Let's break down the problem step by step:

Step 1: The wholesaler cheats the retailer by 10%.

This means that if the retailer pays for 100 kg of rice, the wholesaler gives him only 90 kg.

Step 2: The retailer sells the rice to his customers but cheats by 20%.

This means that for every 100 kg of rice the retailer claims to sell, he actually only gives 80 kg to his customers.

Step 3: Assume the retailer buys rice from the wholesaler at a price of $P$ per kg.

• The retailer pays $100P$ for 100 kg of rice, but due to the wholesaler's cheating, he only receives 90 kg of rice.
• So, the actual price per kg of rice that the retailer gets is: $\text{Cost per kg} = \frac{100P}{90} = \frac{10P}{9}$

Step 4: Now, the retailer sells this rice to his customers at the same price $P$ per kg, but he cheats by 20%.

• This means he claims to sell 100 kg to his customers but actually gives them 80 kg.
• So, for the 80 kg of rice, he charges his customers for 100 kg at the price $P$ per kg.
  • He receives $100P$ for 80 kg of rice.

Step 5: Calculate the selling price per kg for the rice the retailer sells to his customers:

$\text{Selling price per kg} = \frac{100P}{80} = \frac{5P}{4}$

Step 6: The profit per kg is the difference between the selling price and the actual cost price:

$\text{Profit per kg} = \frac{5P}{4} - \frac{10P}{9}$

Step 7: To find the profit per kg, first get a common denominator for the two fractions:

$\text{Profit per kg} = \left( \frac{5P}{4} - \frac{10P}{9} \right) = \frac{45P - 40P}{36} = \frac{5P}{36}$

Step 8: To find the percentage profit, divide the profit per kg by the cost price per kg, then multiply by 100:

$\text{Percentage profit} = \left( \frac{\frac{5P}{36}}{\frac{10P}{9}} \right) \times 100 = \left( \frac{5}{36} \times \frac{9}{10} \right) \times 100 = \frac{45}{360} \times 100 = 12.5\%$

Final Answer:

The retailer's percentage profit is 12.5%.

---

Do you want details on any specific step or have more questions?

Here are 5 relative questions for further practice:

1. What would the retailer's profit be if he cheated his customers by 25% instead of 20%?
2. How does the wholesaler's cheating affect the retailer's total profit?
3. What if the wholesaler reduced the amount of rice by 15%? What would the new profit percentage be for the retailer?
4. If the retailer decided not to cheat his customers, what would his profit or loss be in this scenario?
5. What happens to the profit margin if the retailer increased his selling price by 10%?

Tip: When working with percentages and profit problems, always define variables clearly for easier manipulation of equations.