

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.
(as there is loss on
the whole)
= 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 :

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?

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.

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.
Answers :
1 = 5 : 3, 2 = 25 litres, 15 litres, 3 = 50 litres, 30 litres



