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 :
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 ?
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 ?
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 ?