Mixing Ratios and Quantities: A Mathematical Investigation

This article delves into the intricacies of mixing ratios and quantities through a detailed mathematical problem. We will explore the methods used to solve such problems, providing insights and solutions that are relevant for students and professionals alike. Together, we will break down a complex problem involving the mixing of milk and water from two bottles into a final mixture.

Understanding the Problem

The problem at hand involves two bottles, A and B, containing a mixture of milk and water in specific ratios. Bottle A has a ratio of milk to water of 3:2, while Bottle B has a ratio of 3:1. These two bottles are then mixed in a certain ratio, resulting in a final mixture of 144 ml of milk and 72 ml of water. The question is to find the initial quantity of the mixture in Bottle A.

Educational Significance

Mixing ratios and quantities are essential concepts in a variety of fields, including engineering, chemistry, and food science. Understanding how to solve such problems is crucial for students and professionals working in related fields. This problem serves as a practical application of the principles of mixture and alligation, which are fundamental concepts in algebra.

The Step-by-Step Solution

To solve this problem, we need to follow a systematic approach. Let's break down the solution into manageable steps:

Step 1: Determine the Ratios in Each Bottle

Bottle A: Milk to water ratio is 3:2. Bottle B: Milk to water ratio is 3:1.

This means that in each bottle, the quantities of milk and water can be represented as fractions. For Bottle A, the milk is 3/5 of the total, and water is 2/5. For Bottle B, the milk is 3/4 of the total, and water is 1/4.

Step 2: Understand the Final Mixture Composition

The final mixture is composed of 144 ml of milk and 72 ml of water. The ratio of milk to water in the final mixture is 144:72, which simplifies to 2:1. This means that for every 3 parts of the final mixture, 2 parts are milk, and 1 part is water.

Step 3: Use a System to Solve for Quantities

Let's assume the quantity of milk and water taken from Bottle A is 3x and 2x, respectively. For Bottle B, let it be 3y and y, respectively. The final mixture consists of these quantities mixed in the ratio 3:2, with 144 ml of milk and 72 ml of water.

Using the given final mixture ratio (2:1), we can write the following equations:

Milk equation: 3x 3y 144 Water equation: 2x y 72

From the water equation, we can express y in terms of x:

y 72 - 2x

Substitute this into the milk equation:

3x 3(72 - 2x) 144

3x 216 - 6x 144

-3x -72

x 24

Now, we can find y:

y 72 - 2*24 72 - 48 24

So, the initial quantities taken from each bottle are:

Bottle A: 3x 3*24 72 ml of milk and 2x 2*24 48 ml of water, totaling 120 ml. Bottle B: 3y 3*24 72 ml of milk and y 24 ml of water, totaling 96 ml.

Now, if the two bottles are mixed in the ratio 3:2 to produce a 144 ml and 72 ml final mixture, we need to find the initial quantity:

Step 4: Calculate the Initial Quantity

The given information suggests that the final mixture of 144 ml and 72 ml is formed by mixing the two bottles in a certain ratio. Since the final mixture is a combination of milk and water, let's find the initial quantity:

The total final volume is 216 ml (144 ml of milk 72 ml of water), and it is formed by mixing the two bottles in the ratio 3:2. This means that the total volume of liquids taken from each bottle initially must be consistent with this ratio.

The ratio 3:2 means that for every 5 parts, 3 parts are from Bottle A and 2 parts are from Bottle B. Therefore, if the total volume of the final mixture is 216 ml, we can distribute this volume according to the ratio:

Total initial volume in Bottle A (3/5) * 216 129.6 ml

Total initial volume in Bottle B (2/5) * 216 86.4 ml

Conclusion

In conclusion, the initial quantity of the mixture in Bottle A is 129.6 ml, and in Bottle B, it is 86.4 ml. This in-depth analysis not only helps in solving the problem but also highlights the importance of understanding ratios and their applications in real-world scenarios. By following a systematic approach, we have successfully determined the initial quantities, providing a clear and comprehensive solution.

Key Takeaways

Mastering the principles of ratios and proportions can significantly enhance problem-solving skills in mathematics and practical applications. Understanding the concept of alligation and how to apply it to solve mixture problems is crucial. Consistent practice and application of these principles will improve your ability to solve complex problems involving mixtures.

Additional Resources

For more insights and solutions to similar problems, refer to the following resources:

Ratio and Proportion - Math is Fun Systems of Ratio Word Problems - Khan Academy Exam Solutions