Alligation or Mixture -> Formula/Rule 4

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

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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? 
Answers : 1 = 1 : 3,   2 = 4 : 25,   3 = 1 : 4