Understanding the Problem: Calculating Overall Profit

The question asks us to find the overall profit percentage when Rahul purchases a certain number of items, some of which are defective, and the remaining non-defective items are sold at a profit. We need to calculate the profit based on the total cost price of all items.

Step-by-Step Solution for Profit Calculation

Let's break down the problem and solve it step-by-step:

1. Identify the Total Number of Items: Rahul purchased 80 items.
2. Calculate the Number of Defective Items: 25% of the total items were defective.
   • Number of defective items $= 25\% \text{ of } 80$
   • Number of defective items $= \frac{25}{100} \times 80 = \frac{1}{4} \times 80 = 20$ items.
   These defective items cannot be sold for profit (we assume they have no resale value in this context).
3. Calculate the Number of Non-Defective Items: The remaining items were non-defective.
   • Number of non-defective items $= \text{Total items} - \text{Defective items}$
   • Number of non-defective items $= 80 - 20 = 60$ items.
   These 60 items were sold.
4. Assume a Cost Price (CP) per Item: To make calculations easy, let's assume the cost price of each item is Rs. 1.
   • Total Cost Price of 80 items $= 80 \times \text{CP per item}$
   • Total Cost Price $= 80 \times 1 = \text{Rs. } 80$.
5. Calculate the Selling Price (SP) of Non-Defective Items: The remaining items (60 items) were sold at a 50% profit.
   • Profit percentage on these items $= 50\%$
   • Selling Price per non-defective item $=$ CP per item $+$ Profit per item
   • Profit per item $= 50\% \text{ of CP per item}$
   • Profit per item $= \frac{50}{100} \times 1 = 0.5 \times 1 = \text{Rs. } 0.5$.
   • Selling Price per non-defective item $= 1 + 0.5 = \text{Rs. } 1.5$.
6. Calculate the Total Revenue from Selling Non-Defective Items: Rahul sold 60 non-defective items at Rs. 1.5 each.
   • Total Revenue (Selling Price) $= \text{Number of non-defective items} \times \text{SP per non-defective item}$
   • Total Revenue $= 60 \times 1.5 = \text{Rs. } 90$.
7. Calculate the Overall Profit: The overall profit is the difference between the total revenue from sales and the total cost price of all items purchased.
   • Overall Profit $= \text{Total Revenue} - \text{Total Cost Price}$
   • Overall Profit $= 90 - 80 = \text{Rs. } 10$.
8. Calculate the Overall Profit Percentage: The overall profit percentage is calculated on the total cost price of all 80 items.
   • Overall Profit $\% = \left( \frac{\text{Overall Profit}}{\text{Total Cost Price}} \right) \times 100$
   • Overall Profit $\% = \left( \frac{10}{80} \right) \times 100$
   • Overall Profit $\% = \frac{1}{8} \times 100$
   • Overall Profit $\% = 12.5\%$.

The overall profit percentage on the total purchase is 12.5%.

Summary of Calculations

Revision Table: Key Concepts in Profit and Loss

Additional Information on Profit and Loss Problems

When solving profit and loss problems involving multiple items or different conditions:

• Always calculate profit or loss percentage with respect to the total cost price unless otherwise specified.
• If some items are defective or unsold, their cost price is still included in the total cost price, but they do not contribute to the total selling price.
• Assuming a convenient cost price (like Rs. 1 or Rs. 100) for a single item can simplify calculations, especially when percentages are involved.
• Ensure you distinguish between profit/loss on individual items or a subset of items versus the overall profit/loss on the entire transaction.