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Home->Aptitude->Alligation or Mixture ->Formula/Rule 15

Alligation or Mixture :-

    Direct Formula / Rule 15 :

  • Theorem: Ajar contains a mixture of two liquids A and B in the ratio a : b. When L litres of the mixture is taken out and P litres of liquid B is poured into the jar, the ratio becomes x : y. Then the amount of liquid A, contained in the jar, is given by
    = [ L(y/x - b/a)+P(1 + b/a) X (x/y X a/b) ] litres
    (a/b - b/a X x/y)+(1 - x/y)

    and the amount of liquid B in the jar is given by
    = [ L(y/x - b/a)+P(1 + b/a) X (x/y) ] litres
    (a/b - b/a X x/y)+(1 - x/y)

  • Example :

  • Ajar contains a mixture of two liquids A and B in the ratio 4 : 1. When 10 litres of the mixture is taken out and 10 litres of liquid B is poured into the jar, the ratio becomes 2 : 3. How many litres of liquid A was contained in the jar?

  • Detail Method :
    Let the quantity of mixture in the jar be 5x litres. Then
    = 4x - 10 ( 4 ) x - 10 ( 1 ) + 10 = 2 : 3
    4 + 14 + 1
    or, 4x - 8 : x - 2 + 10 = 2 : 3
    Or, 4x - 8 = 2
    x + 83
    So x = 4
    Then quantity of A in the mixture = 4x = 4 x 4 = 16 litres.

  • Ailigation Method :
    Method I.
    In original mixture, % of liquid B
    = 1 X 100 = 20%
    4 + 1
    In the resultant mixture, % of liquid B
    = 3 X 100 = 60%
    2 + 3
    Replacement is made by the liquid B, so the % of B in second mixture = 100%
    Then by the method of Alligation:
    20%        60% 100%
    40% 40%
    Ratio in which first and second mixtures should be added is 1 : 1. What does it imply? It simply implies that the reduced quantity of the first mixture and the quantity of mixture B which is to be added are the same.
    Total mixture = 10 + 10 = 20 litres.
    and liquid A = 20 X 4 = 16 litres.
    5
    Method II.
    The above method is explained through percentage. Now, method II will be explained through fraction.
    Fraction of B in original mixture = 1
    5
    Fraction of B in second mixture (liquid B) = 1
    Fraction of B in resulting mixture = 3
    5
    So,
    1/5        3/5 1
    2/5 2/5
    Thus, we see that the original mixture and liquid B are mixed in the same ratio. That is, if 10 litres of liquid B is added then after taking out 10 litres of mixture from the jar, there should have been 10 litres of mixture left. So, the quantity of mixture in the jar = 10 + 10 = 20 litres
    Total mixture = 10 + 10 = 20 litres.
    and quantity of A in the jar = 20 X 4 = 16 litres.
    5

  • Quicker Method : Here you can use direct formula :
    Amount of liquid A contained in the jar
    = [ 10(3/2 - 1/4)+10(1 + 1/4) X (2/3 X 4/1) ]
    (4 - 1/4 X 2/3)+(1 - 2/3)
    = 25/2 + 25/2 X 2/3 X 4/1
    23/6 + 1/3
    = 8 X 2 = 16 litres.


    Exercise :
  1. A vessel contains mixture of liquids A and B in the ratio 3 : 2. When 20 litres of the mixture is taken out and replaced by 20 litres of liquid B, the ratio changes to 1 : 4. How many litres of liquid A was there initially present in the vessel?
  2. A can contains a mixture of two liquids in proportion 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the proportion of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially?
  3. Ajar contains a mixture of two liquids A and B in the ratio 3 : 1. When 15 litres of the mixture is taken out and 9 litres of liquid B is poured into the jar, the ratio becomes 3 : 4. How many litres of liquid A was contained in the jar?
  4. Answers : 1 = 18 litres,   2 = 21 litres,   3 = 27 litres