Theorem:
The proportion in which water must be mixed
with spirit to gain or to lose x% by selling it at cost price is given by
=
(
x
)
100
Example :
In what proportion must water be mixed with spirit to
gain 16 ^{2}/_{3}% by selling it at cost price?
Detail Method :
Let the required proportion of water
to spirit be a : b and the cost price of spirit be Rs x per
litre.
As per the question,
Selling price of the mixture = Rs x per litre.
Cost price of the mixture
= x X
100
x X
100
100
= Rs
6x
= per litre.
7
Now, assume that the cost price of water = Rs 0 per litre.
∴ (a X 0 + b X x) = (a + b)
6x
7
or, bx = (a + b) =
6
x
7
or, b
(
1 -
6
)
=
6a
7
7
or,
b
=
6a
7
7
or,
a
=
1
b
6
∴ required ratio = 1: 6
Ailigation Method :
Let CP of sprit be Re 1 per litre.
Then SP of 1 litre o f mixture = Rs 1.
Gain = 16
2
%
3
CP of 1 litre of mixture = Rs
(
100 X 3 X 1
)
= Rs
6
350
7
CP of 1 litre water
CP of 1 litre pure spirit
(Rs 0)
Mean price Rs 6/7
(Rs 1)
1/7
6/7
Quantity of water
=
1/7
=
1
Quantity of spirit
6/7
6
or Ratio of water and spirit = 1 : 6.
Quicker Method : Here you can use direct formula :
the required proportion =
50
= 1 : 6
300
Exercise :
In what proportion must water be mixed with spirit to gain 33 ^{1}/_{3}%by selling it at cost price?
In what proportion must water be mixed with spirit to
gain 16% by selling it at cost price?
In what proportion must water be mixed with spirit to
gain 25% by selling it at cost price?