = (2500 × 5000)½
= 3535.53
ii) Divide it by 10 (to get the ten-year average increase).
3535.534 /10 = 353.53.
As per GM, the average increase is 353.53.
Ques. A series consists of 5 numbers. The arithmetic mean and harmonic mean of the series is 8 and 2. Calculate the geometric mean of the series. (2 marks)
Ans. Arithmetic mean, AM = 8
Harmonic mean, HM = 2
As we know, GM2 = AM x HM
Therefore, GM = √8 x 2 = 4
Therefore, the geometric mean of the series is 4.
Ques. An arithmetic progression is given, a, b, c, whose GM is 4 and HM is 2 and the common difference of the progression is 3. Find out the values of a, b and c. (3 marks)
Ans. Given that,
Geometric mean, GM = 4
Harmonic mean, HM = 2
From formula we know that, AM = GM2/HM
Therefore, AM = 16/2 = 8
From the definition of Arithmetic progression, we know that b = AM of the AP.
Common difference = 3 (given in question)
Therefore, a = b-3 = 8-3 = 5
And, c = b+3 = 8+3 = 11
Therefore, the arithmetic progression is, 5, 8, 11.
Ques. Consider a series of two positive integers, p and q. If x and y are the HM and GM of the series, such that 5x = 4y. Find out how m and n are related to each other. (5 marks)
Ans. GM = √mn = y
Or, mn = y2
HM = 2 / (1/m + 1/n) = x
Or, 2mn / m+n = x
Given in the question, 5x = 4y
Putting the value of x and y,
10mn / (m + n) = 4√mn
Or, 10√mn = 4 (m+n)
Or, 100 mn = 16 (m2 + n2 + 2mn) ……( squaring in both sides)
Or, 16m2 + 16n2 - 68mn = 0
Dividing both sides by mn,
16m/n + 16n/m - 68 = 0
Let, m/n = t
16t + 16/t – 68 =0
16t2 – 68t + 16 = 0
Solving for t,
t = 4 or 1/4
Therefore, m/n = 4 or ¼
Or, m = 4n
Ques. The arithmetic mean of a series is 10, and geometric mean is 5. Calculate the harmonic mean of the series. (2 marks)
Ans. Arithmetic mean, AM = 10
Geometric mean, GM = 5
GM2 = AM x HM
Or, HM = GM2 / AM
Or, HM = 25/ 10
Or, HM = 2.5
Ques. A series has two positive numbers p and q. The AM of p and q is 3 and GM is 2√2. Find the value of p and q. (3 marks)
Ans. AM = p+q / 2 = 3
Or, p+q = 6 ………(1)
GM = √pq = 2√2
Or, pq = 8 ………..(2)
Putting the values of a and b from(1) and (2) in (a – b)2 = (a + b)2 – 4ab, we get
(p – q)2 = 36 – 32 = 4
Or, p – q = ± 2 (approx) …………(3)
From solving (1) and (3), we get
p = 4 or 2 and q = 2 or 4
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