Average

Definition
Simple Average (or Mean) is defined as the ratio of sum of the
quantities to the number of quantities.
Sum of all quantities
Average =
No. of quantities
Putting in symbols,
x1 + x 2 + x 3 + - - - - - - - - - - - + x n
X =
N
Here x1, x2, x3, ----------- xn represent the n values of
quantities under consideration & x is their mean.
Note:
Average or mean is said to be a measure of central tendency.

Example
Find the average of 53, 57, 63 and 71.
Solution:
53 + 57 + 63 + 71 244
Average = = = 61
4 4

Weighted Average or
Mean:
If some body asks you to calculate the combined
average marks of both the sections of class X- A and X- B,
when both sections have 60% and 70% average marks
respectively? Then your answer will be 65% but this is wrong
as you do not know the total number of students in each
sections. So to calculate weighted average, we must have to
know the number of students in both the sections.
Let N1, N2, N3, …. Nn be the weights attached to variable
values X1, X2, X3, …….. Xn respectively.
Then the weighted arithmetic mean is given by
N1X1 + N2 X2 + N3 X3 + ..... + Nn Xn
X =
N1 + N2 + N3 + ..... + Nn

Example
The average marks of 30 students in a section of class X are 20
while that of 20 students of second section is 30. Find the average
marks for the entire class X.
Solution
We can do the question by using both the Simple average &
weighted average method.
Sum of marks of all students 20 × 30 + 30 × 20
Simple average = =
Total number of students 30 + 20
= 24
3 2
By the weighted mean method, Average = × 20 + × 30
5 5
= 12 + 12 = 24.

Example
The average of 11 results is 50. If the average of first 6 results is
49 and that of the last 6 is 52, find the 6th result.
(1) 48 (2) 50 (3) 60 (4) 56
Solution:
The sum of 11 results = 11 × 50 = 550
The sum of the first 6 results = 49 × 6 = 294
The sum of the last 6 results = 52 × 6 = 312
So the 6th result is = (294 + 312) – 550 = 56 Answer (4)

Real Facts A bout
•
Average
If each number is increased / decreased by a certain quantity
n, then the mean also increases or decreases by the same
quantity.
Example:
If the average of x, y, and z is k. Then the average of x+2,
y+2, and z+2 is k+2.

Real Facts A bout
•
Average
If each number is multiplied/ divided by a certain quantity n,
then the mean also gets multiplied or divided by the same
quantity.
Example:
If the average of x, y, z is k, then average of 3x, 3y, 3z will be
3k

Real Facts A bout
•
Average of the quantities and
If the same value is added to half
same value is subtracted from other half quantities then
there will not be any change in the final value of the average.
Example:
If the average of a, b, c, d is k, then average of a + x, b + x,
c – x, d – x is also k.

Real Facts A bout
• Average
The average of the numbers which are in arithmetic
progression is the middle number or the average of the first and
last numbers.
Example:
The average of 10 consecutive numbers starting from 21 is
the average of 5th & 6th number, i.e. average of 25 and 26
which is 25.5
Example:
4 + 34
The average of 4, 7, 10, ……., 34 = = 19
2

Example:
There are 30 consecutive numbers. What is the difference
between the averages of first 10 and the last 10 numbers?
Solution:
The average of first 10 numbers is the average of 5th & 6th
numbers. Where as the average of last 10 numbers is the average
of 25th & 26th numbers. Since all are consecutive numbers,
25th number is 20 more than 5th number. We can say that
the average of last 10 numbers is 20 more than the average
of first 10 numbers. So, the required answer is 20.

Average Speed
Total dis tan ce cov ered
Average Speed =
Total time taken
If d1 & d2 are the distances covered at speeds v1 & v2 respectively
and the time taken are t1 & t2 respectively, then the average speed
over the entire distance (d1 + d2) is given by
Total dis tan ce cov ered d +d2 d +d2
= 1 = 1
Total time taken t 1 +t 2 d1 d
+ 2
v1 v2
TIP:
Average speed can never be double or more than double of
any of the two speeds.

Average
#
Speed
If both the distances are equal i.e. d = d2 = d then,
1
2 v 1v 2
Average speed =
v1 + v 2
# If both the time taken are equal i.e. t1 = t2 = t then
v1 + v2
Average speed =
2

Example:
A man travels at a speed of 60 kmph on a journey from A to B
and returns at 100 kmph. Find his average speed for the
journey (in kmph).
(1) 80 (2) 72
(3) 75 (4) None of these
Solution:
2 × 60 × 100
Average speed = = 75 Answer (3)
60 + 100

Thank you !!