Definition: Mixture and Alligation is a mathematical technique used to determine the resultant price, concentration, or quantity of a mixture when two or more ingredients with different values are combined in a specific ratio. While Alligation provides a shortcut to solve problems involving weighted averages, Mixture problems focus on the physical composition and dilution of substances.
Understanding the Alligation Rule
The core principle of Alligation is based on the concept of weighted averages. Imagine you have two types of sugar—one priced at a higher rate and one at a lower rate. If you mix them, the cost price of the mixture will always lie somewhere between the prices of the two individual components. Alligation allows us to find the ratio in which these ingredients must be mixed to achieve a desired target price.
To apply this, we arrange the prices in a cross-pattern. If we have a cheaper ingredient (C) and a dearer ingredient (D), and we want to achieve a mean price (M), the rule states that the ratio of the quantity of the cheaper ingredient to the dearer ingredient is given by the difference between the mean price and the dearer price, and the mean price and the cheaper price.
“The ratio of the quantity of cheaper ingredient to the quantity of dearer ingredient = (Mean Price – Cheaper Price) : (Dearer Price – Mean Price).”
The Process of Successive Dilution
Often, competitive exams include questions about replacing a part of a mixture with a pure solvent. For instance, if you have a container full of milk and you replace a portion with water, the concentration of milk decreases. This is a classic application of the Successive Dilution Formula.
If a container contains x units of a liquid and we remove y units and replace it with water, and this process is repeated n times, the remaining quantity of the original liquid is calculated as:
- Remaining Quantity = x * (1 – y/x)n
This formula is highly effective for solving problems related to wine barrels, milk adulteration, or chemical solutions where the concentration changes over multiple cycles of removal and replacement.
Important Formulas and Comparison Table
When solving UPSC or SSC level problems, speed and accuracy are paramount. Memorizing the standard relationships between quantities and prices will save you significant time during the examination.
| Scenario | Key Formula / Relationship |
|---|---|
| Two Ingredients Mixture | Qcheap / Qdear = (Pdear – Pmean) / (Pmean – Pcheap) |
| Successive Replacement | Final Concentration = Initial * (1 – Replacement/Total)n |
| Weighted Average | Mean = (n1x1 + n2x2) / (n1 + n2) |
Key Points to Remember
- Always ensure that the units are consistent (e.g., all prices should be in Rupees per kg, or all concentrations in percentage).
- The Mean Price must always be a value between the price of the cheaper and the dearer ingredient.
- When dealing with profit/loss, convert the selling price of the mixture to the Cost Price before applying the alligation rule.
- In replacement problems, the total volume of the mixture remains constant after every operation.
- If the question asks for the ratio of the “dearer to the cheaper,” ensure you invert your final calculated ratio accordingly.
- For mixture problems involving three ingredients, break the problem into two parts or use the weighted average method.
Previous Year Question Hints
A common pattern in UPSC CSAT papers involves a vessel containing a mixture of two liquids in a specific ratio. A student might be asked: “A vessel contains 80 liters of milk. 8 liters are withdrawn and replaced by water. This process is repeated twice. What is the quantity of milk left?”
To solve this, apply the replacement formula: 80 * (1 – 8/80)2. This simplifies to 80 * (0.9)2, resulting in 64.8 liters. Identifying the number of repetitions (n) is the most critical step in these logic-based questions.
Quick Revision Summary
- Alligation is essentially a shortcut for the weighted average formula.
- Always calculate the difference diagonally in the cross-method.
- In dilution, the concentration of the original liquid decreases exponentially with each step.
- The ratio obtained from alligation is the ratio of the quantities of the ingredients, not their prices.
- If profit is given, calculate the Cost Price (CP) of the mixture first using the formula: CP = SP * 100 / (100 + Profit%).
- Always verify if the result makes logical sense (i.e., the mean must be between the extremes).
- Use the replacement formula only when the volume replaced is constant in every iteration.
- Practice converting ratios to percentages for faster mental calculations.
