Introduction to Mixtures and Alligations for RRB Exams
Mixtures and Alligations is a vital topic in the Quantitative Aptitude section for RRB NTPC, Group D, and Technician exams. It provides a shortcut method to solve problems involving two or more ingredients mixed in specific ratios. Mastering this technique helps you solve complex arithmetic questions—ranging from simple ratio problems to advanced profit, loss, and interest calculations—in seconds.
Topic Weightage and Importance
In RRB exams, candidates can expect 2 to 3 questions directly or indirectly related to Mixtures and Alligations. Because these problems often appear in the form of word problems (e.g., milk and water, spirit and alcohol), it is considered a high-weightage topic. Proficiency here significantly reduces the time spent on time-consuming calculations.
Key Concepts and Formulas
The Alligation rule is based on the weighted average concept. It states that if two ingredients are mixed, the ratio of the quantities is given by the cross-difference of the unit values.
The Formula:
| Ingredient 1 (Cheaper) | Ingredient 2 (Dearer) |
|---|---|
| Value 1 (C) | Value 2 (D) |
| (D - Mean) | (Mean - C) |
The ratio of quantities of the two ingredients is: (D - Mean) : (Mean - C).
The Replacement Formula
When a quantity is replaced repeatedly, the remaining amount is: Remaining amount = Initial Quantity * (1 - Removed/Initial)^n, where 'n' is the number of operations.
Solved Examples (Step-by-Step)
Example 1: Basic Mixing
In what ratio must a grocer mix tea at Rs. 60/kg and Rs. 100/kg so that the mixture is worth Rs. 75/kg?
Step 1: Identify the components: C=60, D=100, Mean=75.
Step 2: Calculate (D-Mean) = 100 - 75 = 25. Calculate (Mean-C) = 75 - 60 = 15.
Step 3: Ratio = 25:15 = 5:3. Answer is 5:3.
Example 2: Milk and Water
A container contains 40 liters of milk. 4 liters are removed and replaced with water. If this process is repeated twice more, how much milk remains?
Step 1: Initial = 40, Removed = 4, n=3 (initial + 2 repeats).
Step 2: Remaining = 40 * (1 - 4/40)^3 = 40 * (0.9)^3.
Step 3: 40 * 0.729 = 29.16 liters.
Common Mistakes to Avoid
- Mixing up unit values: Always ensure all values are in the same unit (e.g., price per kg).
- Ignoring the 'Mean' value position: The mean must always be higher than the cheaper value and lower than the dearer value.
- Forgetting the number of operations in replacement problems: Always count the initial operation if stated as 'repeatedly'.
Practice Questions with Solutions
Q1. How much water should be added to 15L of milk at Rs. 20/L to make the mixture worth Rs. 15/L? Ans: 5L.
Q2. Two vessels contain milk and water in ratios 3:2 and 2:3. In what ratio must they be mixed to get a 1:1 mixture? Ans: 1:1.
Q3. A mixture of 80L contains milk and water in 7:3. How much water must be added to make it 2:1? Ans: 4L.
Q4. A man mixes 20kg of rice at Rs. 15/kg with 30kg of rice at Rs. 20/kg. What is the cost price of the mixture? Ans: Rs. 18/kg.
Frequently Asked Questions (FAQs)
Q: Is Alligation useful for Profit and Loss? Yes, it is excellent for finding the ratio of goods bought at two different profit percentages.
Q: Does the Alligation rule apply to percentages? Yes, the logic remains identical.
Conclusion and Final Tips
Mixtures and Alligations is a powerful tool for your RRB preparation. Practice the replacement formula regularly as it is a common stumbling block. Stay consistent, practice daily, and you will certainly clear the cutoff!
