Average

Simple Average (or Mean) is defined as the ratio of sum of thequantities to the number of quantities.

Putting in symbols,

Here x1, x2, x3, ----------- xn represent the n values of

quantities under consideration & is their mean. x

= 3.

Definition

quantitiesofNo

quantitiesallofSum

.Average =

X = N

x----------- x x x n321

Note:Average or mean is said to be a measure of central tendency.

Example

Find the average of 53, 57, 63 and 71.

Solution:

Average = = 614

2444

71635753

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

X =

n321

nn332211

N.....NNN

XN.....XNXNXN

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.

SolutionWe can do the question by using both the Simple average & weighted average method.

Simple average = =

= 24

By the weighted mean method, Average = 20 + 30

= 12 + 12 = 24.

studentsofnumberTotal

studentsallofmarksofSum

2030

20303020

5

3

5

2

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 = 550The sum of the first 6 results = 49 × 6 = 294The sum of the last 6 results = 52 × 6 = 312So the 6th result is = (294 + 312) – 550 = 56 Answer (4)

• 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 About 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 About Average

• If the same value is added to half of the quantities and 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 About 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:

The average of 4, 7, 10, ……., 34 = = 192344

Real Facts About Average

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

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

takentimeTotal

eredcovcetandisTotalAverage Speed =

takentimeTotal

eredcovcetandisTotal

21

21

tt

dd

2

2

1

1

21

v

d

v

ddd

TIP:

Average speed can never be double or more than double of any of the two speeds.

# If both the time taken are equal i.e. t1 = t2 = t then

Average Speed

Average speed 2vv 21

# If both the distances are equal i.e. d1 = d2 = d then,

Average speed21

21

vv

vv2

10060100602

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:

Average speed = = 75 Answer (3)

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