Average is one of the most important topics for RRB NTPC and Other Bank/Govt. Exams. Though there are some questions in any exam that are directly based on the topic – average, it also finds application Data Interpretation and other Topics too. This makes this topic as one of the most important topics for any exam. Through this article let’s try and learn some very useful tricks for average, which will make it easier and faster to calculate average in any type of question.
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1. Introduction to the Concept of Average
Average: Average is the sum of different terms (data) divided by the total number of terms.
Average = Sum of given terms)/(Total number of terms)
For example: Find the average of given terms 12, 16, 17, 19, 21.
Solution:
Total number of terms = 5
Average = (12 + 16 + 17 + 19 + 21)/5 = 17
1.1. Some Basic Formulae
The list given below, states the basic formulae for the concept of Average. Let us take a look at them before we learn some awesome tricks for average.
(1) Average of first ‘n’ natural number = (n + 1)/2
(2) Average of first ‘n’ even numbers = (n + 1)
(3) Average of first ‘n’ odd numbers = n
(4) Average of ‘n’ consecutive natural numbers = (First number + Last number)/2
(5) Average of squares of first ‘n’ natural numbers = (n + 1)(2n + 1)/6
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2. Some Important Tips and Tricks for Average
Here are some handy tricks for Average which will make your calculation faster and efficient with practice:
(1) If the value of each number is increased by the same value ‘a’, then the average of all numbers will also increase by ‘a’.
(2) If the value of each number is decreased by the same value ‘a’, then the average of all numbers will also decrease by ‘a’.
(3) If the value of each number is multiplied by the same value ‘a’, then the average of all numbers will also get multiplied by ‘a’.
(4) If the value of each number is divided by the same value ‘a’, then the average of all numbers will also get divided by ‘a’.
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3. A few more Useful Shortcuts and Tricks for Average
(1) If the average of ‘x’ numbers is ‘a’ and the average of ‘y’ numbers is ‘b’, then the average of (x + y) numbers = (xa + yb)/(x + y).
Example:
The average of 18 numbers is 24 and the average of 7 numbers is 21, then what will be the average of all 25 numbers?
Solution:
Average = (18 * 24 + 17 * 21)/(18 + 7) = 789/25 = 31.56
(2) If the average of ‘n’ quantities is equal to ‘x’, when a quantity is removed the average becomes ‘y’. Then the value of removed quantity = [n(x – y) + y].
Example:
The average age of 25 students is equal to 15 years. The average age of 24 students becomes 14 years, if one of the students leave, Find the age of the student who left the group?
Solution:
Age of the left student = [25(15 – 14) + 14] = 39
(3) If the average of marks obtained by ‘n’ students in an exam is ‘m’. If the average marks of passed students is ‘p’ and total of failed students is ‘f’. Then the number of students who failed in the exam is equal to [n(p – m)/(p – f)].
Example: The average marks obtained by 125 students in an exam is 29. If the average marks of passed students is 36 and that of failed students is 11. What is the number of failed students?
Solution:
No. of failed students = 125(36 – 29)/(36 – 11) = 125 * 7/25 = 35
(4) If a batsman in his nth innings makes a score of ‘s’ and thereby increased his average by ‘t’, then the average after ‘n’ innings is equal to [s – t(n-1)].
Example: A batsman in his 44th innings makes a score of 86 and thereby increases his average by 1, Find the average after 44 innings?
Solution:
Average after 44th innings = [86 – 1 * (44 – 1)] = 86 – 43 = 43
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