Solving a Mixture Problem: A Logical Analysis for Maximum Profit

This article delves into a classic mixture problem involving the calculation of the cost price, selling price, and gain to determine the feasibility of a given scenario. We will explore the mathematical solution and provide a logical analysis to understand why certain conditions might not yield a valid solution.

Problem Statement

The original problem states:

How many kilograms of sugar costing Rs. 3 per kg must be mixed with 9 kg of sugar that costs Rs. 7 per kg so that there may be a gain of 10 by selling the mixture at Rs. 9.24 per kg?

Let's denote:

x as the amount of sugar costing Rs. 3 per kg to be mixed. Cost Price (CP) of 1 kg of sugar costing Rs. 3: Rs. 3 Cost Price (CP) of 1 kg of sugar costing Rs. 7: Rs. 7 Selling Price (SP) of 1 kg of mixture: Rs. 9.24

Solution Process

To solve the problem, we first need to consider the cost and selling price of the mixture.

Step 1: Calculate the Cost Price of the Mixture

Given that a gain of 10% is expected, we can calculate the Cost Price (CP) of the mixture:

Let the total Cost Price of the mixture be C. C Cost Price of sugar costing Rs. 3 x Cost Price of sugar costing Rs. 79. C Rs. 3 x Rs. 63.

We know that the selling price SP is given by:

SP C 10% of C SP C 0.1C SP 1.1C

Given that the selling price of 1 kg of the mixture is Rs. 9.24, we can write:

9.24 1.1C C 9.24 / 1.1 C 8.4

Step 2: Substitute Back into the Cost Price Equation

Now we can substitute the values back into the Cost Price equation:

8.4 3x 63 8.4 - 63 3x -54.6 3x x -54.6 / 3 x -18

The negative value for x indicates that the initial assumption may have been incorrect or that the problem may not have a feasible solution with the given conditions. It is possible that a gain of 10% cannot be achieved with the given prices and selling price.

Logical Analysis and Conclusion

The negative value for x suggests that the given numbers might be incorrect. Let's analyze the problem logically:

We have two quantities of sugar costing Rs. 3 and Rs. 7 to start with. The cost of the resultant mixture can only be between these two values of 3 and 7. Even taking the highest value 7, a 10% profit means the sale price per kg is 7.70 only. The question states the sale price is 9.24, which is impossible. When we calculate using the given numbers, we get a negative value for x.

Final Analysis

The problem statement is incorrect because the given prices and selling price do not align with the desired profit margin. The negative value for x is a clear indication that the conditions provided in the problem do not allow for a valid solution.

Summary

This article has shown that solving a mixture problem involves careful consideration of the cost and selling prices. In this specific case, the problem statement is flawed, and a gain of 10% cannot be achieved with the given prices and selling price. Understanding such logical inconsistencies is crucial for applying mathematical concepts accurately in real-world scenarios.