Mathematical problems on average for school mathematics and competative examination/all over world- India USA and other countries
Math for students and competative examination on Average topics
An average or an arithmetic mean of given data is the sum of the given observations divided by number of observations.
Average =(sum of given observations)÷(number of observations)
Important formulae
1. Average of first n natural numbers =(n+1)/2
2. What is the first n even numbers= n+1
3. Average of first n odd numbers= n
4. Average of consecutive numbers= ( first number + last number)/2
5. Average of 1 to n odd numbers = ( last odd number+1)/2
6. Average of 1 to n even numbers= ( last even numbers+2)/2
7. Average of squares of first n natural numbers = (n+1)(2n+1)/6
8. Average of the cubes of first n natural numbers= n(n+1)²/4
9. Average of n multiples of any number= Number x (n+1)/2
Short tricks:
1. If the average of n1 , observations is a1,, the average of n₂ observations is a2, and so on, then average of all the observations = (n1a1+ n2a₂+...)/( n₁+ n₂+...)
2. If the average of m observations is a and the average of n observations taken out of m is b, then
Average of rest of the observations=
(ma-nb)/(m-n)
3. If average of n observations is a but the average becomes b,when one observation is eliminated, then value of eliminated observation
= n(a - b) + b
4. If average of n observations is a but the average becomes b, when a new observation is added, then value of added observation
= n(b - a) + b
5. If the average of n students in a class is a, where average of passed students is x and average of failed students is y, then
Number of students passed
= [Total students (Total average- Average of those failed)] ÷(Average of those passed- Average of those failed)
= n(a - y)÷(x-y)
6. If we have 'n' observations out of which some observations (a1, a2, a3,...) are replaced by some other new observations and in this way, if the average, increases or decreases by 'b' then, value of new observations = a±nb where,
a = a1 + a2+ a3+ .......
7. If the average of total components in a group is a ,where average of remaining components (IInd part) is 'c', then Number of remaining components (IInd part) = n(a - b)÷( c-a)
Some problems:
1. The arithmeating mean of the following numbers 1, 2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7 is ? (SSC CGL 2014)
Ans: Arithmetic mean=(1+2x2+3x3+4x4+5x5+6x6+7x7)/(1+2+3+4+5+6+7)
= 5
2. After replacing and old member by a new member, it was found that the average age of 5 members of a club is the same as it was 3 years ago. The difference between the ages of the replaced and the new member is=?
Ans: let the average age of 5 person be 10 years
Then, sum of their age = 5 x10 = 50 yr
Average age of 5 members 3 years ago= (50-3x5)/5 = 7 yr
So, after adding the new member average age become 7 years.
Difference in the age of old member and new member= 5x10-7x5 =50-35= 15 yr
3. The average age of first 25 member is 10 and the average of the next 25 member is 12. Find the average age of all 50 members?
Ans: average age of first 25 members is 10 and then next 25 is 12.
By short tricks
Overall average (25x10+25x12)/(25+25)
= 550/50 = 11
4. The average of 30 mumbers is 40 and that of other 40 numbers is 30. The average of all the numbers is ?
Ans: 
5. If the average of m numbers is n², and that of n numbers is m², then the average of (m+n) numbers is =?
Ans: Given average of m numbers=n²
Therefore sum of the m numbers =n² x m =mn²
Again sum of n numbers= m²n
Now average of all numbers = (mn²+nm²)/(m+n) = mn(m+n)/(m+n) = mn
6. The average of 9 numbers is 50. The average of first five numbers is 54 and that of the last three numbers is 52. Then the sixth number is=?
Ans: average of 9 number is 50
Therefore some of 9 numbers= 50 x 9= 450
Now Everest of first five numbers=54
Therefore some of first five numbers = 54x5 = 270
Similarly some of last three number= 52 x 3= 156
Hence the 6th number is= 450-(270+156)
= 450-426 = 24
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