Quantity I: Overall profit percentage if the cost prices of two shirts are equal. One shirt is sold for 20% profit and the other is sold for 10% loss.
Quantity II: Profit % made in selling each meter if the profit made in selling 20 m of a cloth equals the cost price of 5 m of that cloth.

Options

A. Quantity I ` gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge `Quantity II
D. Quantity I `le` QuantityII

Answer

To solve the problem, we need to analyze both quantities step by step.

### Quantity I: Overall Profit Percentage

1. **Define the Cost Price (CP)**:
Let the cost price of each shirt be \( CP \).

2. **Calculate Selling Price (SP) for Shirt 1**:
Shirt 1 is sold at a 20% profit.
\[ SP_1 = CP + 0.20 \times CP = 1.20 \times CP \]

3. **Calculate Selling Price (SP) for Shirt 2**:
Shirt 2 is sold at a 10% loss.
\[ SP_2 = CP - 0.10 \times CP = 0.90 \times CP \]

4. **Calculate Total Cost Price (TCP)**:
The total cost price for both shirts is:
\[ TCP = CP + CP = 2CP \]

5. **Calculate Total Selling Price (TSP)**:
The total selling price for both shirts is:
\[ TSP = SP_1 + SP_2 = (1.20 \times CP) + (0.90 \times CP) = 2.10 \times CP \]

6. **Calculate Overall Profit**:
Overall profit can be calculated as:
\[ \text{Profit} = TSP - TCP = (2.10 \times CP) - (2CP) = 0.10 \times CP \]

7. **Calculate Overall Profit Percentage**:
Overall profit percentage is given by:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{TCP} \right) \times 100 = \left( \frac{0.10 \times CP}{2CP} \right) \times 100 = 5\% \]

### Quantity II: Profit Percentage on Cloth

1. **Define the Cost Price (CP) per meter**:
Let the cost price of 1 meter of cloth be \( CP = 1 \) (for simplicity).

2. **Calculate Cost Price for 20 meters**:
The cost price for 20 meters of cloth is:
\[ TCP = 20 \times CP = 20 \times 1 = 20 \]

3. **Calculate Profit**:
The profit made from selling 20 meters of cloth equals the cost price of 5 meters of cloth:
\[ \text{Profit} = CP \text{ of 5 meters} = 5 \times 1 = 5 \]

4. **Calculate Profit Percentage**:
Profit percentage is calculated as:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{TCP} \right) \times 100 = \left( \frac{5}{20} \right) \times 100 = 25\% \]

### Conclusion:
- Quantity I: Overall profit percentage = 5%
- Quantity II: Profit percentage on cloth = 25%

### Comparison:
Since \( 5\% < 25\% \), we conclude that Quantity I is less than Quantity II.

### Final Answer:
**Option B: Quantity I is less than Quantity II.**

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