☼ Alligations  is the rule that helps us to find the amount in which two different quantities of  a substance are mixed so as to arrive at average cost.

☼ If  the gradients are mixed in some ratio,   then

  = 

where
a = CP of unit quantity of cheaper
b = CP of unit quantity of dearer
m = Mean  price  of  Mixture

 e.g. In what proportion must rice at Rs. 3.50 per kg. be mixed with rice at Rs. 4.30 per kg., so that the mixture be worth Rs. 4.00 a kg.  ?

 Soln.

They must be mixed in the ratio  0.30  :  0.50  =  3  :  5

☼ If  the gradients are mixed in some ratio,   then

1^st Quantity  :  2^nd Quantity  :  3^rd Quantity  :  4^th Quantity  :  5^th Quantity

=     :       :      :     :    +

Where,

 a = CP of  1^st  quantity

 b = CP of  2^nd  quantity

 c = CP of  3^rd  quantity

 d = CP of  4^th  quantity

 e = CP of  5^th  quantity

 m = Mean  price  of  Mixture

 Let,  a  <  b  <  c  <  m  <  d  <  e

(  In this selection the m belongs to the interval of end-points of each pair.  i.e. one of the quantity of pair must be greater than m and another must be less than m. )

Note :  Different ways of linking will give different solutions.

e.g.      In what ratio must a person mix three kinds of  wheat costing him Rs. 5.20, Rs. 5.70 and Rs. 7.20 per kg., so that the mixture may be worth Rs. 5.30 per kg.

They must be mixed in the ratio  2.30  :  0.10  :  0.10   =  23  :  1  :  1

☼ Amount of liquid left after n operations, when the container originally contains x units of liquid from which y units is taken out each time is :

x  =   x

 e.g. A container contain 60 kg. of  milk. From this container 12 kg. of  milk was taken out and replaced by water. This process was further repeated two times. How much milk is now contained by the container ?

 Soln. Here,  x  =  60  kg.,  y  =  12  kg.,  n  =  3.

Milk  is  now  contained  by  the  container   =   x

=   60

=   60   =   30.72  kg.

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