Physics and Mathematics
Arithmetic Mean
1. Concept Overview
When two or more quantities are given, we often want to find a single representative value that lies between them.
In Arithmetic Progression, this representative value is called the Arithmetic Mean (A.M.).
The Arithmetic Mean is the number which, when placed between two numbers, forms an A.P.
2. Arithmetic Mean Between Two Numbers
If two numbers are [a] and [b], then their Arithmetic Mean is:
[ A.M. = \dfrac{a + b}{2} ]
Explanation:
The middle term of an A.P. must be equally distant from both ends.
Hence, it is simply the average of the two numbers.
3. Arithmetic Mean Between More Than Two Numbers
If [a₁, a₂, a₃, … , aₙ] are [n] numbers, then their Arithmetic Mean is:
[ A.M. = \dfrac{a₁ + a₂ + a₃ + … + aₙ}{n} ]
This definition is commonly used in statistics and real-life applications.
4. Inserting Arithmetic Means Between Two Numbers
If [a] and [b] are two numbers and we want to insert [n] Arithmetic Means between them:
- Total number of terms = [n + 2]
- First term [a₁ = a]
- Last term [aₙ₊₂ = b]
Common difference:
[ d = \dfrac{b − a}{n + 1} ]
The inserted Arithmetic Means are:
[a + d, a + 2d, a + 3d, … , a + nd]
5. Key Properties of Arithmetic Mean
- Arithmetic Mean always lies between the smallest and largest values
- If all numbers are equal, A.M. equals that common value
- Arithmetic Mean is affected by extreme values
- Arithmetic Mean of two numbers is the middle term of the A.P.
6. Examples with Solutions
Example 1
Find the Arithmetic Mean of 6 and 14.
Step-by-Step Solution:
The Arithmetic Mean of two numbers is given by
[ A.M. = \dfrac{a + b}{2} ]
Here, the two numbers are:
[a = 6] and [b = 14]
Substitute the values:
[ A.M. = \dfrac{6 + 14}{2} ]
Add the numbers in the numerator:
[6 + 14 = 20]
Divide by 2:
[ A.M. = \dfrac{20}{2} = 10 ]
Final Answer:
The Arithmetic Mean is 10.
Example 2
Find the Arithmetic Mean of 3, 7, and 11.
Step-by-Step Solution:
Arithmetic Mean of [n] numbers is:
[ A.M. = \dfrac{\text{Sum of all numbers}}{n} ]
Here, the numbers are: 3, 7, 11
Number of terms: [n = 3]
Find the sum:
[3 + 7 + 11 = 21]
Divide the sum by number of terms:
[ A.M. = \dfrac{21}{3} = 7 ]
Final Answer:
The Arithmetic Mean is 7.
Example 3
Insert 3 Arithmetic Means between 2 and 18.
Step-by-Step Solution:
Number of Arithmetic Means to be inserted: [n = 3]
Total number of terms in the A.P.:
[n + 2 = 5]
First term: [a = 2]
Last term: [l = 18]
Formula for common difference:
[ d = \dfrac{l − a}{n + 1} ]
Substitute values:
[ d = \dfrac{18 − 2}{4} = \dfrac{16}{4} = 4 ]
Now write the A.P.:
[2, 6, 10, 14, 18]
Final Answer:
The three Arithmetic Means are 6, 10, 14.
Example 4
Find the Arithmetic Mean of −4 and 8.
Step-by-Step Solution:
Use the A.M. formula for two numbers:
[ A.M. = \dfrac{a + b}{2} ]
Here:
[a = −4], [b = 8]
Substitute:
[ A.M. = \dfrac{−4 + 8}{2} ]
Simplify numerator:
[−4 + 8 = 4]
Divide:
[ A.M. = \dfrac{4}{2} = 2 ]
Final Answer:
The Arithmetic Mean is 2.
Example 5
If the Arithmetic Mean of two numbers is 12 and one number is 8, find the other number.
Step-by-Step Solution:
Formula for A.M. of two numbers:
[ \dfrac{a + b}{2} = 12 ]
One number is given as 8.
Let the other number be [x].
Substitute values:
[ \dfrac{8 + x}{2} = 12 ]
Multiply both sides by 2 to remove denominator:
[8 + x = 24]
Subtract 8 from both sides:
[x = 16]
Final Answer:
The other number is 16.
7. Conceptual Questions with Solutions
1. Why is Arithmetic Mean called the middle term?
The Arithmetic Mean lies exactly halfway between two numbers. The difference between the first number and the A.M. is the same as the difference between the A.M. and the second number, which is the defining property of an A.P.
2. Can the Arithmetic Mean be equal to one of the numbers?
Yes, but only when both numbers are equal. In that case, the A.M. is equal to each number.
3. Is Arithmetic Mean always an integer?
No. The Arithmetic Mean depends on the sum of numbers. If the sum is not divisible by the number of terms, the A.M. will be a fraction or decimal.
4. Why is A.M. affected by extreme values?
Because it is based on the sum of all observations, very large or very small values significantly change the result.
5. Can negative numbers have an Arithmetic Mean?
Yes. The Arithmetic Mean can be positive, negative, or zero depending on the given numbers.
6. Is Arithmetic Mean related to A.P.?
Yes. The Arithmetic Mean of two numbers is the middle term of the Arithmetic Progression formed by them.
7. Can Arithmetic Mean be used for ratios?
No. Ratios are better represented using Geometric Mean, not Arithmetic Mean.
8. What happens if we insert many A.M.s between two numbers?
The numbers form a longer Arithmetic Progression with smaller common difference.
9. Is Arithmetic Mean the only type of mean?
No. There are also Geometric Mean and Harmonic Mean.
10. Why is A.M. important in mathematics?
Arithmetic Mean is widely used in statistics, averages, physics, economics, and real-life data analysis.
8. FAQs / Common Misconceptions
1. Arithmetic Mean is always the average.
Yes. The term “average” generally refers to the Arithmetic Mean unless stated otherwise.
2. A.M. cannot be negative.
Incorrect. If all values are negative or the sum is negative, the A.M. will also be negative.
3. A.M. always represents all data well.
Incorrect. In skewed data, Arithmetic Mean may give a misleading picture.
4. Inserting A.M.s changes the endpoints.
Incorrect. The first and last terms remain the same.
5. Arithmetic Mean must be one of the data values.
Incorrect. It may or may not belong to the given set.
6. Arithmetic Mean depends only on largest value.
Incorrect. It depends on the sum of all values.
7. Arithmetic Mean is always whole number.
False. It can be fractional.
8. Arithmetic Mean and median are same.
Incorrect. They are different measures of central tendency.
9. Arithmetic Mean applies only to A.P.
No. It applies to any numerical data.
10. Arithmetic Mean has no real-life use.
Incorrect. It is used in marks, income, speed, economics, physics, etc.
9. Practice Questions with Step-by-Step Solutions
Question 1. Find the Arithmetic Mean of 12 and 20.
Step-by-Step Solution:
The Arithmetic Mean of two numbers [a] and [b] is given by the formula:
[ A.M. = \dfrac{a + b}{2} ]
Here, the two given numbers are:
[a = 12] and [b = 20]
Substitute these values in the formula:
[ A.M. = \dfrac{12 + 20}{2} ]
Add the numbers in the numerator:
[12 + 20 = 32]
Divide the sum by 2:
[ A.M. = \dfrac{32}{2} = 16 ]
Final Answer:
The Arithmetic Mean is 16.
Question 2. Find the Arithmetic Mean of 5, 9, and 13.
Step-by-Step Solution:
When more than two numbers are given, the Arithmetic Mean is calculated using:
[ A.M. = \dfrac{\text{Sum of all numbers}}{\text{Number of numbers}} ]
The given numbers are: 5, 9, and 13
Number of terms: [n = 3]
First, find the sum of all the numbers:
[5 + 9 + 13 = 27]
Now divide the sum by the number of terms:
[ A.M. = \dfrac{27}{3} = 9 ]
Final Answer:
The Arithmetic Mean is 9.
Question 3. Insert 4 Arithmetic Means between 3 and 23.
Step-by-Step Solution:
Number of Arithmetic Means to be inserted:
[n = 4]
Total number of terms in the Arithmetic Progression will be:
[n + 2 = 6]
The first term of the A.P. is:
[a = 3]
The last term of the A.P. is:
[l = 23]
To find the common difference, we use the formula:
[ d = \dfrac{l − a}{n + 1} ]
Substitute the values:
[ d = \dfrac{23 − 3}{5} = \dfrac{20}{5} = 4 ]
Now write all the terms of the A.P. using the common difference:
[3, 7, 11, 15, 19, 23]
The numbers between the first and last terms are the Arithmetic Means.
Final Answer:
The four Arithmetic Means are 7, 11, 15, 19.
Question 4. If the Arithmetic Mean of two numbers is 15 and one number is 11, find the other number.
Step-by-Step Solution:
Let the unknown number be [x].
The formula for Arithmetic Mean of two numbers is:
[ \dfrac{a + b}{2} = \text{A.M.} ]
Substitute the given values:
[ \dfrac{11 + x}{2} = 15 ]
Multiply both sides by 2 to remove the denominator:
[11 + x = 30]
Subtract 11 from both sides to isolate [x]:
[x = 19]
Final Answer:
The other number is 19.
Question 5. Find the Arithmetic Mean of −6 and 10.
Step-by-Step Solution:
Use the Arithmetic Mean formula for two numbers:
[ A.M. = \dfrac{a + b}{2} ]
Here,
[a = −6] and [b = 10]
Substitute the values:
[ A.M. = \dfrac{−6 + 10}{2} ]
Simplify the numerator:
[−6 + 10 = 4]
Divide by 2:
[ A.M. = \dfrac{4}{2} = 2 ]
Final Answer:
The Arithmetic Mean is 2.
Question 6. Insert 2 Arithmetic Means between 4 and 16.
Step-by-Step Solution:
Number of Arithmetic Means to be inserted:
[n = 2]
Total number of terms in the A.P.:
[n + 2 = 4]
First term:
[a = 4]
Last term:
[l = 16]
Find the common difference using:
[ d = \dfrac{16 − 4}{3} = \dfrac{12}{3} = 4 ]
Write the full A.P.:
[4, 8, 12, 16]
The numbers between the first and last terms are the Arithmetic Means.
Final Answer:
The two Arithmetic Means are 8 and 12.
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