Equal Proportions of Milk and Water: A Mathematical Approach

Understanding the mathematical principles behind mixture problems is crucial for various fields, including chemistry, engineering, and even cooking. This article delves into a specific problem where we need to adjust a milk and water mixture to achieve equal proportions of the two components. We will provide a detailed step-by-step solution, accompanied by an explanation of the algebraic methods used.

Problem Description

We start with a 24 liters of milk and water mixture that contains milk and water in a ratio of 3:5. The question is: how much of this mixture should be removed and replaced with pure milk to achieve an equal proportion of milk and water in the final mixture?

Step-by-Step Solution

Step 1: Calculate Initial Amounts

The total parts in the ratio 3:5 is 3 5 8 parts.

- Amount of milk: Milk ( frac{3}{8} times 24 9 ) liters

- Amount of water: Water ( frac{5}{8} times 24 15 ) liters

Step 2: Define the Amount to be Removed

Let x be the amount of the mixture (in liters) to be removed and replaced with pure milk.

- Milk removed: Milk removed ( frac{3}{8}x ) liters

- Water removed: Water removed ( frac{5}{8}x ) liters

Step 3: Calculate Remaining Amounts After Removal

- Remaining milk: Remaining milk 9 - ( frac{3}{8}x ) liters

- Remaining water: Remaining water 15 - ( frac{5}{8}x ) liters

Step 4: Add x Liters of Pure Milk

- New amount of milk: New milk 9 - ( frac{3}{8}x ) x 9 ( frac{5}{8}x ) liters

- Amount of water remains the same: New water 15 - ( frac{5}{8}x ) liters

Step 5: Set Up the Equation for Equal Proportions

We want milk and water in equal proportions:

9 ( frac{5}{8}x ) 15 - ( frac{5}{8}x )

Step 6: Solve the Equation

Combine like terms:

( 9 frac{5}{8}x - frac{5}{8}x 15 )

( 9 15 - frac{10}{8}x )

( 9 15 - frac{5}{4}x )

Isolate x:

( frac{5}{4}x 15 - 9 )

( frac{5}{4}x 6 )

( x 6 times frac{4}{5} frac{24}{5} 4.8 ) liters

Conclusion

To achieve equal proportions of milk and water in the final mixture, 4.8 liters of the mixture should be taken out and replaced with pure milk.

Algebraic Equations for Milk and Water

As an additional exercise, let's solve the problem using the provided algebraic equations.

Milk Equation

( frac{1}{2} frac{43}{52a} - frac{43}{52a}X )

( X frac{9}{86} )

Water Equation

( frac{1}{2} frac{14}{43a} - frac{14}{43a}X )

( X frac{7}{29} )

We can see that the algebraic equations provide different values for X, indicating that the initial conditions or approach might need to be reconsidered for consistency.