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 Alligation or Mixture :-
 ::Important Formulas ::Examples

Direct Formula / Rule 4 :

• 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 :
1. In what proportion must water be mixed with spirit to gain 33 1/3%by selling it at cost price?
2. In what proportion must water be mixed with spirit to gain 16% by selling it at cost price?
3. In what proportion must water be mixed with spirit to gain 25% by selling it at cost price?
4. Answers : 1 = 1 : 3,   2 = 4 : 25,   3 = 1 : 4

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