A shopkeeper marks his goods at 20% more and offers discount of 10%. He cheats to the extent of 10% while buying as well as selling by using false weight. Find his over all profit %.

Answer

To find the overall profit percentage of the shopkeeper who marks his goods at 20% more, offers a discount of 10%, and cheats by using false weights, we can follow these steps:

### Step-by-Step Solution:

1. **Assume the Cost Price (CP):**
Let's assume the cost price of 100 grams of goods is ₹100. Therefore, the cost price per gram is ₹1.

2. **Calculate the Marked Price (MP):**
The shopkeeper marks his goods at 20% more than the cost price.
\[ \text{Marked Price} = \text{CP} + 20\% \text{ of CP} = 100 + 20\% \times 100 = 100 + 20 = ₹120 \]

3. **Calculate the Selling Price (SP) after Discount:**
The shopkeeper offers a discount of 10% on the marked price.
\[ \text{Discount} = 10\% \text{ of MP} = 10\% \times 120 = 12 \]
\[ \text{Selling Price} = \text{MP} - \text{Discount} = 120 - 12 = ₹108 \]

4. **Cheating While Buying:**
The shopkeeper cheats by 10% while buying. If he wants to buy 100 grams, he actually gets 110 grams.
- He pays for 100 grams but receives 110 grams.
- The effective cost price for 110 grams is still ₹100, so the cost price per gram is:
\[ \text{Effective CP per gram} = \frac{100}{110} = \frac{10}{11} \approx ₹0.909 \]

5. **Cheating While Selling:**
The shopkeeper cheats by 10% while selling. If he wants to sell 100 grams, he actually sells only 90 grams.
- He sells 90 grams at ₹108 (the selling price for 100 grams).
- Therefore, the effective selling price for 90 grams is:
\[ \text{Effective SP for 90 grams} = ₹108 \]

6. **Calculate the Total Cost Price and Selling Price:**
- For 90 grams, the cost price is:
\[ \text{CP for 90 grams} = 90 \times \frac{10}{11} = \frac{900}{11} \approx ₹81.82 \]
- The selling price for 90 grams is ₹108.

7. **Calculate the Profit:**
\[ \text{Profit} = \text{SP} - \text{CP} = 108 - 81.82 = 26.18 \]

8. **Calculate the Profit Percentage:**
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 = \left( \frac{26.18}{81.82} \right) \times 100 \approx 32% \]

### Final Answer:
The overall profit percentage of the shopkeeper is approximately **32%**.