Mixture & Alligation



Mixing of two or more than two type of quantities gives us a mixure.

Quantities of these elements can be expressed as percentage or ratio.
i.e. Percentage (20% of sugar in water)
Fraction ( A solution of sugar and water such that sugar : water = 1:4)


Alligation is a rule which is used to solve the problems related to mixture and its ingredient.

It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price.


When two elements are mixed to make a mixture and one of the elements is cheaper and other one is costlier then,

Here Mean Price is CP of mixture per unit quantity.
Above rule can be written as,

Cheaper Quantity : Costlier Quantity = (D – M) : (M – C)

Mean Price

The cost price of a unit quantity of the mixture is called the mean price.

Mixture of more than two elements

This method is a bit tricky initially but if you practice it then it becomes quite easy.

If the mixture is of more than two ingredients, then write the prices of each ingredient below one another in ascending order. Write the mean price to the left of the list. Now make couples of prices in such a way that one price of the couple is below mean price and another price of the couple is above mean price. Now find the difference between each price and mean price and write it opposite to the price linked to it. This difference is required answer.

Shortcut Formulas

Rule 1

Rule 2

Rule 3
p gram of ingredient solution has a% ingredient in it. To increase the ingredient content to b% in the solution

Given below are few questions to apply the concepts learnt above:

1.A 20 litre mixture of milk and water contains milk and water in the ratio 3 : 2. 10 litres of the mixture is removed and replaced with pure milk and the operation is repeated once more. At the end of the two removal and replacement, what is the ratio of milk and water in the resultant mixture?

(1)  17 : 3       (2)  9 : 1        (3)  3 : 17      (4)  5 : 3

2.A merchant mixes three varieties of rice costing Rs.20/kg, Rs.24/kg and Rs.30/kg

and sells the mixture at a profit of 20% at Rs.30 / kg. How many kgs of the second

variety will be in the mixture if 2 kgs of the third variety is there in the mixture?

(1) 1 kg (2) 5 kgs (3) 3 kgs (4) 6 kgs

3.How many litres of a 12 litre mixture containing milk and water in the ratio of 2: 3 be replaced with pure milk so that the resultant mixture contains milk and water in equal proportion?

(1)4 litres (2) 2 litres  (3) 1 litre (4) 1.5 litres

4 A sample of x litres from a container having a 60 litre mixture of milk and water containing milk and water in the ratio of 2 : 3 is replaced with pure milk so that the container will have milk and water in equal proportions. What is the value of x?

(1)6 litres (2) 10 litres (3) 30 litres (4) None of these

5.A zookeeper counted the heads of the animals in a zoo and found it to be 80. When he counted the legs of the animals he found it to be 260. If the zoo had either pigeons or horses, how many horses were there in the zoo?

(1)40  (2) 30  (3) 50  (4) 60



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