# Average and Ages Formulas

An average is defined as the sum of different data (terms) divided by the total number of terms.

Average = Sum of given terms(S) / Total number of terms(N)

Example: Find the average given terms 2,3,4,5,6

Sol:          Total number of terms = 5

Average = (2+3+4+5+6) / 5

= 4

Some basic average formulas

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

2. Average of 'n' even numbers = (n+1)

3. Average of 'n' odd numbers = n

4. Average of 'n' consecutive natural numbers = (First number + last number) / 2

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

Some Important points:

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 the 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 the numbers will also get divided by 'a'.

Some useful formulas:

1. If the average of 'x' numbers is a and that of 'y' numbers is b, then the average of (x+y) numbers = (xa + yb) / (x +y)
Example: The average of 10 numbers is 15 and that of 15 numbers is 20. Find the average of all 25 numbers?

Sol: Average = (10*15+15*20) / (10+15) = (150 + 300) / 25 = 18

2. If the average of 'n' quantities is equal to 'x' when a quantity is removed the average becomes 'y'. Then the value of the removed quantity is = [n(x-y)+y]

Example: The average of 24 men and 1 woman is equal to 35 years. If 1 woman left, the average becomes 34 years. Find the age of women who left the class?

Sol: Age of women = [25 (35 - 34) + 34] = 59 years

3. If the average of marks obtained by 'n students in an exam is 'M'. If the average marks of passed students in 'P' and that of failed students is 'F'. Then the number of students who failed in exam is = n(P - M) / P - F

No. of failed students = (Total students( Passed average - Total average) / Passed average - FAiled average

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. Find the numbers of failed students?

Sol: No. of failed students = 125 (36 - 29) / 36-11 = 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 = s -t(n - 1)

Example: A batsman in his 44th innings makes a score of 86 and thereby increase his average by 1. Find the average after 44 innings?

Sol: Average after 44th innings = (86 - 1(44 - 1)) = 86 - 43 = 43

Now as you have known the basics about this chapter, lets solve some Problems on Average and Ages

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