

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 + 1  4 + 1 
or, 4x  8 : x  2 + 10 = 2 : 3
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
In the resultant mixture, % of liquid B
Replacement is made by the liquid B, so the % of B in
second mixture = 100%
Then by the method of Alligation:
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,
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 :

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?

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?

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?
Answers :
1 = 18 litres, 2 = 21 litres, 3 = 27 litres



