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

Direct Formula / Rule 16 :

• Theorem: L litres of a mixture contains two liquids A and B in the ratio a : b. The amount of liquid B, that is added to get a new mixture containing liquid A and B in the ratio x : y, is given by  = [ ( y/x ) - ( 1 ) ] L litres. 1 + b/a 1 + a/b

• Example :

• 729 litres of a mixture contains milk and water in the ratio 7 : 2. How much water is to be added to get a new mixture containing milk and water in the ratio 7 : 3 ?

• Detail Method :
Let the amount of water be x litres.
 The original mixture contains ( 729 X 7 ) litres of milk 7 + 2
 and ( 729 X 2 ) litres of water. 7 + 2
Now, from the question,  x litres of water is added. Therefore = 729 X 7/9 = 7 729 X 2/9 + x 3
or, 729 X 7 X 3 = 729 X 2 X 7 + 9 X 7x
or, 63x = 7 X 729  So x = 7 X 729 = 81 litres 63

• Ailigation Method :
Method I. Percentage
Change the ratio into percentage.
% of water in the original mixture  = 2 X 100 = 200/9% 7 + 2
% of water in the resulting mixture  = 3 X 100 = 30% 7 + 3
 200/9% 30% 100% 70% 70/9%
Therefore, the ratio in which the mixture and water are to be added is 1 : 1/9 or 9 : 1
Then quantity of water to be added  = 729 X 1 = 81 litres. 9
Method II. Fractional
Change the ratio into fraction.  Fraction of water in the original mixture = 2 9
 Fraction of water in the resulting mixture = 3 10
So,  2/9 3/10 1 7/10 7/90
Therefore, the ratio in which the mixture and water are to be added is
 = 7 : 7 = 1 : 1/9 = 9 : 1 10 90
Then quantity of water to be added to the mixture  and quantity of A in the jar = 729 = 81 litres. 9

• Quicker Method : Here you can use direct formula :
 = [ ( 3/7 ) - ( 1 ) ] X 729 1 + 2/7 1 + 7/2
= [3/7 X 7/9 - 2/9] X 729
= (1/3 - 2/9) X 729 = 729/9 = 81 litres.

Exercise :
1. 56 litres of a mixture contains milk and water in the ratio 5 : 2. HOW much water is to be added to get a new mixture containing milk and water in the ratio 5 : 3 ?
2. 36 litres of a mixture contains milk and water in the ratio 2 : 1. How much water is to be added to get a new mixture containing milk and water in the ratio I : 1 ?
3. 25 litres of a mixture contains milk and water in the ratio 3 : 2. How much water is to be added to get a new mixture containing, milk and water in the ratio 3 : 4 ?
4. Answers : 1 = 8 litres,   2 = 12 litres,   3 = 10 litres

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