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

Direct Formula / Rule 23 :

• Theorem: A container contains x part milk and y part water. From this container, 'a' part of the mixture is taken out and replaced by water. Now, half of the container contains milk and another half contains water. The value of 'a' is given by  = 1 [ x - y ] part. 2 x

• Example :

• A container contains 7 part milk and 3 part water. How many parts of mixture should be taken out and replaced by water so that container contains half milk and half water.

• Detail Method :
Let the container contain 1 litre of mixture
Amount of milk = 7/10 litre and the amount of water = 3/10
Now, let us suppose that x part of the mixture is taken out. In the container amount of milk = (7/10 - 7x/10) litres and the amount of water = (3/10 - 3x/10) litres.
If container is replaced by x part of water, then the amount of water in the container becomes = (3/10 - 3x/10 + x) litres.
As per the question  = 7/10 - 7x/10 = 1/2 3/10 - 3x/10 + x 1/2
or, 7/10(1 - x) = 3/10(1 - x) + x
or, (1 - x)2/5 = x
or, 7/5x = 2/5
x = 2/7
So 2/7 part of the mixture is taken out.

• Ailigation Method :
Let us suppose that initially container contains x litres of the mixture, then
Milk : Water = 7x/10 : 3x/10 = 7 : 3
Now, applying the alligation method,  Mixture Water
 3/10 1/2 1 1/2 1/5
= 5 : 2
Now, according to the question,
taken out mixture = replaced water = 2/7 part.

• Quicker Method : Here you can use direct formula :
the required answer  = 1 [ 7 - 3 ] = 2 part. 2 7 7

Exercise :
1. A container contains 8 parts milk and 4 parts water. How many parts of mixture should be taken out and replaced by water so that container contains half milk and half water.
2. A container contains 9 parts milk and 6 parts water. How many parts of mixture should be taken out and replaced by water so that container contains half milk and half water.
3. A container contains 4 parts milk and 1 parts water. How many parts of mixture should be taken out and replaced by water so that container contains half milk and half water.
4. Answers : 1 = 1/4 parts,   2 = 1/6,   3 = 3/8

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