Mixing Milk and Water: A Complex Mixture Problem

Diluting and reconstituting mixtures is a common problem in many industries, including dairy. Understanding how to solve these problems is vital for ensuring the quality of the final product. This article will walk you through solving a complex mixture problem involving milk and water, demonstrating the step-by-step process and final solution.

Problem Statement

A milkman mixed 30 liters of water with 150 liters of milk. After mixing, he replaced 35 liters of the mixture with milk. The final quantity of milk in the mixture is 160 liters. We need to find the value of y.

Solution

To solve this problem, we will follow a systematic approach:

Step 1: Determine the Total Volume After Adding Water

Initially, the milkman has 150 liters of milk. He adds 30 liters of water:

Total Volume 150 30Y

Step 2: Calculate the Mixture After Removing 35 Liters

When 35 liters of the mixture is removed, the proportion of milk in the mixture before removal is:

Proportion of Milk 150 / (180 30Y)

The amount of milk removed with 35 liters of the mixture is:

Milk Removed 35 * (150 / (180 30Y))

Step 3: Calculate the Remaining Quantity of Milk

The remaining quantity of milk after removing 35 liters of the mixture is:

Remaining Milk 150 - 35 * (150 / (180 30Y))

Step 4: Add Back the 35 Liters of Milk

After removing 35 liters of the mixture, the milkman adds back 35 liters of milk. The final quantity of milk is:

Final Milk Quantity 150 - 35 * (150 / (180 30Y)) 35

Step 5: Set the Final Quantity of Milk Equal to 160 Liters

Setting the final quantity of milk equal to 160 liters, we have:

150 - 35 * (150 / (180 30Y)) 35 160

Step 6: Solve for y

Rearranging the equation:

150 - 35 * (150 / (180 30Y)) 35 160

185 - 35 * (150 / (180 30Y)) 160

185 - 160 35 * (150 / (180 30Y))

25 35 * (150 / (180 30Y))

Dividing both sides by 35:

25 / 35 150 / (180 30Y)

Simplifying the fraction:

5 / 7 150 / (180 30Y)

Cross multiplying:

5 * (180 30Y) 7 * 150

900 150Y 1050

150Y 1050 - 900

150Y 150

Y 30

Conclusion

The value of y is 30. This means that the milkman initially added 30 liters of water to the 150 liters of milk to achieve the desired final quantity of 160 liters of milk.

To summarize:

Final Quantity of Milk