Deductive Reasoning in Profit Calculations: A Case Study on Cost Price Ratio and Profit Distribution

In the vast realm of business and mathematical problem solving, few questions are as intriguing as those concerning the interplay between cost price, profit, and the associated financial gains. This detailed exploration revolves around a classic example, illustrating the application of algebraic reasoning and ratio analysis to solve a problem involving cost price and profit distribution.

The Problem at Hand

Consider a scenario where two products, A and B, are sold. The ratio of the cost prices of A and B is 5:7, indicating that from a proportional standpoint, the cost of B is 7/5 times that of A. Moreover, A is sold at a profit of Rs 80, while B is sold at a profit of Rs 20. The total profit earned on selling both products is Rs 296. The question is, what is the difference in the cost prices of A and B?

Step-by-Step Solution

To solve this problem, we will follow a structured approach, employing algebraic equations to derive the necessary values.

Step 1: Denoting Cost Prices Using Variables

Let the cost price of A be 5x.
Let the cost price of B be 7x.

Step 2: Calculating Selling Prices

For product A:
Cost Price of A 5x
Profit of A 80% of 5x 0.8 × 5x 4x
Selling Price of A Cost Price Profit 5x 4x 9x For product B:
Cost Price of B 7x
Profit of B 20% of 7x 0.2 × 7x 1.4x
Selling Price of B Cost Price Profit 7x 1.4x 8.4x

Step 3: Calculating Total Profit

The total selling price for both products is:
Total Selling Price Selling Price of A Selling Price of B 9x 8.4x 17.4x

The total cost price for both products is:
Total Cost Price Cost Price of A Cost Price of B 5x 7x 12x

Step 4: Determining Total Profit Earnings

Total Profit Total Selling Price - Total Cost Price 17.4x - 12x 5.4x

Given the total profit is Rs 296, we have:
5.4x 296

Step 5: Solving for x

x 296 / 5.4 ≈ 54.74

Step 6: Calculating Cost Prices

Cost Price of A 5x 5 × 54.74 ≈ 273.70 Cost Price of B 7x 7 × 54.74 ≈ 383.18

Step 7: Finding the Difference Between Cost Prices

The difference between the cost prices of A and B is:

Difference Cost Price of B - Cost Price of A ≈ 383.18 - 273.70 ≈ 109.48

Conclusion

The difference between the cost price of A and B is approximately Rs 109.48.

This method of solving problems using algebraic equations is not only a testament to the power of mathematical reasoning but also a practical tool for businesses to assess and optimize their product pricing strategies. Understanding such calculations can help in making informed decisions and optimizing profit margins.