Mixing Milk and Water Mixture for Desired Milk Content

Understanding how to mix different concentrations of milk and water to achieve a specific milk content is a fundamental skill in many fields, including culinary arts, pharmacy, and chemical engineering. This article explores the problem of mixing two milk mixtures with different milk concentrations to achieve a desired concentration, providing a detailed solution and method to reach the final answer.

Problem Definition

Consider two mixtures of milk and water, one containing 40% milk and the other containing 80% milk. We aim to determine the ratio in which these two mixtures should be mixed to obtain a new mixture containing 50% milk. Let's break down the problem step by step.

Setting Up the Problem

Let the volume of the 80% milk mixture be x kg. Let the volume of the 40% milk mixture be 1 kg, as that was initially given in the problem.

The total volume of the mixture formed will be x 1 kg.

The amount of milk in the 80% mixture is 0.8x kg.

The amount of milk in the 40% mixture is 0.4 kg.

Formulating the Equation

The total amount of milk in the new mixture is 0.5 times the total volume, which is 0.5(x 1) kg.

We set up the equation based on the total milk in the new mixture:

0.8x 0.4 0.5(x 1)

Solving for x

Expand the right side of the equation:

0.8x 0.4 0.5x 0.5

Subtract 0.5x from both sides:

0.3x 0.4 0.5

Subtract 0.4 from both sides:

0.3x 0.1

Solve for x by dividing both sides by 0.3:

x 1/3 or approximately 0.333 kg

Determining the Ratio

The ratio of the 40% milk mixture to the 80% milk mixture is thus 1 : 0.333, or in its simplest form, 3 : 1.

Alternative Approach

Individual Volume Ratios

Define the volumes taken from mixture A (40% milk) as a liters and from mixture B (80% milk) as b liters.

The amount of milk in each part is:

Milk from A: 40/100 * a 2a/5

Milk from B: 80/100 * b 4b/5

The total milk in the new mixture should be 50% of the total volume (a b):

(2a/5 4b/5) / (a b) 50/100

Formulating and Solving the Equation

Multiply both sides by (a b) to eliminate the denominator:

2a 4b 0.5(a b)

Multiply through by 2 to clear the decimal:

4a 8b a b

Subtract (a b) from both sides:

3a -7b

Since b cannot be negative, we can simplify to:

a 3b

Therefore, the ratio of a to b is 3:1, which confirms the previous result.

Conclusion

Whether you use the direct method or the alternative approach, the ratio of the 40% milk mixture to the 80% milk mixture that results in a 50% milk content is 3:1. This means that three parts of the 40% mixture should be mixed with one part of the 80% mixture to achieve the desired concentration.