Let's solve this step by step.

Given:

• A dealer sold $\frac{2}{3}$ of his items at a gain of 20%.
• The remaining $\frac{1}{3}$ of the items were sold at cost price.
• We are tasked with finding the overall gain in the whole transaction.

Step 1: Assume the cost price (C.P.) of all the items is $100x$ units (just to make calculations easier).

So:

• The cost price of $\frac{2}{3}$ of the items is $\frac{2}{3} \times 100x = 66.67x$.
• The cost price of $\frac{1}{3}$ of the items is $\frac{1}{3} \times 100x = 33.33x$.

Step 2: Selling price of the items sold at a gain of 20%

The selling price for $\frac{2}{3}$ of the items, sold at a gain of 20%, will be:

= 66.67x + 0.2 \times 66.67x = 66.67x + 13.33x = 80x$$ ### Step 3: Selling price of the items sold at cost price The selling price for $$\frac{1}{3}$$ of the items, sold at cost price, is equal to the cost price itself: $$\text{Selling price} = 33.33x$$ ### Step 4: Total selling price The total selling price from both parts is: $$80x + 33.33x = 113.33x$$ ### Step 5: Total cost price The total cost price of all the items is $$100x$$. ### Step 6: Gain The gain is calculated as the difference between the total selling price and the total cost price: $$\text{Gain} = \text{Total selling price} - \text{Total cost price} = 113.33x - 100x = 13.33x$$ ### Step 7: Percentage gain The percentage gain is given by: $$\text{Percentage gain} = \frac{\text{Gain}}{\text{Cost price}} \times 100 = \frac{13.33x}{100x} \times 100 = 13.33\%$$ ### Final Answer: The dealer earned a **13.33% gain** in the whole transaction. --- Would you like further details on any step? Here are 5 related questions: 1. What is the difference between cost price and selling price in profit calculations? 2. How do we calculate the percentage gain or loss in a transaction? 3. What happens to the overall gain if the remaining items are sold at a loss? 4. Can the gain formula be derived using algebraic variables instead of percentages? 5. How does selling part of the goods at a loss affect the overall transaction? **Tip**: Always break down complex problems into smaller parts for easier and more accurate calculations.