Mixture and Allegation

Mixture and alligation is a mathematical technique used to solve problems related to mixtures of different ingredients or components with varying concentrations or prices. It is commonly used in areas such as chemistry, finance, and pharmacy to determine the proportions of different substances needed to create a mixture with a desired concentration or cost.

The key components of mixture and alligation problems are:

1. Mixtures: Mixtures refer to combinations of two or more substances or ingredients mixed together.

2. Alligation: Alligation is the process of finding the ratio in which different components with different strengths or prices should be mixed to obtain a desired average strength or price.

Here's how the alligation method works:

1. Identify the Components: Determine the different components or ingredients present in the mixture and their respective strengths or prices.

2. Set up a Table: Create a table with columns for the components, their strengths or prices, and the desired average strength or price.

3. Calculate the Mean Value: Calculate the mean or average value of the components based on the desired average strength or price.

4. Apply the Alligation Rule: Place the components in the table in such a way that their distances from the mean value are proportional to their strengths or prices. Use the following formula:

\[ \text{Component's distance from mean} = \frac{(\text{Component's strength or price}) - (\text{Mean value})}{(\text{Difference between the two components' strengths or prices})} \]

5. Determine the Ratios: The ratios obtained from the alligation process represent the proportions in which the components should be mixed to achieve the desired average strength or price.

6. Verify the Solution: Check that the ratios obtained from the alligation process are correct and make sense in the context of the problem.

Mixture and alligation problems are often encountered in competitive exams, such as those for government jobs or entrance exams, as well as in real-life scenarios where mixing different ingredients or components is required to achieve specific objectives.