Profit Maximization through Orange Purchase and Sale

In the field of business and retail, understanding the principles of cost price and selling price is essential for maximizing profits. This article explores a scenario involving the purchase and sale of oranges, illustrating how to achieve a desired profit percentage using mathematical calculations.

Purchase and Cost Calculations

A man buys a certain number of oranges at different rates: some at 3 for a rupee and an equal number at 2 for a rupee. To determine the selling price per dozen that would yield a desired 20% profit, we first calculate the cost price per dozen.

Step-by-Step Calculation

Identify the number of oranges purchased: Let's assume he buys an equal number of oranges in both cases, which we will simplify to 100 for ease of calculation. Calculate the cost price (CP) for each set of oranges: For 100 oranges at 3 for a rupee: 100 ÷ 3 33.33, hence the CP is 33.33 rupees. For the other 100 oranges at 2 for a rupee: 100 ÷ 2 50, hence the CP is 50 rupees. Combine the costs:

Total CP for 200 oranges 33.33 50 83.33 rupees.

Profit Calculation

To achieve a 20% profit, the total selling price (SP) should be 120% of the CP.

Mathematical Calculation

Calculate the required total SP:

Total SP 120% of 83.33 rupees 1.20 × 83.33 100 rupees.

Calculate the SP per dozen:

Since 200 oranges are equivalent to 16.67 dozens (200 ÷ 12), the SP per dozen 100 ÷ 16.67 ≈ 6 rupees.

Alternative Calculation

Let’s consider a simpler approach by using a smaller, even number for ease of mental calculation, like 100.

Alternative Method

Assume 100 oranges for each rate: For 100 oranges at 50 for a rupee: 100 ÷ 50 2 rupees. For 100 oranges at 20 for a rupee: 100 ÷ 20 5 rupees. Total cost for 200 oranges 2 5 7 rupees. To achieve 20% profit:

Total SP 1.20 × 7 8.4 rupees.

SP per dozen 8.4 ÷ 200 × 12 0.48 rupees.

Generalized Formula for Future Reference

For a more generalized approach, if the number of oranges purchased on each occasion is N:

Generalized Calculation

Total cost price (CP) for 2N oranges:

Total CP N/50 N/20 7N/100.

Required SP to achieve 20% profit:

Total SP 1.20 × CP 1.20 × 7N/100 8.4N/100.

SP per dozen:

SP per dozen 8.4N/100 ÷ 200 × 12 12 × 8.4N/100 × 1/200 8.4N/2000.

By applying the above formula, the man can ensure a 20% profit on the cost of 2N oranges.

This method simplifies the process of determining the selling price per dozen, making it mentally simpler and easier to apply in similar scenarios.