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?