Mixtures and Allegations

Different sums in the general and quantitative aptitude are a part of almost every competitive exam. Using the traditional school method to solve the mixture problems consumes a lot more time, resulting in unsuccessful results. The mixture and alligations give us the formula to solve mixture problems based on ratio, proportions, average, time and distance, profit and loss, etc. If the formulas are implemented properly with regular practice, it helps solve the problems in much less time. Thus the need to understand and master the concept of mixture and alligation arises. If the rules are followed properly, the interpretation of the answers will be correct in a few seconds.

Concept of Mixtures and Alligations

A mixture is the amalgamation of two or more elements to create a third element. The alligations is the order that helps us calculate a mixture’s price, given that the elements of a mixture have different prices. The mixture can be of two types, either simple or complex mixture. A simple mixture is usually formed by mixing two or more different elements, while a complex mixture is made by combining two or more different mixtures. A simple mixture is mixing rice at two different prices or mixing milk and water. A complex mixture is mixing two alloys containing two or more different types of metals in a different ratio. Mixtures and alligations are repeatedly asked in various competitive exams and it is important to understand its concept and application. If the formulas and the tricks to solve the problems of mixture and alligations are unknown, solving them can be pretty time-consuming. The concept of alligations are different from that of the mixture, they can seem confusing, but if solved with correct guidance, it is easy. The concept of allegation can be applied to a series of different problems like finding average speed, average weight or price, profit and loss, and application in the compound mixture. 

Mixtures and Alligations formulas 

There are two basic mixture and alligation formulas to solve problems, also called allegation rules. The allegation rules help us solve and find the ratio of complex and simple mixtures based on two different methods. The first method is called the formula method and the second method is the diagram method. The formula method can be expressed as :

Quantity of cheaper/Quantity of expensive = (CP of expensive – Mean Price)/(Mean Price- CP of cheaper)

The method two or diagram method can be represented as:

Cost of the cheaper(C)                                       Cost of the Dearer(D)

Mean price or value(M)

Quantity of cheaper(D-M)                                 Quantity of dearer(M-C)

It is to be noted that the rules of allegation having methods one and two can only be applied to the topics mentioned:

• Ratio and proportion

• Simple and compound interest

• Profit and loss 

• Average 

• Percentage

An example is showing the application of rules of the allegation by solving some mixture and alligation questions. The example below is solved using method one :

Q) A shopkeeper decides to sell a mixture of two rice varieties priced at Rs. 16 per Kg. In what ratio should he mix them to reach the selling price if the cost of one variety is Rs. 14 per kg and the other is Rs. 24 per kg?

If we see the above question, it has a cheaper entity and a dearer or expensive entity. The price of the mixture is the mean value for the calculation. We will get the ratios if we put the values in the formula method. The ratio is:

Quantity of expensive : Quantity of cheaper = 24-16/16-14 = 8:2 = 4:1 

Thus the answer is 4:1.

Solving another mixture and alligation problem with the help of method two or diagram method.

Q) Mixture X of milk and water contains 75% milk and mixture Y of milk and water contains 10% water, then find the ratio in which these two mixtures are mixed to form a third mixture which contains 15% water in it?

Water percentage in mixture X = 100% – 75% = 25%

The mean value = 15% = M

X = 25% = cost of cheaper = C

Y = 10% = cost of expensive or dearer = D

Now putting the values in the below diagram

Cost of the cheaper(C)                                     Cost of the Dearer(D)

Mean price or value(M)

Quantity of cheaper(D-M)                            Quantity of dearer(M-C)

 25%                                        10%

15%

(10-15)                                     (15-25)

Required ratio = -5 : -10 = 1: 2

Tips and tricks

Some major tips and tricks are:

• The use of allegation rules.

• Allegation rules find the ratio and the rate at which articles are sold.

• Placing the values correctly in the allegation rules.

Conclusion

Mixture and alligation is an important concept to understand as it helps solve a series of problems. The rules or the methods are easy and with practice and the knowledge of putting the right value to the formula and diagram, the process can become a lot easier. Almost all competitive exams include the mixture and alligation problems in the general or quantitative aptitude section. Knowledge is important to crack competitive exams successfully. If the traditional method of solving the problems is followed, it takes more time. Thus, the rules and methods help in saving time.