By selling 45 lemons for Rs. 40, a man incurs a loss of 20%. How many lemons should he sell for Rs. 24 so that he makes a profit of 20% on that transaction?

Introduction / Context:
This is a profit and loss question involving unit cost and unit selling price. The seller first sells lemons at a loss, giving us information about the cost price per lemon. Using this cost price, we must determine how many lemons he should sell for a fixed total selling price in order to achieve a desired profit percentage.

Given Data / Assumptions:
- He sells 45 lemons for Rs. 40 at a loss of 20%.
- He wants to sell some number of lemons for Rs. 24 and make a profit of 20%.
- We assume all lemons are identical with the same cost price per lemon.
- There are no other expenses.

Concept / Approach:
First, we find the selling price per lemon in the loss-making case. Using the loss percentage, we recover the cost price per lemon. Next, we use the desired profit percentage to find the required selling price per lemon in the gain-making case. Finally, we divide the total desired selling amount by this new selling price per lemon to find the number of lemons to be sold.

Step-by-Step Solution:
Step 1: Selling price (SP) for 45 lemons is Rs. 40, so SP per lemon = 40 / 45 = 8 / 9 rupees.Step 2: He sells at a loss of 20%, which means SP = 80% of cost price (CP).Step 3: So 8 / 9 = 0.8 * CP per lemon.Step 4: Hence CP per lemon = (8 / 9) / 0.8 = (8 / 9) / (4 / 5) = (8 / 9) * (5 / 4) = 40 / 36 = 10 / 9 rupees.Step 5: For the second scenario, he wants a profit of 20%, so SP = 120% of CP = 1.2 * CP.Step 6: SP per lemon for 20% profit = 1.2 * (10 / 9) = (6 / 5) * (10 / 9) = 60 / 45 = 4 / 3 rupees.Step 7: Total SP desired in the second case is Rs. 24, so number of lemons = total SP / SP per lemon = 24 / (4 / 3).Step 8: Simplify: 24 / (4 / 3) = 24 * (3 / 4) = 72 / 4 = 18 lemons.

Verification / Alternative check:
Confirm using cost and profit. Cost per lemon is 10 / 9 rupees, so cost of 18 lemons = 18 * (10 / 9) = 20 rupees. Selling them for Rs. 24 gives a profit of 24 - 20 = Rs. 4. Profit percent = (4 / 20) * 100 = 20%. This matches the required profit, confirming that 18 lemons is correct.

Why Other Options Are Wrong:
16 lemons would cost 16 * (10 / 9) = 160 / 9 ≈ Rs. 17.78, giving a profit larger than 20%. 20 lemons cost 200 / 9 ≈ Rs. 22.22, so the profit would be less than 20% at Rs. 24. 22 and 24 lemons also fail to give exactly 20% profit when you compute cost and compare with Rs. 24.

Common Pitfalls:
Some learners confuse loss and profit percentages and treat them symmetrically, which is incorrect. Others skip computing cost price per lemon and try to work directly with total amounts, which often leads to mistakes. Always find CP first when both loss and profit scenarios are involved.

Final Answer:
The man should sell 18 lemons for Rs. 24 to gain 20%.