A trader marked his goods at 20% above the cost price. He sold half the stock at the marked price, one quarter at a discount of 20% on the marked price and the rest at a discount of 40% on the marked price. His total gain percent is:

Let's break down the problem step by step.

### 1. Marked Price and Cost Price Relationship:
The trader marked his goods at 20% above the cost price. Let the **cost price** of the goods be \( C \).

So, the **marked price** (MP) is:
\[
MP = C + 0.2C = 1.2C
\]
Thus, the marked price is 120% of the cost price.

### 2. Calculation of Sales:
The trader sold the stock in three parts:

- **Half the stock** (50%) was sold at the marked price.
- **One quarter** (25%) was sold at a 20% discount on the marked price.
- **The rest** (25%) was sold at a 40% discount on the marked price.

#### a. Half the stock at the marked price:
Since half the stock was sold at the marked price, the selling price for this portion is:
\[
SP_1 = 1.2C \times \frac{1}{2} = 0.6C
\]

#### b. One quarter at a 20% discount:
The selling price for this portion is at a 20% discount on the marked price. Thus, the selling price for this part is:
\[
SP_2 = (1.2C - 0.2 \times 1.2C) \times \frac{1}{4} = 0.8 \times 1.2C \times \frac{1}{4} = 0.24C
\]

#### c. The rest (one quarter) at a 40% discount:
The selling price for this portion is at a 40% discount on the marked price. Thus, the selling price for this part is:
\[
SP_3 = (1.2C - 0.4 \times 1.2C) \times \frac{1}{4} = 0.6 \times 1.2C \times \frac{1}{4} = 0.18C
\]

### 3. Total Selling Price:
The total selling price is the sum of \( SP_1 \), \( SP_2 \), and \( SP_3 \):
\[
SP_{\text{total}} = 0.6C + 0.24C + 0.18C = 1.02C
\]

### 4. Total Gain:
The total gain is the difference between the total selling price and the total cost price. Since the total cost price of all goods is \( C \), the gain is:
\[
\text{Gain} = SP_{\text{total}} - C = 1.02C - C = 0.02C
\]

### 5. Gain Percent:
The gain percent is the ratio of the gain to the cost price, multiplied by 100:
\[
\text{Gain Percent} = \left( \frac{0.02C}{C} \right) \times 100 = 2\%
\]

Thus, the trader's total gain percent is **2%**.

Would you like any more details or further clarification?

Here are 5 related questions you could explore:
1. What would the gain percent be if he sold everything at a 30% discount on the marked price?
2. How would the total profit change if the goods were marked at 25% above the cost price?
3. What would happen if the trader sold the remaining stock at cost price instead of a discount?
4. How does changing the discount percentages affect the overall gain?
5. What is the loss percent if the trader sells all the goods at 50% below the marked price?

**Tip:** When calculating profits, always consider the cost price as your baseline to avoid confusion between discounts and marked prices.

Learn how to calculate the total gain percent when a trader marks his goods 20% above the cost price and sells them in different portions with varying discounts. This problem covers key concepts such as profit, loss, marked price, and selling price. Suitable for Grades 9-12 students.

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