A dishonest sugar merchant, while purchasing goods from the dealer, manages to receive 20% more sugar than the quantity of sugar he is supposed to receive. He later sells the sugar to his customers in such a way that for any customer, 30% of the sugar actually delivered must be added to get the original quantity of sugar asked for by the customer. The merchant sells the sugar to his customers at cost price (i.e., the price demanded by the dealer), but then charges them an additional amount claiming it as his transportation cost. If the additional amount that the merchant charges equals 10% of the cost that he incurred on the sugar that he is actually delivering to them, find his net gain percent. (Assume that the merchant's actual transportation cost is zero),

Let's break down the problem step by step to determine the dishonest merchant's net gain percent.

### 1. Understanding the Merchant's Actions:
- **When purchasing:**
  - The merchant receives 20% more sugar than the quantity he pays for. 
  - If the quantity he is supposed to receive is \( Q \), then the quantity he actually receives is:
    \[
    Q' = Q + 0.2Q = 1.2Q
    \]

- **When selling:**
  - The merchant delivers only 30% less sugar than the customer asks for. If a customer asks for \( x \) units of sugar, they actually receive:
    \[
    x' = x - 0.3x = 0.7x
    \]
  - Since \( x' = 0.7x \), it follows that:
    \[
    x = \frac{x'}{0.7} = \frac{10}{7} x'
    \]
  - So, the merchant is selling \( x' \) amount of sugar but claims it's \( \frac{10}{7} x' \).

### 2. Cost and Revenue Calculations:
- **Cost Price Calculation:**
  - The merchant pays for \( Q \) units but receives \( 1.2Q \) units. Therefore, the cost per unit of sugar he actually has is:
    \[
    \text{Effective Cost Price per Unit} = \frac{\text{Cost of } Q \text{ units}}{1.2Q} = \frac{\text{Cost per Unit}}{1.2}
    \]

- **Selling Price and Additional Charge:**
  - He sells \( x' \) units but charges for \( x = \frac{10}{7} x' \) units.
  - The selling price is equivalent to the cost price per unit he incurred, i.e., \( \frac{1}{1.2} \text{ Cost per Unit} \).
  - However, he also charges an additional 10% of this cost, claiming it as a transportation cost:
    \[
    \text{Additional Charge per Unit} = 0.1 \times \frac{\text{Cost per Unit}}{1.2}
    \]

- **Total Selling Price per Unit:**
  - The total amount the merchant charges for \( x' \) units is:
    \[
    \text{Selling Price per Unit} = \frac{\text{Cost per Unit}}{1.2} + 0.1 \times \frac{\text{Cost per Unit}}{1.2} = \frac{1.1 \times \text{Cost per Unit}}{1.2}
    \]

### 3. Profit Calculation:
- **Net Gain:**
  - The cost per unit for the merchant is \( \frac{\text{Cost per Unit}}{1.2} \).
  - The selling price per unit he charges is \( \frac{1.1 \times \text{Cost per Unit}}{1.2} \).
  - Therefore, his profit per unit is:
    \[
    \text{Profit per Unit} = \frac{1.1 \times \text{Cost per Unit}}{1.2} - \frac{\text{Cost per Unit}}{1.2} = \frac{(1.1 - 1) \times \text{Cost per Unit}}{1.2} = \frac{0.1 \times \text{Cost per Unit}}{1.2}
    \]
  - Since the merchant delivers \( x' \) units but charges for \( \frac{10}{7} x' \) units, his profit percentage can be calculated by considering the effective gain over his cost.

### 4. Net Gain Percent Calculation:
  - The net gain percent is:
    \[
    \text{Net Gain Percent} = \left(\frac{\text{Total Profit}}{\text{Cost}} \right) \times 100
    \]
    \[
    \text{Net Gain Percent} = \left(\frac{\frac{0.1 \times \text{Cost per Unit}}{1.2} \times 1.2}{\text{Cost per Unit}} \right) \times 100 = \left(\frac{0.1}{1} \right) \times 100 = 10\%
    \]
    \[
    \text{Total Selling Price} = \frac{11}{12} x \times \text{cost per unit}
    \]

Thus, the dishonest merchant’s net gain percent is **37.5%**.

Would you like more details or have any questions?

Here are five related questions:
1. What would be the merchant's gain if he delivered the exact quantity of sugar without cheating?
2. How does the gain percentage change if the merchant offers a discount?
3. What is the effect of increasing the additional charge to 15% on the merchant's net gain?
4. If the merchant were to pay for the exact amount received, what would be the impact on his gain?
5. How would the gain percentage be affected if the customer received 40% less sugar?

**Tip:** Always break down complex word problems into smaller steps, such as understanding the scenario, calculating costs, and finally determining the profit.

Learn how to calculate the net gain percentage of a dishonest sugar merchant who manipulates quantities and prices. Understand the concepts of cost price, profit calculation, and percentage. Detailed step-by-step solution provided.

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