Direction: In the given question, read the given statement and compare the two given quantities on its basis.
Rutuja bought two articles-article A at ₹X and article B at ₹X+50. She sold article A at `20%` profit and article B at `10%` loss, and earned ₹35 as profit on the whole deal.
Quantity I. Profit earned by Rutuja on selling Article A (in ₹)
Quantity II. Loss incurred (in ₹)when an article which costs ₹480 is sold at `20%` loss

Options

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

Answer

To solve the problem, we need to find the profit earned by Rutuja on selling Article A and compare it with the loss incurred when an article costing ₹480 is sold at a 20% loss.

### Step-by-Step Solution:

1. **Identify the cost prices:**
- Cost price of Article A = ₹X
- Cost price of Article B = ₹X + ₹50

2. **Calculate the total cost price:**
\[ \text{Total Cost Price} = X + (X + 50) = 2X + 50 \]

3. **Calculate the selling price of Article A:**
- Profit on Article A = 20%
- Selling Price of Article A = Cost Price + Profit
\[ \text{Selling Price of A} = X + 0.20X = 1.20X = \frac{6X}{5} \]

4. **Calculate the selling price of Article B:**
- Loss on Article B = 10%
- Selling Price of Article B = Cost Price - Loss
\[ \text{Selling Price of B} = (X + 50) - 0.10(X + 50) = 0.90(X + 50) = \frac{9(X + 50)}{10} = \frac{9X + 450}{10} \]

5. **Calculate the total selling price:**
\[ \text{Total Selling Price} = \text{Selling Price of A} + \text{Selling Price of B} \]
\[ = \frac{6X}{5} + \frac{9X + 450}{10} \]
To add these fractions, we need a common denominator, which is 10:
\[ = \frac{12X}{10} + \frac{9X + 450}{10} = \frac{12X + 9X + 450}{10} = \frac{21X + 450}{10} \]

6. **Set up the profit equation:**
Rutuja earned a profit of ₹35 on the whole deal, so:
\[ \text{Total Selling Price} - \text{Total Cost Price} = 35 \]
\[ \frac{21X + 450}{10} - (2X + 50) = 35 \]

7. **Clear the fraction by multiplying through by 10:**
\[ 21X + 450 - 20X - 500 = 350 \]
\[ X - 50 = 350 \]
\[ X = 400 \]

8. **Calculate the profit earned on selling Article A:**
- Selling Price of Article A = \(\frac{6X}{5}\)
\[ = \frac{6 \times 400}{5} = 480 \]
- Profit on Article A = Selling Price - Cost Price
\[ = 480 - 400 = 80 \]

9. **Calculate the loss incurred on selling an article costing ₹480 at 20% loss:**
- Loss = 20% of ₹480
\[ = 0.20 \times 480 = 96 \]

### Final Comparison:
- Quantity I (Profit on Article A) = ₹80
- Quantity II (Loss on ₹480 at 20% loss) = ₹96

### Conclusion:
Since ₹80 < ₹96, we conclude that:
**Quantity I is less than Quantity II.**