Optimizing Profit from Orange Sales: A Mathematical Insight

In the complex realm of retail, maximizing profits often requires a keen understanding of mathematical principles. The scenario of selling oranges to achieve a desired profit margin is a classic application of these principles. By analyzing the given examples, we can derive practical insights that can be used in real-world business scenarios.

Example 1: Loss of 40 Rupees

A vendor sells oranges at the rate of 34 for a rupee and incurs a loss of 40 rupees. To understand the situation, we first calculate the cost price (CP) per orange. Given that 34 oranges cost 1 rupee, the cost price for one orange would be:

CP per orange 1 / 34 0.0294 rupees

Now, let's determine the price per orange to achieve a 20% profit. To achieve a 20% profit, the selling price (SP) per orange should be:

SP CP × (1 Profit Percentage)

SP 0.0294 × 1.20 0.03528 rupees

Therefore, to gain a 20% profit, the vendor should sell each orange for approximately 0.03528 rupees. However, the practical approach would be to sell 8 oranges for 1 rupee to achieve the desired profit margin:

No of oranges 12 / (1.00 / 0.03528) 8

Example 2: Gain of 20 Rupees

Another vendor sells 32 oranges at a loss of 40 rupees. To find the selling price per orange that would result in a 20% gain, we first calculate the cost price for 32 oranges. Given that the loss is 40 rupees:

Cost price for 32 oranges 40 / (1 - 0.20) 50 rupees

Next, to achieve a 20% gain, the selling price for 32 oranges should be:

SP for 32 oranges 50 × 1.20 60 rupees

Finally, to find the price per orange:

SP per orange 60 / 32 1.875 rupees

Therefore, to gain a 20% profit, the vendor should sell each orange for approximately 1.875 rupees, which translates to 8 oranges for 1 rupee:

Number of oranges 32 / (1.875 / 1) 8

Mathematical Derivation

The core of the problem involves determining the selling price per unit (orange) to achieve a specific profit margin. Below is the step-by-step mathematical derivation:

1. Loss Cost Price - Selling Price

2. Gain Selling Price - Cost Price

3. To calculate the number of oranges to be sold for 1 rupee to achieve a 20% profit, we use:

SP per orange CP × (1 Profit Percentage)

4. Finally, the number of oranges per rupee is given by:

Number of oranges Total Cost Price / (Total Selling Price × (1 Profit Percentage))

By applying these principles, vendors can optimize their pricing strategies to enhance profitability.

Summary

Understanding the principles of cost and selling price is crucial for optimizing profit margins in the retail sector. Through careful calculations and practical applications, vendors can determine the optimal selling price to achieve desired profit margins, leading to increased profitability.

Conclusion

The examples provided illustrate the practical application of mathematical principles in real-world business scenarios. By leveraging these insights, merchants can make informed decisions that lead to enhanced profitability. Whether it's loss optimization or profit maximization, a deep understanding of cost and selling price is invaluable.