Alligation and Mixture

 Alligation and Mixture

                            Alligation is the rule that find the ratio in which two or more ingredients 

                            at the given price must be mixed to produce a mixture of a desired price.

Mean Price :-The cost price of unit quantity of the mixture.

Rule of alligation :-If two ingredients are mixed,then

      Cheaper quantity : Dearer quantity = (D-M) : (M-C)                

Replacement of part of mixture :-Suppose a container contains a solution from which
                                                     some quantity of solution is taken out and replaced with
                                                    one of the ingredients.This process is repeated n times then  

Example 1.:-In what ratio must rice at Rs.40 /kg to mix rice at Rs.60 /kg,so that  the average cost of mixture is Rs.55 /kg ?

Explanation :-

Hence the answer is 60-55 : 55-40 = 5:15

                                                             1:3

 Example 2.:-

In what ratio must a grocer mix two varieties of pulses costing Rs. 15 and  

                       Rs. 20 per kg respectively so as to get a mixture worth Rs. 16.50 kg?

Explanation :-

              

Hence the answer is 20-16.50:16.50-15=3.50 :1.50

                                                             7:3

Example 3.:-How many kilogram of sugar costing Rs. 9 per kg must be mixed with 27 kg

                      of sugar costing Rs. 7 per kg so that there may be a gain of 10% by selling

                      the mixture at Rs. 9.24 per kg?

Explanation :-S.P. of 1 kg of mixture = Rs. 9.24, 

                        Gain 10%.

C.P. of 1 kg of mixture = Rs.		100	x 9.24		= Rs. 8.40
		110

By the rule of alligation, we have:

           Ratio of quantities of 1^st and 2^nd kind = 1.40 : 0.60 = 7 : 3.

           Let x kg of sugar of 1^st be mixed with 27 kg of 2^nd kind.

          Then, 7 : 3 = x : 27

               x = (27✖7)/3 = 63

Example 4.:-Two containers A and B contain milk and water in the ratio of 5 : 2 and 

                     7 : 6 respectively. Find the ratio in which these two mixtures can be mixed

                     so that a new mixture formed in the container C is in the ratio of 8 : 5. 

Explanation :-Let the cost price of milk be Rs. 1 per liter.

              Therefore, cost of milk in 1 liter of mixture in

Container A (Milk : Water = 5 : 2) =5/7 ✖Rs.1

=5/7

Container B (Milk : Water = 7 : 6) =

7/13✖Rs.1 = 7/13

Container C (Milk : Water = 8 : 5) =

8/13✖Rs.1 = 8/13

             Now by alligation  rule , 

The required ratio of milk and water = 1/13 : 9/91
                                                               7 : 9

Example 5.:-A container contains 40 litres of milk. From this container 4 litres of milk

                      was taken out and replaced by water. This process was repeated further

                      two times. How much milk is now contained by the container?

Explanation :-

Example 6.:-8 liters are drawn from a  container full of milk. then filled with water

                        This process was repeated three more times. The ratio of the quantity of

                        milk now left in container to that of water is 16:65.How much milk did

                        the container hold originally?

Explanation :-Let the quantity of milk in the container is x liters.

                         Then  the quantity of milk in the container after 4 operations 
                                                                           

                                                                    x = 24 liters