Proportioning Milk and Water to Achieve Desired Profit

When a milk seller aims to achieve a specific profit margin by mixing milk with free water and selling the mixture at the cost price of milk, understanding the correct proportion is crucial. This article explores the method for adding water to 15 liters of pure milk to achieve a 50% profit, as well as the steps to calculate the required volume of water for a 60% profit margin.

Understanding the Problem

The initial problem involves a milk seller with 15 liters of pure milk, aiming to achieve a 50% profit by selling the mixture at the cost price of pure milk. This requires a precise calculation of the volume of water to be added to the milk.

Calculation for 50% Profit

To achieve a 50% profit, the selling price of the mixture should be 1.5 times the cost price of the pure milk. Here’s a detailed breakdown of the steps involved:

Determining the Cost and Selling Prices

Assume the cost price (CP) of 1 liter of pure milk is C.

The cost price of 15 liters of pure milk is 15C. To achieve a 50% profit, the selling price (SP) of the mixture must be 1.5 times the cost price of pure milk.

Calculating the Total Selling Price

The selling price SP for 15 liters of pure milk is:

SP 1.5 times; 15C 22.5C.

Mixing Milk with Water

Let M and W denote the volumes of pure milk and water, respectively, to be mixed to achieve the 50% profit.

The total volume of the mixture after adding x liters of water is 15 x liters. Selling this mixture at the cost price of pure milk:

SP (15 x)C.

Setting Up the Equation

To achieve the desired profit:

(15 x)C 22.5C.

Solving for x:

15 x 22.5

x 22.5 - 15 7.5

Therefore, to achieve a 50% profit, the milk seller must add 7.5 liters of water to the pure milk.

Calculation for 60% Profit

If the seller wants to achieve a 60% profit, the selling price should be 160% of the cost price. Since he doesn't want to increase the selling price, the quantity of pure milk needs to be increased to 160% of the original volume, i.e., 15 liters multiplied by 1.6, which equals 24 liters.

Thus, he needs to add:

24 - 15 9 liters of water.

Conclusion

By understanding the relationship between the cost price, selling price, and the desired profit margin, the milk seller can accurately mix milk with water to maximize his earnings without increasing the selling price. For a 50% profit, add 7.5 liters of water, and for a 60% profit, add 9 liters of water to 15 liters of pure milk.