A shopkeeper sold one-fourth of his goods at a loss of 10%. He sold the remaining at a higher per cent of profit to get 12 `1/2`% profit on the whole. transaction. The higher profit per cent is

Options

A. 17 `1/2` %
B. 33 `1/3`%
C. 22 `1/2` %
D. 0.2

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step-by-Step Solution:

1. **Define the Cost Price (CP)**:
Let the total cost price of the goods be \( C \).

2. **Calculate the Cost Price of One-Fourth of the Goods**:
The cost price of one-fourth of the goods is:
\[ \text{Cost Price of } \frac{1}{4} \text{ goods} = \frac{C}{4} \]

3. **Determine the Selling Price of One-Fourth of the Goods**:
Since the shopkeeper sold this portion at a loss of 10%, the selling price (SP) of one-fourth of the goods can be calculated as:
\[ \text{Selling Price} = \text{Cost Price} \times \left(1 - \frac{\text{Loss \%}}{100}\right) = \frac{C}{4} \times \left(1 - \frac{10}{100}\right) = \frac{C}{4} \times \frac{90}{100} = \frac{9C}{40} \]

4. **Calculate the Cost Price of the Remaining Goods**:
The remaining goods constitute three-fourths of the total goods:
\[ \text{Cost Price of } \frac{3}{4} \text{ goods} = \frac{3C}{4} \]

5. **Let the Profit Percentage on the Remaining Goods be \( P\% \)**:
The selling price of the remaining three-fourths of the goods can be expressed as:
\[ \text{Selling Price} = \frac{3C}{4} \times \left(1 + \frac{P}{100}\right) = \frac{3C}{4} \times \frac{100 + P}{100} \]

6. **Calculate the Total Selling Price**:
The total selling price of all goods is the sum of the selling prices of both portions:
\[ \text{Total Selling Price} = \frac{9C}{40} + \frac{3C}{4} \times \frac{100 + P}{100} \]

7. **Determine the Total Selling Price for a 12.5% Profit on the Whole**:
The total selling price for a 12.5% profit on the whole cost price is:
\[ \text{Total Selling Price} = C \times \left(1 + \frac{12.5}{100}\right) = C \times \frac{112.5}{100} = \frac{112.5C}{100} \]

8. **Set Up the Equation**:
Now, we equate the total selling price from both calculations:
\[ \frac{9C}{40} + \frac{3C}{4} \times \frac{100 + P}{100} = \frac{112.5C}{100} \]

9. **Eliminate \( C \) from the Equation**:
We can cancel \( C \) from both sides (assuming \( C \neq 0 \)):
\[ \frac{9}{40} + \frac{3}{4} \times \frac{100 + P}{100} = \frac{112.5}{100} \]

10. **Convert Fractions to a Common Denominator**:
Convert \( \frac{3}{4} \) to a fraction with a denominator of 100:
\[ \frac{3}{4} = \frac{75}{100} \]
Thus, the equation becomes:
\[ \frac{9}{40} + \frac{75 + 3P}{400} = \frac{112.5}{100} \]

11. **Multiply Through by 400 to Eliminate Denominators**:
Multiply the entire equation by 400:
\[ 9 \times 10 + 75 + 3P = 450 \]
This simplifies to:
\[ 90 + 75 + 3P = 450 \]

12. **Combine Like Terms**:
Combine the constants:
\[ 165 + 3P = 450 \]

13. **Isolate \( P \)**:
Subtract 165 from both sides:
\[ 3P = 450 - 165 \]
\[ 3P = 285 \]
Divide by 3:
\[ P = 95 \]

14. **Conclusion**:
The higher profit percentage is:
\[ P = 95\% \]