40:51
Alligation and Mixture || LESSON-1 || Removed and Replaced
Feel Free to Learn
Overview
This video introduces the concept of 'alligation and mixture' specifically focusing on problems involving removing and replacing quantities from a mixture. It explains two methods for solving these problems: the traditional step-by-step method and a formula-based approach. The presenter emphasizes that while the formula is quicker for exams, understanding the traditional method is crucial for handling variations and complex problems. Key distinctions are made regarding when the formula is applicable (when the removed and replaced amounts are equal) versus when the traditional method is necessary (when these amounts differ). The video then walks through five example problems, demonstrating both methods where applicable.
How was this?
Save this permanently with flashcards, quizzes, and AI chat
Chapters
- The 'removed and replaced' concept is a common and important topic in aptitude tests for competitive exams.
- Two methods exist: traditional (step-by-step) and formula-based.
- The formula method is faster for exams, but the traditional method is essential for understanding and solving complex variations.
- The formula is only applicable when the quantity removed and the quantity replaced are the same.
Understanding these methods is crucial for efficiently solving quantitative aptitude problems in competitive exams, saving time and improving accuracy.
If 20 liters are removed and 20 liters are replaced, the formula can be used. If 20 liters are removed and 12 liters are replaced, the traditional method must be used.
- The formula is: Final Quantity / Initial Quantity = (1 - x/c)^t
- 'x' represents the quantity of mixture removed.
- 'c' represents the total capacity of the container.
- 't' represents the number of times the removal and replacement process is repeated.
This formula provides a direct and efficient way to calculate the final quantity of a component in a mixture after repeated removal and replacement operations, provided the conditions are met.
For a vessel with capacity 'c', if 'x' liters are removed and replaced 't' times, the final quantity of the original substance can be calculated using the formula.
- The core of the traditional method is accurately calculating the amount of each component (e.g., milk, water) removed in each step.
- This involves determining the ratio of components in the mixture at the time of removal.
- After removal, the replaced liquid is added, and the new quantities and ratios are calculated for the next step.
- This method requires careful tracking of quantities and ratios through each operation.
Mastering the traditional method allows you to solve problems where the formula doesn't apply and deepens your understanding of mixture dynamics.
If 20 liters of an 80-liter mixture (ethanol and water) are removed, you first determine the proportion of ethanol and water in those 20 liters based on the current mixture ratio, then subtract those amounts.
- Initial state: 80 liters of ethanol.
- Process: 20 liters removed, 20 liters water added (repeated twice).
- Traditional method: Calculate ethanol and water after step 1, then recalculate after step 2.
- Formula method: Apply the formula with initial ethanol quantity, x=20, c=80, t=2 to find final ethanol, then subtract from total to get water.
This example demonstrates how to apply both methods when the removed and replaced quantities are equal, highlighting the formula's efficiency for such cases.
After two cycles of removing 20 liters and adding 20 liters of water to an initial 80-liter ethanol vessel, the final mixture contains 45 liters of ethanol and 35 liters of water.
- Initial state: 20 liters of acid.
- Process: 4 liters removed, 4 liters water added (repeated twice).
- Goal: Find the ratio of final acid to initial acid.
- Both methods yield the same result, but the formula is quicker.
This problem reinforces the application of both methods for equal removal and replacement, focusing on calculating ratios.
After two cycles of removing 4 liters and adding 4 liters of water to an initial 20-liter acid vessel, the ratio of acid left to the initial acid is 16:25.
- Initial state: 10 liters of acid.
- Process: 2 liters removed, 2 liters water added (repeated twice).
- Goal: Find the ratio of final acid to initial acid.
- This problem is solvable by both methods as removed and replaced quantities are equal.
This example further solidifies the understanding of the formula and traditional method for equal replacement scenarios.
After two cycles of removing 2 liters and adding 2 liters of water to an initial 10-liter acid vessel, the ratio of acid left to the initial acid is 16:25.
- Initial state: 30 liters of wine and 10 liters of water (total 40 liters).
- Process: 4 liters removed, 4 liters water added (repeated twice).
- Goal: Find the ratio of final wine to initial wine.
- The formula method is demonstrated for efficiency.
This problem illustrates the application of the formula when dealing with a mixture of two substances, emphasizing the calculation of the remaining quantity of one specific component.
In a 40-liter mixture (30L wine, 10L water), after removing 4 liters and adding 4 liters of water twice, the ratio of remaining wine to initial wine is 81:100.
- Initial state: 120 liters milk, 24 liters water.
- Process 1: 12 liters mixture removed, 3 liters water added.
- Process 2: 27 liters mixture removed.
- The formula cannot be used because removed (12L) and replaced (3L) quantities differ.
- The traditional method is required to track component quantities accurately.
This problem highlights the critical limitation of the formula and the necessity of the traditional method when the removed and replaced quantities are unequal.
Starting with 120L milk and 24L water, after removing 12L mixture and adding 3L water, then removing another 27L mixture, the final quantity of water in the jar is 20 liters.
Key takeaways
- The 'removed and replaced' concept in mixtures is vital for quantitative aptitude tests.
- Always check if the removed and replaced quantities are equal before applying the formula.
- The formula Final Quantity / Initial Quantity = (1 - x/c)^t is efficient when removed and replaced amounts are identical.
- The traditional method is essential for problems with unequal removed and replaced quantities or when the formula's conditions aren't met.
- Understanding the ratio of components within the mixture is key to solving traditional method problems.
- Practice is crucial to master both methods and identify which to use for any given problem.
- The formula calculates the final quantity of the substance initially present, not necessarily the substance added later.
Key terms
Alligation and MixtureRemoved and ReplacedTraditional MethodFormula MethodQuantity Removed (x)Container Capacity (c)Number of Operations (t)RatioMixture Components
Test your understanding
- Under what specific condition can the formula for removed and replaced quantities be applied?
- How does the traditional method differ from the formula method in terms of problem-solving approach?
- What information does each variable (x, c, t) represent in the formula Final Quantity / Initial Quantity = (1 - x/c)^t?
- Why is it important to understand the traditional method even if the formula method is faster for exams?
- Describe the steps involved in calculating the amount of each component removed when using the traditional method for a mixture problem.