Example.1: A sum of Rs 39 was divided among 45 boys and girls. Each girl gets 50 paise, whereas a boy gets one rupee. Find the number of boys and girls.
 Average amount of money received by each =
 Amount received by each girl = 50 paise = Rs
 Amount received by each boy = Re. 1
Number of girls = 45 – 33 = 12.
Important Funda
 Always identify the ingredients as cheaper & dearer to apply the alligation rule.
In the alligation rule, the variables c, d & m may be expressed in terms of percentages (e.g. A 20% mixture of salt in water), fractions (e.g. twofifth of the solution contains salt) or proportions (e.g. A solution of milk and water is such that milk : Water = 2 : 3). The important point is to remember is that c & d may represent pure ingredients or mixtures.
Example.2: A jar contains a mixture of two liquids A and B in the ratio 4 : 1. When 10 litres of the liquid B is poured into the jar, the ratio becomes 2: 3. How many litres of liquid A were contained in the jar?
Method 1: (weighted average or equation method)
 Let the quantities of A & B in the original mixture be 4x and x litres.
 According to the question
12x = 2x + 20
⇒ 10x = 20
⇒ x = 2 The quantity of A in the original mixture = 4x = 4 × 2 = 8 litres.
Method 2: (Alligation with composition of B)
 The average composition of B in the first mixture is 1/5.
 The average composition of B in the second mixture = 1
 The average composition of B in the resultant mixture = 3/5
 Hence applying the rule of Alligation we have
[1 – (3/5)]/[(3/5) – (1/5)] = (2/5)/(2/5) = 1
So, initial quantity of mixture in the jar = 10 litres.
And, quantity of A in the jar = (10 × 4)/5 = 8 litres.
Method 3: (Alligation with percentage of B)
 The percentage of B in 1st mixture = 20%
 The percentage of B in 2nd mixture = 100%
 The percentage of B in Final Mixture = 60%
 By rule of allegation we have
Volume 1^{st} : Volume 2^{nd} = (100%  60%) : (60%  20%)
V_{1} : V_{2} = 1 : 1
Volume of mixture 1^{st} = 10 litres
Volume of A in mixture 1^{st} = 80% of 10 litres = 8 litres. Answer
Example.3: Nine litres of solution are drawn from a cask containing water. It is replaced with a similar quantity of pure milk. This operation is done twice. The ratio of water to milk in the cask now is 16 : 9. How much does the cask hold?
 Let there be x litres in the cask
 After n operations:
(1 – 9/x)^{2} = 16/25
∴ x = 45 litres.
Example.4: There are two containers A and B of milk solution. The ratio of milk and water in container A is 3 : 1 and in container B, it is 4 : 1. How many liters of container B solution has to be added to 20. lts of container ‘A’ solution such that in the resulting solution; the ratio of milk to water should be 19 : 6?
 In container A, the part of milk =
 It is given 20 lts of container A is added. So, the quantity of container B should be 5 lts.
Example.5: There are two alloys A and B. Alloy A contains zinc, copper and silver, as 80% 15% and 5% respectively, whereas alloy B also contains the same metals with percentages 70%, 20%, 10% respectively. If these two alloys are mixed such that the resultant will contain 8% silver, what is the ratio of these three metals in the resultant alloy?
 Since the resultant alloy contains 8% silver, first we will find, in what ratio these two alloys A and B were mixed to form the resultant.
 Then the resultant zinc percentage is
 So, copper percentage = 100 – (74 + 8) = 18
∴ The ratio of these metals = 74 : 18 : 8 = 37 : 9 : 4.
Example.6: The cost of an apple is directly proportional to square of its weight in a fruit bazaar. Two friends A and B went there to purchase apples. A got exactly 5 apples per kg and each apple is of same weight. Where as B got exactly 4 apples per kg each weight is exactly same. If B paid Rs. 10 more than A per kg apples, what is the cost of an apple which weighs 1 kg?
 It is given cost ∝ (weight)2
⇒ c = k w^{2}. A got 5 apples per kg and each apple is of same weight. ⇒ Each apple is 200 gm. = 1/5 kg.
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1. What is the concept of mixtures and alligation? 
2. How is mixtures and alligation useful in real life? 
3. Can you explain the basic formula or equation used in mixtures and alligation? 
4. How can mixtures and alligation be applied to solve practical problems? 
5. Can you provide an example of mixtures and alligation in a reallife scenario? 

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