Definition: Mixture and Alligation is a mathematical technique used to determine the resultant value or concentration of a mixture when two or more ingredients of different prices or quantities are combined. It serves as a shortcut method for solving problems involving weighted averages and ratios in a single, efficient step.
The Conceptual Foundation of Alligation
At its core, the Alligation rule is an application of the weighted average concept. When you mix two quantities—let us call them Quantity A and Quantity B—with different price points or concentrations, the cost of the final mixture will always lie somewhere between the prices of the two individual components.
The method provides a visual framework to find the ratio in which these ingredients must be mixed to achieve a specific target price. By arranging the prices in a cross-pattern, you can subtract the smaller value from the larger value diagonally, which directly reveals the ratio of the quantities required.
The Standard Alligation Rule
To apply this method, you need three values: the price of the cheaper ingredient (C), the price of the dearer ingredient (D), and the target mean price (M). The rule dictates that the ratio of the cheaper quantity to the dearer quantity is equal to (D – M) : (M – C).
“The mean price must always be greater than the price of the cheaper ingredient and smaller than the price of the dearer ingredient. If it is not, the mixture is impossible to create at that specific target price.”
This method is highly effective for competitive exams like UPSC CSAT or SSC because it bypasses long algebraic equations. Instead of setting up variables like x and y, you simply subtract diagonally, saving precious time during the examination.
Advanced Applications: Replacement and Repeated Dilution
A common variation of this topic involves the replacement of liquids. Imagine a container filled with a liquid, where a portion is removed and replaced with another liquid (like water). If this process is repeated n times, the final concentration of the original liquid can be calculated using a specific formula.
- Identify the initial quantity (V).
- Identify the quantity removed or replaced in each step (x).
- Apply the formula: Final Concentration = V(1 – x/V)n.
Students often struggle with the logic behind this formula. Think of it as a percentage decrease. Each time you remove a portion, you are effectively reducing the remaining amount of the original substance by a fraction, making it a geometric progression problem disguised as a mixture problem.
Important Formulas for Competitive Exams
| Scenario | Key Formula |
|---|---|
| Basic Alligation Ratio | (Dearer – Mean) : (Mean – Cheaper) |
| Repeated Replacement | Final Amount = Initial × (1 – Removed/Initial)n |
| Profit/Loss in Mixtures | Mean Price = Cost Price × (100 + Profit%) / 100 |
Bridging Logic and Calculation
While formulas are useful, the UPSC CSAT has shifted toward logical reasoning. Do not just memorize the cross-subtraction method; understand that alligation is essentially balancing the surplus and deficit. If one ingredient is “too expensive” compared to the target, it must be balanced by an ingredient that is “too cheap.”
When dealing with percentage concentrations, ensure all units are consistent. For example, if you are mixing a 20% acid solution with a 50% acid solution, the mean must also be expressed as a percentage. Mixing a percentage with a raw quantity will lead to errors; always standardize your inputs before starting the alligation cross.
Key Points to Remember
- Consistency: Ensure all values (price, weight, or percentage) represent the same unit.
- The Mean Rule: The mean price always sits between the cheaper and dearer prices.
- Diagonal Subtraction: Always subtract the smaller value from the larger one to keep ratios positive.
- Efficiency: Use alligation for problems involving two ingredients; for three or more, consider weighted averages.
- Units: If the question asks for the ratio of “water to milk,” ensure your final answer matches that order, as alligation usually gives “cheaper to dearer.”
Quick Revision Summary
- Alligation is a shortcut for weighted average problems.
- The ratio obtained is always between the quantity of the first and second ingredients.
- The mean value must be an intermediate value.
- For repeated replacement, use the power n to represent the number of operations.
- Always check if the mean price is calculated on Cost Price (CP) or Selling Price (SP).
- If the mixture is sold at a profit, convert the SP to CP before applying alligation.
- Practice problems involving “replacement” are high-yield for SSC and UPSC.
- Logical deduction is often faster than algebraic substitution in complex mixture scenarios.
