Mixture & Alligations

strong>Alligation helps us
To find the average value of a mixture when the prices of two or more varieties are mixed together.
To find the proportion in which they are mixed and,
To find the proportion in which these - at given prices - must be mixed to produce a mixture at a given price.


Example: A grocer mixes 18 kg of tea costing Rs. 6.20/kg with 22 kg costing Rs. 8.40/kg. Find the cost of the mixture per kg.
Solution
Cost of 18 kg @ 6.20 = 6.20 × 18 = Rs. 111.60
Cost of 22 kg @ 8.40 = 8.40 × 22 = Rs. 184.80
Cost 40 kg of the mixture = Rs. 111.60 + Rs. 184.80 = Rs. 296.40
? Cost of 1 kg of the mixture = Rs. 7.41

Example: In what proportion must a grocer mix teas @ 1.22/kg and Rs.1.94/kg so as to make a mixture worth Rs. 1.66/kg.
Solution
Using the formula given above,


Example: A merchant has 40 kg sugar, part of which he sells at 12% gain and the rest at 28%. He gains 18% on the whole. Find the quantity sold at 12% profit.
Solution
Going by the above Example No. 2 we get the ratio of mixtures as 5 : 3
Therefore the quantity sold at 12% profit is 25 kg.

Example: Three glasses each of equal size were filled with a mixture of wine and soda. The proportion of wine to soda in each glass is as follows:2 : 3; 3 : 4; & 4: 5.
The contents of the three glasses are emptied into a single vessel. What is the proportion of spirit and water in it.
Solution
Let each glass be of the size 315 ml (LCM of the sums of the ratio i.e., 5, 7, 9)


Example: From a can of super quality oil containing 96 litres, 12 litres are drawn out, and the can is filled up with inferior quality of oil. The process is repeated thrice, finally what shall the number of litres of super oil left in the can?
Solution
8 Litres = of 96 Litres = of the can.
? The quantity of super oil after 1st operation = of the whole.
? The quantity after the 2nd operation = of the whole.
? The quantity after the 3rd operation = of the whole = of the whole.
? The number of litres left = of 96 = 45.

A very important result: If a vessel contains 'a' litres wine, and if 'b' litres be withdrawn and replaced then if 'b' litres of the mixture be withdrawn and replaced, and the operation repeated 'n' times in all, then:-