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.