This document provides examples and explanations of calculating averages. It defines average as the total of quantities divided by the number of quantities. Various methods for finding averages are presented, including using the standard formula, accounting for excess or deficiency, and handling arithmetic progressions. Examples include finding averages of data sets with different numbers of items and values, as well as word problems involving changes to averages when new data is included or removed.

1. Averages
Dr. S. Murali
Professor & Head
Dept. of Mechanical Engineering
1

2. Definition
• The average of any set of quantities
can be found by adding all the
numbers together then dividing by
the number of quantities in the set
• average = total of quantities /
number of quantities
2

3. Math Formula
Formula
Sum of Observations
Number of Observations
3

4. Initial Avg.
Final Avg.
In a Crude way- Simple
Final Avg. =Initial + Diff.
4

5. In a Crude way- Simple
Excess
Final Avg.
Deficiency
Excess loss = Deficiency gain
5

6. 1. Find the average of 4,6,9 and 11
Sol:
given numbers are small
apply conventional formula
Sum of the numbers is = 30
Average
= Sum/No. of Numbers
=30/4
=7.5
6

7. 2. Find the average of 146,152, 149,
157 and 162
Sol:
Numbers are larger: avoid using conventional method
Method 1
146
152
149
162
157
assume average 146 and excess available at other numbers
are
Sum
0
+6
+3
+16
11
36
This excess is shared among 6 numbers =36/5=7.2
Average of 6numbers is =Assumed avg. + avg. of excess
146+7.2=153.2
7

8. 146,152, 149, 157 and 162
Method 2
arranging the numbers in ascending order
146
149
152
157
162
assume average 152 and find excess/less available at other
numbers are
-6
-3
0
5
10
Sum
This excess is shared among 6 numbers =6/5=1.2
6
Average of 5 numbers is =Assumed avg. + avg. of excess
152+1.2=153.2
8

9. 3. Find the average of 16,24,32,40
and 48
Sol:
8
16
8
8
24
32
8
40
48
Series in Arithmetic Progression with
odd numbers(5)
Hence middle number 32 is the average
9

10. 4. Find the average of 16,24,32,40,48
and 56
Sol:
8
16
8
8
24
32
8
40
8
48
56
Series in Arithmetic Progression with
even numbers(6)
Average of middle numbers is the
average
=(32+40)/2=36
10

11. 5.The avg. of 7 consecutive numbers
is 33. the largest of these numbers is?
Sol:
7 consecutive numbers
X-3
X-2
X-1
X
X+1
X+2
X+3
X
X+2
X+4
X+6
X
X+1
X+3
X+5
7 consecutive even numbers
X-6
X-4
X-2
7 consecutive odd numbers
X-5
X-3
X-1
All these combinations are in AP and as 7 being odd number, the
middle number equals to the average
The largest of numbers is
x=33+3=36
11

12. 6. The avg. of 30 students in a class is 12 years. The avg. age
of a group of 5 of the students is 10 years and that of another
group of 5 of them is 14 years. What is the avg. of the
remaining students?
10
14
10
12
20
10
5
5
Excess of 5 students =less required by
5 students
No change in remaining average, hence
average of remaining is 12
12

13. 7. The average of eight numbers is 14. the
average of six of these numbers is 16.What
is the average of remaining two numbers.
16
14
8
Gain by 6 = loss by 2
12
= x 2
Average loss=6
Average of remaining two = 14-6=8
13

14. 8. If the average mark of a student in four
examinations is 40. If he got 80 marks in 5th
Examination. What is the New average ?
Sol:
40
80-40=40
80
Assume Avg. of 5 books is 40
Excess available at 5th book is 40
Increase in average =40/5=8
New average
=40+8 =48
14

15. 9. If the average mark of a student in four
examinations is 40. If he got 20 marks in 5th
Examination. What is the New average ?
Sol:
40
20-40=-20
20
Assume Avg. of 5 books is 40
less shared by 5th book is 20
Decrease in average =-20/5=-4
New average
=40-4 =36
15

16. 10. If the avg. age of 30 boys in a class is 15 yrs.
When the age of class teacher is included the avg.
is 16 years. What is the age of the class teacher?
Sol:
16
X
15
15
Un contributed age of teacher =Avg. of students= 15
Avg increase due to teacher addition is
=New avg – initial avg = 16-15=1
Distributed age of teacher =X=1×31=31
Teacher age=assumed average + surplus
=15+X=15+31=46
16

17. 11. The average of 50 numbers is 38. What
is the average of the numbers, if two
numbers namely 45 and 55 are discarded.
Excess at 45 and 55
is =7+17=24
55
38
45
7
17
Less available at 48 numbers is
=24/48=0.5
The average of 48 numbers is
=38-0.5 =37.5
17

18. 12. In a group the Avg. income of 6 men is
Rs. 500 And that of 5 women is Rs. 280.
What is the average income of the group?
Sol:
Assume average of 11 members
as 280
500-280=220
280
Then excess average available
with 6 men=500-280=220
Total excess income available with
6 men=6×220
500
280
6
5
Increase in average of 11
members =6×220/11=120
New average =280+ 120=400
18

19. 13. A class has 2 sections, in one of which there are 40
students with an average of 14.5 years. The average of the
class is 14.2 years . If there are be 32 students in the other
section, its average is ?
14.5
Excess available at 40 students
=0.3 × 40=12
Increase in average 32 students
=12/32=0.375
Average of 32 students
x = 14.2-0.375
x = 13.825
14.2
0.375
x
40
32
19

20. 14. The average of 11 observations is 60. if the
average of first five observations is 58 and that of
last five is 56, then the sixth observations is
Excess
60 /11
56 /5
Excess supplied by sixth observation
to first five and last five is
= 4*5+2*5 =30
Sixth observation=60 + 30=90
20

21. 15. Mukhesh has twice money as sohan and sohan
has 50% more money than pankaj has. If the
average money of with them is Rs. 110, then
muksh has
3x
1.5x
Total money available
110
x
=110*3
3x+1.5X+x =110*3
5.5 x
= 110 *3
x
= 110/5.5*3
x
=60
Mukhesh money= 1.5 x 60=90.
21

22. 16. Three years ago, the average age of a family of 5
members is 17 years. A baby having been born. the average
age of the family is the same today. What is the present age
of the baby
20
Excess
17
15
2
Excess supplied 5 family
members to the baby is = 3*5 =15
Age of the baby is=17 - 15=2
22

23. Our head weights have been Averaged ?
Thank you
Contact
smuralichinna@gmail.com
23

24. Puzzle 1. A series from 1-9 is arranged in the boxes
as shown such that the sum along vertical and
horizontal direction is 13. find the number at G
block?
Sol:
1 2 34 5 67 8 9
A
B
C
D
E
F
G
H
I
24

25. Thank you for patience
Contact
smuralichinna@gmail.com
25