Points to Remember:
- The initial ratio of milk to water is 7:5.
- 9 liters of the mixture are replaced with 9 liters of water.
- The final ratio of milk to water is 7:9.
- We need to find the initial quantity of milk.
Introduction:
This is a problem involving ratios and proportions. It requires a step-by-step approach to solve for the unknown quantity of milk initially present in the mixture. We will use algebraic methods to solve this problem. Problems of this type are common in mathematics and have applications in various fields, such as chemistry (mixing solutions) and finance (calculating proportions of investments).
Body:
1. Understanding the Initial Mixture:
Let’s assume the initial quantity of milk is 7x liters and the initial quantity of water is 5x liters. The total quantity of the mixture is therefore 7x + 5x = 12x liters.
2. The Replacement Process:
When 9 liters of the mixture are removed, the proportion of milk and water removed remains the same as the initial mixture (7:5). Therefore, the amount of milk removed is (7/12) * 9 = 5.25 liters, and the amount of water removed is (5/12) * 9 = 3.75 liters.
3. The Mixture After Replacement:
After removing 9 liters of the mixture, the remaining milk is 7x – 5.25 liters, and the remaining water is 5x – 3.75 liters. Then, 9 liters of water are added. This means the new quantity of water is (5x – 3.75) + 9 = 5x + 5.25 liters.
4. The Final Ratio:
The final ratio of milk to water is given as 7:9. Therefore, we can set up the equation:
(7x – 5.25) / (5x + 5.25) = 7/9
5. Solving for x:
Cross-multiplying the equation, we get:
9(7x – 5.25) = 7(5x + 5.25)
63x – 47.25 = 35x + 36.75
28x = 84
x = 3
6. Calculating the Initial Quantity of Milk:
The initial quantity of milk was 7x liters. Substituting x = 3, we get:
Initial quantity of milk = 7 * 3 = 21 liters
Conclusion:
Therefore, there were initially 21 liters of milk in the bucket. This problem highlights the importance of understanding ratios and proportions and applying algebraic techniques to solve for unknown variables. Similar problems can be encountered in various real-world scenarios requiring precise calculations involving mixtures or proportions. A clear understanding of these concepts is crucial for accurate problem-solving in many fields. The solution demonstrates a systematic approach to solving such problems, emphasizing the importance of breaking down complex problems into smaller, manageable steps. This approach ensures accuracy and promotes a deeper understanding of the underlying mathematical principles.
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