Alligation

A mixture combines two or more elements to form a third element. We use the concepts of ratio and proportion to solve questions related to mixtures. Alligation is an efficient method devised to calculate the price of a mix that is formed using two or more elements with different prices. 

It helps us achieve the desired proportion and percentage of different elements in a mixture. Alligation is helpful when calculating the price of a mix with different ingredients. We calculate the cost of the mixture with the help of the mean price and alligation method in maths. 

Alligation Meaning And Definition

Alligation is the method used to calculate the price of a mixture. It uses the concept of weighted average. Alligation is a rule that defines the ratio in which two or more elements should be mixed to produce the desired results at the given price. 

Let us look at some essential terms to understand the alligation meaning better.

Weighted Average

A weighted average is the average of a data set where some data points are treated as more important than others. We determine the weight of each point and then multiply the weight by each value. 

Example: We buy 50 pens at the cost of 10 rupees and 20 pens at 15. What will be the average cost?

Using the concept of weighted average:

=[50(10)+20(15)]/( 50+20)

=[500+300]/70

=800/70

=11.4

Mean Price

Mean price is the cost price of a unit quantity of the mixture. When we mix elements of different prices to form a mixture of a mean cost known as average cost, the ratio of quantities becomes inversely proportional to their difference from the mean cost.

Rule Of Alligation

Let us look at a mixture of two ingredients with concentration “a” and “b”. ‘“a” is the cheaper, and “b” is the costlier component. 

Quantity of cheaper / Quantity of costlier

=[CP of costlier one- Mean Price]/[Mean Price-CP of cheap one]

Alligation Rule For Compound Mixture

In a compound mixture, a mixture of the same ingredients is mixed in different proportions resulting in a new combination. 

For a mixture containing the ingredients A and B, 

Mixture 1 contains the ingredients in the ratio a:b.

Mixture 2 contains the ingredients in the ratio x:y.

Now M units of mixture 1 are mixed with N units of mixture 2 to form the compound. The resultant mix contains the ingredients A and B in the ratio:

Quantity of ingredient A/Quantity of Ingredient B=qa/qb

={M[a/(a+b)]+N[x/(x+y)]}/{M[b/(a+b)]+N[y/(x+y)]}

Alligation Method

Alligation method in maths used for a mixture of two elements with different cost prices is given below:

Quantity of Cheaper Substance/Quantity of Dearer substance:

=[CP of dearer substance- Mean Price]/[Mean Price-CP of cheaper substance]

Example of alligation method problem:

A shopkeeper sells rice at 8% and 18% profit. The average gain he received is 14%. What is the amount of rice he sells at 18% profit if the total quantity sold is 100kg?

Solution: 

Dearer gain: 18%

Lower gain: 8%

Mean gain: 14%

Using the formula:

Quantity of Cheaper Substance/Quantity of Dearer substance

=(18-14)/(14-8)

=2/3

Quantity of Dearer substance becomes 60kg. 

Another alligation method that is commonly used is the method of repeated dilution.

A container has “x” units of a liquid. “y” units of the liquid are taken out and changed with water. After repeating this process n times, the amount of pure liquid left is given by:

Quantity of Pure Liquid=x[1-(y/x)]n

Example problem: From a 40 litres container of milk, 4 litres of milk was taken out and changed with water. The exact process is further repeated twice. What amount of milk is left in the container?

Solution: 

Using the formula:

Quantity of Pure Liquid=x[1-(y/x)]n

x=40

y=4

n=3

Quantity of Pure Liquid=40[1-(4/40)]3

=40[9/10]3

=40(9/10)(9/10)(9/10)

=29.16 litres

Conclusion

The alligation method is used to determine the quantity and price of the desired mixture. This method has various daily life applications as well. This method was devised to simplify the long calculative approach to obtain the desired quantity efficiently.