Average
The average of the number of quantities of observations of the same kind is their sum divided by their number. The average is also called average value or mean value or arithmetic mean.
Average = 
For observations x1 ,x2, x3, ……………xn
Average = 
Ex: The average of 6 consecutive even number is 21. Find the largest number?
Largest no. = A + (n−1)
A = average
n = no. of terms
Largest no. = 21 + ( 6 -1) = 26
Ex: The average of 6 consecutive odd number is 22. Find the smallest number?
Smallest no. = A - (n - 1)
A = average
n = no. of terms
Smallest no. = 22 - ( 6 - 1)
= 17
Ex: The average of 5 consecutive even number is 46. Find the smallest number?
Smallest no. = A - (n - 1)
A = average
n = no. of terms
Smallest no. = 46 -( 5 -1)
= 42
IMPORTANT FACTS
The average of first ‘n’ natural numbers = 
The average of first ‘n’ even numbers = ( n+1)
The average of first ‘n’ odd numbers = n
The average of square of first ‘n’ natural numbers = 
Ex: Find the average of first 100 natural numbers.?
Sol: Average = 
= 
= 50.5
Ex: The average of 5 numbers is 29. If one number is excluded, the average becomes 27.Find the excluded number?
Sol:
Excluded no. = ( 5 x 29 - 4 x 27 )
= 
= 37
Ex: The average age of 36 students is 15 years. When teacher’s age is included to it, the average increased by 1.What is the teacher’s age?
Sol:
Teacher’s age = ( 37 x 16 - 36 x 15 )
= 
= 52
Ex: The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing 40 kg . What is the weight of new person?
Sol:
Total weight increased = 8 x 2.5
= 20 kg
weight of the new person = 40 + 20
= 60 kg
Ex: The average weight of 10 persons decreases by 2.5 kg when a new person comes in place of one of them weighing 70 kg . What is the weight of new person?
Sol:
Total weight decreased = 10 x 2.5
= 25 kg Weight of the new person = 70 - 25
= 45 kg
Ex: A batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3 runs. Find his average after 17th inning.
Sol:
Let the average after 17th inning = X
and average after 16th inning = (X - 3)
16(X - 3) + 87 = 17X
16X - 48 +87 = 17X
X = 39
Ex: The average of 11 results is 60. If the average of first 6 results is 58 and that of last 6 results is 63. Find the 6th result?
Sol:
A11 = 60
Average of first 6 (A6) = 58
Average of last 6 (A6) = 63
6th result = (58 × 6 + 63 × 6 - 60 × 11)
= (348+378) - 660
= 726 -660
= 66
Ex: The average of a, 11,23 and 17 is 15 and the average of a,b,12 and 25 is 16. Find the value of a : b?
Sol:
a + 11 + 23 +17 = 15 x 4
a = 9
a + b + 12 + 25 = 16 x 4
a + b = 27
9 + b = 27
b = 18
a : b = 9 : 18
= 1 : 2
Ex: The average age of all the 100 employees in an office is 29 years , where 2/5 employees are female and the ratio of average age of male to female is 5 : 7. Find the average age of female employees?
Sol:
60 
5x + 40 
7x = 29 
100
5x + 40 7x = 29 100
300x + 280x = 2900
x = 5
average age of female employees
= 7x
= 7 x 5
= 35 years
Ex: The average of two numbers A & B is 20, an average of B & C is 19 and average of C & A is 21,So find the value of A?
Sol% A + B = 40
B + C = 40
C + A = 42
On adding above three
2(A + B + C)=40 + 38 + 42= 120
= A + B + C = 60
A = (A + B + C) – (B + C)
= 60 – 38= 22
Ex: Three maths classes X, Y and Z, take an algebra test. The average score of class X is 83. The average score of class Y is 76. The average score of class Z is 85. The average score of class X and Y is 79 and average score of class Y and Z is 81. What is the average score of classes X, Y and Z?
Sol: Let the number is student in classes X, Y and Z be a, b and c respectively then total score of
X=83a, Y = 76b and Z =85c and
a : b : c = 3 : 4 : 5
Average score of X, Y, Z
Ex: Three years ago, the average age of a family of 5 members was 17 years. A baby having been born, the average age of the family is the same today. The present age of the baby is:
Sol: Total age of 5 members, 3 years ago
=(17×5) = 85 years
Total age of 5 members, now =[85+(3×5)]
= 85 +15 = 100 years
Total age of 6 members now = (17×6)
= 102 years
The age of the baby = (102-100)
= 2 years.
Ex: The average temperature of a town in the first four days of a month was 58 degrees. The average for the second, third, fourth and fifth days was 60 degrees. The temperature of the first and fifth days was in the ratio 7:8, then what is the temperature on the fifth day?
Sol: First four days average Temperature =580
1,2,3, 4th days total temp. = 58×4 = 232
Then 2,3,4,5 days total temp. = 60×4 = 240
Let the unknown temp, be x
5th day – 1st day = 240-232=8(2,3,4 days temp. is common)
Given the ratio of first and fifth day is 7 : 8
8x-7x=8
x=8
Fifth day's temperature = 8x=8x8=64
Ex: There were 35 students in a hostel. If the number of students is increased by 7 the expenditure on food increased by Rs. 42 per day while the average expenditure of students is reduced by Re.1. What was the initial expenditure on food per day?
Sol: Suppose the initial expenditure per day = Rs.x
x = 210+210 = Rs. 420
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MAHENDRA GURU