Maths Tricks to Calculate Average of Mathematical Quantities

AVERAGE

An Average or an Arithmetic Mean is the sum of 'n' different data divided by 'n'.

Average = (Sum of observations/Number of observations)

For example: if a batsman scores 40, 50,60 and 70 runs in 4 innings, then his average runs will be:
Sum of runs = 40 + 50 + 60 + 70 = 220
Total No. of Innings = 4
Average = 220/4 = 55 Runs

It means we can say batsman scores 55 runs in every inning. Concept of average is the distribution of the overall value of thing among the thing or persons present there.


 METHODS 

Average of First n natural numbers

Formula: Average = (n + 1)/2

Example: Find the average of first 5 natural numbers.
Solution: Sum of first 5 natural numbers = 1 + 2 + 3 + 4 + 5 =15
                       Total numbers = 5         
                       Average = 15/5 = 3

By formula: (5 + 1)/2 = 3

Average of First n Even & Odd numbers

• Average of first n even no. = (n + 1)

• Average of first n odd no. = n

Example: Find the average of first 5 even numbers.
Solution: Sum of first 5 even numbers = 2 + 4 + 6 + 8 + 10 = 30
                       Total numbers = 5
                       Average = 30/5 = 6

By formula: 5 + 1 = 6

Consecutive Numbers

Consecutive numbers are numbers that follow each other in order. They have a fixed difference between every two numbers. For example: 2, 4, 6, 8, 10 are consecutive.

Formula: Average = {(First number + Last number)/2}

Increment of members

If there are some persons whose average of certain quantity is ‘a’, then after including some new members ‘n’ average becomes ‘b’, then sum of that quantity of new entrant will be given as:

In case of Age:-
 1.  If avg. increased: Age of new entrant = |n x New average + No. of old members x (change in average)|

 2.  If avg. decreased: Age of new entrant = |n x New average - No. of old members x (change in average)|

NOTE: Above formulas are applicable on other quantities like weight, height, marks etc. too.

Example: The average age of 30 boys of a class is equal to 14 years. When the age of the principal is included the average becomes 15 years. Find the age of principal.
Solution: Age of principal = 15 + 30 x (15 - 14) = 45 years

Decrement of members

If there are some persons whose average of certain quantity is ‘a’, then after excluding some members ‘n’ average becomes ‘b’, then sum of that quality of removed persons will be given as:

In case of marks:-
 1.  If avg. increased: Marks of removed candidates = |n x New avg. – old number of candidates x Change in avg.|

 2.  If avg. decreased: Marks of removed candidates = |n x New avg. + Old number of candidates x Change in avg.|

NOTE: Above formulas are applicable on other quantities like weight, height, marks etc. too.

REPLACEMENT

If there are 'n' persons whose average of weight is ‘A’, when some members got replaced with new members then average becomes ‘B’, then weight of new members will be given as:

 1.  If avg. increased: Weight of new persons = |Weight of removed persons + No. of persons x change in avg.|

 2.  If avg. decreased: Weight of new persons = |Weight of removed persons - No. of persons x change in avg.|

NOTE: Above formulas are applicable on other quantities like weight, height, marks etc. too.

Increment, Decrement or Replacement

We can use this single method because remembering above formulas may create confusion for some people.

Example: Average weight of 25 students of a class is 50 kg. If the weight of the class teacher is included, the average is increased by 1 kg. The weight of the teacher is?
Solution:  Step 1:  First write total students and average:

 Step 2:  Now write new number of students and new average:

 Step 3:  Now write the difference of number of students and average:

 Step 4:  Multiply original number with difference of average and multiply difference of number of students with new average. Now add or subtract as per the sign.

Weight of the teacher = 76 kg

Multiple Formulas

 1.  Average of 1 to n odd numbers = (Last odd no. + 1)/2

 2.  Average of 1 to n even numbers = (Last even no. + 2)/2

 3.  Average of squares of first 'n' natural numbers = (n + 1)(2n + 1)/6

 4.  Average of cubes of first 'n' natural numbers = n(n + 1)2/4

 5.  Average of 1 to n multiples of any numbers = Number x (n + 1)/2

 6.  If average of 'n' numbers is ‘x’ and each number is multiplied by ‘y’, then new avg. = xy

These are few mathematical formulas to calculate average quickly. It is possible to calculate average using basic method but that waste a lot of time. So its better to adopt these methods. Good Luck 👍