Adjusting Ratios in Milk and Water Mixtures: A Detailed Guide

The process of adjusting the ratio of milk to water in a mixture is a fundamental problem often encountered in both practical and academic settings. This article delves into the steps and methods to solve such problems using a practical example involving a 729-liter mixture of milk and water with a current ratio of 7:2. Additionally, we will explore another example to further clarify the concept and solution methodology.

The First Scenario: 729 Liters of Milk and Water in a 7:2 Ratio

Consider a 729-liter mixture of milk and water with a ratio of 7:2. We need to determine how much water needs to be added to achieve a new mixture with a ratio of 7:3.

Step 1: Calculate the Initial Amounts of Milk and Water

Total parts in the initial ratio 7:2 7 2 9 parts. Amount of milk (7/9) × 729 567 liters. Amount of water (2/9) × 729 162 liters.

Step 2: Determine the New Desired Ratio

We aim to achieve a new mixture with a ratio of 7:3. Let x be the amount of water to be added.

Step 3: Set Up the Equation for the New Mixture

After adding x liters of water, the new amount of water becomes 162 x liters. The amount of milk remains 567 liters. The new ratio of milk to water is: (567 / (162 x) 7 / 3)

Step 4: Cross-Multiply to Solve for x

Cross-multiplying gives: 567 × 3 7 × (162 x) Calculating the left side: 1701 7 × (162 x) Expanding the right side: 1701 1134 7x

Step 5: Solve for x

Isolating x: 1701 - 1134 7x 567 7x x 567 / 7 81

Conclusion: To achieve a new mixture containing milk and water in the ratio of 7:3, 81 liters of water need to be added.

The Second Scenario: 40 Liters of Milk and Water in a 7:1 Ratio

Consider another example where a 40-liter mixture of milk and water has a 7:1 ratio. We need to determine how much water should be added to achieve a new mixture with a 3:1 ratio.

Step 1: Calculate the Initial Amounts of Milk and Water

Taking the ratio 7:1, we can calculate the amount of milk and water in the mixture. Amount of milk (7/8) × 40 L 35 L. Amount of water (1/8) × 40 L 5 L.

Step 2: Determine the New Desired Ratio

Now, let W denote the amount of water to be added.

Step 3: Set Up the Equation for the New Mixture

Adding W liters of water results in a new mixture with a 3:1 ratio: 35 / (5 W) 3 / 1

Cross-multiplying gives:

35 × 1 3 × (5 W)

Further simplifying:

35 15 3W

Isolating W:

35 - 15 3W 20 3W W 20 / 3

Therefore, the amount of water to be added is approximately 6.67 liters.

Conclusion

Both examples demonstrate the process of adjusting the ratio of milk to water in mixtures. The first example involved a larger quantity with a more straightforward ratio calculation, while the second example showcased the application of the concept to solve practical problems. Understanding these methods is essential for professionals and students in chemistry, biology, and related fields.

Keywords: mixture ratio, milk and water, solution problems