Mixture and alligation is one of the most important and frequently tested topics in quantitative aptitude. It is asked in almost every competitive exam including CAT, SSC CGL, SSC CHSL, Bank PO, Bank Clerk, Railway RRB and CSAT. The mixture and alligation concept helps students solve problems related to mixing ingredients at different prices, replacing liquids in a container, finding the mean price of a mixture and calculating the proportion of different components in a mixture. A strong understanding of alligation rule, replacement formula and weighted average is essential for scoring well in these exams. In this post we cover everything from simple mixture and compound mixture to alligation rule, graphical representation of alligation, replacement formula, ratio unchanged rule and alligation rule for three ingredients — all explained with clear diagrams and solved examples.
⬤ Mixture are generally of two types. When different ingredients are mixed together then it is known as Simple Mixture. e.g. a mixture of water and milk; water and pure spirit.
⬤ When two or more Simple mixtures (made of same ingredients of same or different proportions) are mixed together to form another mixture, it is known as Compound Mixture.
Alligation Rule:- Two groups of elements are mixed together to form a third group containing the elements of both the groups.
If the average of first group is A₁ and the number of elements is n₁ and the average of second group is A₂ and the number of elements is n₂, then to find the average of the new group formed , we can use either the weighted average equation or the Alligation Equation.
As a convention we take A₁<A₂, then by the principle of average, we get A₁<Aᵥᵥ<A₂
⦿ The mean value is found out with respect to which unit, the ratio comes out to be of that unit.
⬤ From a container having x units of liquid, suppose y units are taken out and replaceed by water. After n operations, quantity of pure liquid
⬤ If a vessel contains ‘a’ litre of liquid A and if ‘b’ litres be withdrawn and replaced by liquid B then if ‘b’ litres of mixture be withdrawn and replaced by liquid B and the operation is repeated ‘n’ times in all, then
● A vessel, full of li quid A, containing ‘a’ litres of it of which several litres are withdrawn. The vessel is then filled with liquid B. Next the same volume of mixture withdrawn and again the vessel is filled with liquid B. This process is repeated ‘n’ times. As a result, the vessel contains ‘b’ litres of liquid A then
● If a vessel contains liquid A and liquid B in the ratio a:b and iif some quantity of the mixture (or vessel) are withdrawn, then in the remaining mixture, liquid A and liquid B will be in the ratio a:b i.e. ratio will not change
Alligation rule for a mixture of three ingredients:- When three ingredients with cost prices c₁, c₂ and c₃ respectively are mixed to form a mixture of mean price cₘ then
q₁ = Proportion of 1ˢᵗ ingredient = (c₂ – cₘ)(c₃ – c)
q₂ = Proportion of 2ⁿᵈ ingredient = (cₘ – c₁)(c₃ – cₘ)
q₃ = Proportion of 3ʳᵈ ingredient = (c₂ – cₘ)(cₘ – c₁)