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 Alligation or Mixture :-
 ::Important Formulas ::Examples

Direct Formula / Rule 13 :

• Theorem: A vessel of L litres is filled with liquid A and B. x% of A and y% of B is taken out of the vessel. It is found that the vessel is vacated by z%. Then the initial quantity of liquid A and B is given by  = ( z - y ) L litres and ( x - z ) L litres respectively. x - y x - y

• Example :

• A vessel of 80 litres is filled With milk and water, 70% of milk and 30% of water is taken out of the vessel. It is found that the vessel is vacated by 55%. Find the initial quantity of milk and water

• Detail Method :
Let the initial quantity of milk be x litres. Therefore, initial quantity of water = (80 - x) litres.
According to the question,
70% of x + 30% of (80 - x) = 55% of 80
or, 70x + 2400 - 30x = 4400
or, 40x = 2000
x = 50 litres.
Initial quantity of water = (80 - 50) = 30 litres,

• Ailigation Method :
Here the % values of milk and water that is taken from the vessel should be taken into consideration.  Milk Water
(as there is loss on the whole)  70% 55% 30% 25% 15%
= 5 : 3
Ratio of milk to water = 5 : 3
 quantity of milk = 80 X 5 = 50 litres. 5 + 3
 and quantity of water = 80 X 3 = 30 litres. 5 + 3

• Quicker Method : Here you can use direct formula :
 Initial quantity of milk = ( 55 - 30 ) X 80 = 25 X 80 = 50 litres. 70 - 30 40
 Initial quantity of water = ( 70 - 55 ) X 80 = 15 X 80 = 30 litres. 70 - 30 40

Exercise :
1. A vessel of 120 litres is filled with milk and water. 80% of milk and 40% of water is taken out of the vessel. It is found that the vessel is vacated by 65%. What is the ratio of milk to water?
2. A vessel of 40 litres is filled with milk and water. 75% of milk and 35% of water is taken out of the vessel. It is found that the vessel is vacated by 60%. Find the initial quantity of milk and water.
3. A vessel of 80 litres is filled with milk and water. 65% of milk and 25% of water is taken out of the vessel. It is found that the vessel is vacated by 50%. Find the initial quantity of milk and water.
4. Answers : 1 = 5 : 3,   2 = 25 litres, 15 litres,   3 = 50 litres, 30 litres

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