The document discusses the concept of average and different types of averages. It defines average as the sum of all quantities divided by the number of quantities. It also discusses weighted average, which is used when quantities have different weights. The properties and concepts of mixtures, including mixing with and without replacement, are explained through examples. Alligation, which is a type of weighted average used in mixtures, is also introduced.

2. AVERAGE
Average is sum of all the quantities divided by number of quantities
푨풗풆풓풂품풆 =
푺풖풎 풐풇 풒풖풂풏풕풊풕풊풆풔/풏풖풎풃풆풓풔
푵풖풎풃풆풓 풐풇 풒풖풂풏풕풊풕풊풆풔/풏풖풎풃풆풓풔
Example :
Average of 18,20,24,26
퐴푣푒푟푎푔푒 =
18 + 20 + 24 + 26
4
= 22

3. AVERAGE = CENTRAL VALUE
Find the average of 12, 14,22,28,42,44
Now let us assume the central value of above numbers to be 28
Deviations : -16, -14, -6 , 0 , 14 , 16
Net deviation =
−16−14−6+0+14+16
6
= −1
Hence Average = Central Value + Deviation = 28 – 1 = 27

4. PROPERTIES OF AVERAGE
• Average always lies in between the maximum and the minimum value. It can
be equal to the maximum or minimum value if all the numbers are equal.
• Average is the resultant of net surplus and net deficit, as used in the central
tendency method. When weights of different quantities are same, then simple
method is used to find the average. However, when different weights of
different quantities are taken, then it is known as weighted average. Here the
method of weighted average is used to find the average.
• If the value of each quantity is increased or decreased by the same value S,
then the average will also increase or decrease respectively by S.
• If the value of each quantity is multiplied by the same value S, then the
average will also be multiplied by S.
• If the value of each quantity is divided by the same value S (S ≠ 0) then the
average will also be divided by S.

5. WEIGHTED AVERAGE
Simple average can be calculated if weight of each quantity is equal. For
example :
• A class with 20 students average age 16 , another class with 30 students
average age 6 are combined to form a single class. What will be the average
age of the students in the new class.
• Let n1= 20 m1 = 16
• Let n2= 30 m2 = 6
• Average m =
푚1푛1+푚2푛2
푛1+푛2
=
20푋16 +30푋6
20+30
= 8 푦푒푎푟푠
• If you’d have applied (16+6)/2 it would have resulted in 11 years which is
wrong

6. ALLIGATION- WEIGHTED AVERAGE

7. MIXTURE
Mixture without replacement
• In this particular type of mixing, two or more than two substances are mixed
without any part of any mixture being replaced.
How many litres of fresh water should be mixed with 30 litres of 50% milk solution
so that resultant solution is a 10% milk solution?
So, the ratio of fresh water added: milk solution = 4:1
Hence, 120 litres of fresh water should be added.

8. MIXTURE
Mixing with replacement
• Case 1 -> Quantity withdrawn = Quantity replaced
If V is the initial volume of milk (or any liquid), and x litres of milk is always replaced by
water, then quantity of milk left after n such operations =
• Case 2 -> quantity withdrawn and the quantity replaced are of the same volume, but the
total volume before replacement does not remain the same.
• Initially, there are 40 litres of milk, and 4 litres of milk is taken out and 4 litres of water is
poured in
• So, there will be 36 litres of milk and 4 litres of water.
• Now, 5 litres of mixture is taken out and 5 litres of water is poured in.
• The quantity of milk and water being withdrawn here will be in the ratio of 36:4. So, the
quantity of milk withdrawn = (36/40) X 5
Milk left = 40 X (36/40) X (35/40)
• Again, if now 6 litres of mixture is taken out and 6 litres of water is poured in
Milk left = 40 X (36/40) X (35/40) X (34/40)

9. MIXTURE
Mixing with replacemment
• quantity withdrawn and the quantity replaced are not of the same volume.
• Initially there are 40 litres of milk, and 4 litres of milk is taken out and 5 litres
of water is poured in
Obviously, there will be 36 litres of milk and 5 litres of water.
• Now, 5 litres of mixture is taken out and 6 litres of water is poured in then the
quantity of milk and water being withdrawn will be in the ratio of 36:5. So,
the quantity of milk withdrawn = (36/41) X 5
Milk left = 40 X (36/40) X (36/41)
• Again 6 litres of mixture is taken out and 7 litres of water is poured in.
• Thus, the volume of milk in the final mixture
= 40 X (36/40) X (36/41) X (36/42)

10. Thank You
Study further http://prepvelvet.com/alligation and solve the quiz.