A dealer sold two third of his items at a gain of 20% and the remaining at cost price. Find the gain earned by him in the whole transaction.


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:
\[
\text{Selling price} = \text{Cost price} + 20\% \text{of Cost price}
= 66.67x + 0.2 \times 66.67x = 66.67x + 13.33x = 80x
\]

### 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.

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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.

A dealer sold two third of his items at a gain of 20% and the remaining at cost price. Find the gain earned by him in the whole transaction.

Learn how to calculate the overall percentage gain when a dealer sells two-thirds of his items at a 20% gain and the rest at cost price. This problem breaks down key profit and loss concepts, including how to compute gain percentage, with step-by-step guidance. Suitable for students in Grades 8-10.

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