Solving a Complex Ratio Problem: Milk and Water Mixture

This article explores a classic problem in mathematics and chemistry: the mixture of milk and water. Specifically, it presents a complex ratio problem and provides a detailed explanation of how to solve it using algebraic principles. The problem not only exercises basic algebraic skills but also provides insights into practical applications of mathematics.

Problem Statement

Imagine you have a mixture of 20 kg of milk and water, in which 90% of the content is milk. This problem requires us to determine how many kilograms of water should be added to the mixture so that the water comprises 40% of the mixture.

Initial Condition and Key Observations

Let's begin by understanding the initial condition of the mixture. The total weight of the mixture is 20 kg, of which 90% is milk and the remaining 10% is water.

Total mixture: 20 kg Milk: 90% of 20 kg 18 kg Water: 10% of 20 kg 2 kg

Algebraic Solution

To solve this problem, let's define (x) as the amount of water to be added. Once water is added, the total weight of the mixture will increase to (20 x) kg. We need to find (x) such that the water comprises 40% of this new mixture.

Setting Up the Equation

After adding (x) kg of water, the total weight of the mixture will be (20 x) kg. The original 2 kg of water will now be a part of this mixture, and it needs to make up 40% of the new mixture. We can set up the following equation:

[ frac{2 x}{20 x} 0.4 ]

Solving the Equation

Now, let's solve this equation step by step.

[ 2 x 0.4(20 x) ]

[ 2 x 8 0.4x ]

[ x - 0.4x 8 - 2 ]

[ 0.6x 6 ]

[ x frac{6}{0.6} ]

[ x 10 ]

Therefore, 10 kg of water should be added to the mixture to achieve the desired 40% water content.

Conclusion and Practical Application

This problem is not just an academic exercise; it has real-world applications in various fields, including food processing, chemical engineering, and environmental sciences. Understanding how to manipulate ratios and solve complex mixture problems is crucial in many practical scenarios.

For those interested in the mathematical principles behind such problems, the solution demonstrates the simplicity yet power of algebraic methods. It also highlights the importance of translating real-world problems into mathematical terms and solving them systematically.

Further Reading and Resources

Khan Academy - Solving Mixture Problem Using Ratios MathPlanet - Solving Word Problems Using Mixture Purplemath - Ratio and Mixture Word Problems

Explore these resources to deepen your understanding of mixture problems and their broader applications in mathematics and beyond.