A shopkeeper cheats to the extent of 10% both while buying and while selling goods by using false weights. What is his overall gain percentage on the transaction?

Introduction / Context:
This question involves successive gains achieved by cheating on both the buying and selling sides using false weights. The shopkeeper effectively gets more goods than he pays for when buying and gives less goods than he charges for when selling. We assume the market price per true kilogram is the same in both directions, and we compute his net gain based purely on quantity misrepresentation.

Given Data / Assumptions:
- Market price of the commodity is assumed to be Rs. P per true kg (P is a positive constant).
- While buying, he cheats by 10% using false weights to his advantage.
- While selling, he again cheats by 10% using false weights to his advantage.
- The cheating is purely in terms of weight, not in the quoted price per kilogram.
- We must find his overall percentage gain.

Concept / Approach:
A convenient way is to assume the quoted price per kg remains P for both buying and selling, and then model how much true quantity he receives or gives in each case. A common interpretation is that while buying he uses weights that are 10% heavier (so he receives 1.1 kg of true goods for the price of 1 kg), and while selling he uses weights that are 10% lighter (so he gives only 0.9 kg of true goods while charging for 1 kg). From this we compute his effective cost price and selling price per true kilogram and then calculate the overall profit percentage.

Step-by-Step Solution:
Step 1: Assume market price per kg = P rupees.Step 2: While buying, he uses a weight 10% heavier than the true kg, that is 1.1 kg, but calls it 1 kg.Step 3: He pays P rupees for what he calls 1 kg, but actually receives 1.1 kg of goods.Step 4: So his effective cost price (CP) per true kg = P / 1.1 = P * (10 / 11).Step 5: While selling, he uses a weight 10% lighter than true, i.e., 0.9 kg, but calls it 1 kg.Step 6: He charges P rupees for what he calls 1 kg, but actually gives only 0.9 kg of goods.Step 7: So his effective selling price (SP) per true kg = P / 0.9 = P * (10 / 9).Step 8: Now compute the profit per true kg: profit = SP - CP = P * (10 / 9) - P * (10 / 11).Step 9: Factor out P * 10: profit = P * 10 * (1 / 9 - 1 / 11) = P * 10 * ((11 - 9) / 99) = P * 10 * (2 / 99) = 20P / 99.Step 10: Effective cost price CP per kg = P * (10 / 11) = 10P / 11.Step 11: Profit percent = (profit / CP) * 100 = (20P / 99) / (10P / 11) * 100.Step 12: Simplify: (20P / 99) * (11 / 10P) * 100 = (220 / 990) * 100 = (22 / 99) * 100 ≈ 22.22%.

Verification / Alternative check:
Take P = Rs. 11 for easier numbers. Then CP per true kg = 10 rupees and SP per true kg ≈ 12.22 rupees. Profit per kg ≈ 2.22 rupees on a cost of 10 rupees, giving a profit percent of about 22.2%. This numerical check matches the algebraic result. Since the options are in whole percentages, we select the closest value, which is 22%.

Why Other Options Are Wrong:
20% is slightly less than the actual approximate 22.22% gain. 21% is also an underestimate. 23% is slightly too high, and 18% is much lower than the derived figure. Only 22% is reasonably close to the true value of about 22.2%, and is the standard answer used in such questions.

Common Pitfalls:
Some learners treat the 10% cheating as being on price instead of weight and use factors like 0.9P and 1.1P directly, leading to a different result. Others forget to compute the gains per true kilogram and instead mix the quoted and actual quantities incorrectly. Always keep clear which quantity (true kg vs. quoted kg) is used for buying and selling, and then derive effective CP and SP per true unit.

Final Answer:
The shopkeeper's overall gain on the transaction is approximately 22%.