Circumstances of Alligations and Mixtures

Allegation and mixtures are based on the concept of ratio and proportions, and they don’t have a specific formula. Different rules of allegations help in the quick calculation of the price of the mixture. 

The concept of mixtures and allegations is useful in many fields, and their techniques are also applied in important topics such as mixtures, profits, loss, simple interest, time and distance, percentage, etc.

Mixtures

The process of the mixture is when the mixing of two different qualities of things takes place. When the mixture of two different products with different qualities occurs, the quality of the final mixture will lie between the quality of the original constituent products. 

For calculating the weighted average of the mixture, the formula used is:

P= (p1×q1)+(p2×q2)/q1+q2

For example, 

When 15 kg of wheat atta that costs ₹30 per kg is mixed with 10 kg of rice atta that costs ₹30 per kg, then the final resultant mixture price will be:

(15×30)+(10×3)/25= ₹19.2

There are two types of mixtures

• Simple mixtures

A simple mixture refers to the mixing of two or more ingredients together. 

• Compound mixtures 

Compound mixtures refer to the combination of two or more mixtures. 

Allegations

Allegations refer to the rule that helps us to find the ratio of the mixture. The cost of a unit quantity of a mixture is called the mean price. 

The Rule of Allegation refers to the connection between quantities and prices in mixtures.

The Rule of Allegation is written as:

Quantity of cheaper/quantity of dearer= Rate of dearer – Average rate/ Average Rate- Rate of Cheaper

Or,

q1/q2= p2-p/p-p1

Example

The shopkeeper has 150 kg of wheat from which he sells one part of it at 18% profit and the rest at 9% loss. On the wholesale, his total losses were 3%. What is the quantity of wheat he sold at a 9% loss?

Quantity of wheat sold at 9% loss= 100/9×7

=350/3kg.

The major applications of the Rule of Allegation are:

• It helps in finding the mean value of the mixture that consists of two more ingredients mixed, and also the proportion in which they are mixed is provided
• The rule of allegation also helps in finding the proportion in which the ingredients must be mixed at the given prices to produce the mixture provided at the given price. 

The Rule of Allegation makes it very easy for calculating the weighted average quality for a mixture of two different ingredients. 

The types of questions for mixtures and allegations topic 

There are many types of questions that can be based on mixtures and allegations concepts such as:

• The number of ingredients, as well as their price, will be provided, and you will be asked to find the average price of the final mixture.
• The desired quantity or price of the mixture, as well as the price of the ingredients, will be provided. You have to find out the ratio or percentage of ingredients that are used to get the final amount.
• To find the resulting amount of the concentrating or diluting mixture 
• To find out the ratio in which the two quantities are mixed to get the final mixture to be sold at a profit of x%.
  

Conclusion

Therefore, mixtures and allegations have many applications in day-to-day life and also help in finding out different values. Along with this, mixtures and allegations are a key part of many mathematical questions in important examinations.