Alligation or Mixture -> Formula/Rule 1

Direct Formula / Rule 1 :
• Theorem: The proportion in which sugar at Rs. x per kg must be mixed with rice at Rs. y per kg, so that the mixture be worth Rs. z a kg, is given by

  =	(	y - z	)
  		z - x

• Example :
The proportion in which sugar at Rs. 3.10 per kg must be mixed with sugar at Rs. 3.60 per kg, so that the mixture be worth Rs. 3.25 a kg ?
• Detail Method : Let the required ratio be x. y.
   As per Question,
   310x + 360y = 325(x+y)
   or 310x + 360y = 325x + 325y
   325x - 310x = 360y - 325y
   150x = 350y
   15x = 35y
   3x = 7y

  =	x	:	7
  	y		3

   so x : y = 7 : 3    [here y goes to left side & 3 goes to right side]
• Ailigation Method : By the rule of alligation we have :

  C.P.of 1 kg cheaper sugar

  C.P.of 1 kg dearer sugar

  (310 paise)		Mean price (325 paise)		(360 paise)
  35				15

  So, ratio of 1st and 2nd quantities = 35 : 15 = 7 : 3 
  so, they must be mixed in the ratio of = 7 : 3
• Quicker Method : Here you can use direct formula : 

  =	(	y - z	)
  		z - x

  =	(	CP of dearer - Mean Price	)
  		Mean Price - CP of cheaper

  =	(	3.60 - 3.25	)	=	(	0.35	)
  		3.25 - 3.10				0.15

  so, they must be mixed in the ratio of = 7 : 3

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Exercise :
1. In what proportion must rice at Rs 21 per kg be mixed with rice at Rs 28 per kg, so.that the mixture be worth Rs. 25 a kg ? 
2. In what ratio, must coffee at Rs. 62 per kg Is mixed With coffee at Rs. 72 per kg so that the mixture must be worth RS. 64.50 per kg ? 
3. In what ratio must water be mixed with oil costing Rs. 12 per litre to obtain a mixture worth of Rs. 8 per litre ? 
Answers :
1 = 5 : 4,   2 = 3 : 1,   3 = 1 : 2