In the given questions, two quantities are given. One as Quantity-I and another is Quantity-II. You have to determine relationship between these two quantities and choose the appropriate options as given below:
Ram sold a pen at `20%` discount thereby earning a profit of Rs60. Had he sold the pen without any discount he would have earned a profit of Rs 150.
Quantities:
I. Profit earned (in Rs) in the pen is sold at Rs400.
II. Profit earned (in Rs) if the pen is sold at `25%` profit.

Options

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

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Determine the Cost Price (CP) of the Pen
Let the cost price of the pen be \( CP = x \).

### Step 2: Selling Price with 20% Discount
When Ram sold the pen at a 20% discount, he earned a profit of Rs. 60. Therefore, the selling price (SP) can be expressed as:
\[ SP = CP + 60 = x + 60 \]
Since he sold it at a 20% discount, we can express the selling price in terms of the marked price (MP):
\[ SP = MP \times (1 - 0.20) = MP \times 0.80 \]
Thus, we have:
\[ x + 60 = MP \times 0.80 \quad (1) \]

### Step 3: Selling Price without Discount
Had he sold the pen without any discount, he would have earned a profit of Rs. 150. Therefore, the selling price without discount can be expressed as:
\[ SP_{no\ discount} = CP + 150 = x + 150 \]
This selling price is equal to the marked price:
\[ MP = x + 150 \quad (2) \]

### Step 4: Equate the Two Expressions for MP
From equations (1) and (2), we can set them equal to each other:
\[ \frac{x + 60}{0.80} = x + 150 \]

### Step 5: Solve for x
Cross-multiplying gives us:
\[ x + 60 = 0.80(x + 150) \]
Expanding the right side:
\[ x + 60 = 0.80x + 120 \]
Rearranging gives:
\[ x - 0.80x = 120 - 60 \]
\[ 0.20x = 60 \]
\[ x = \frac{60}{0.20} = 300 \]
Thus, the cost price \( CP = 300 \).

### Step 6: Calculate Profit if Sold at Rs. 400
If the pen is sold at Rs. 400, the profit can be calculated as:
\[ \text{Profit} = SP - CP = 400 - 300 = 100 \]
So, Quantity I is 100.

### Step 7: Calculate Profit if Sold at 25% Profit
To find the selling price at a 25% profit:
\[ \text{Profit} = 25\% \text{ of } CP = 0.25 \times 300 = 75 \]
Thus, the selling price at 25% profit is:
\[ SP = CP + \text{Profit} = 300 + 75 = 375 \]
So, Quantity II is 75.

### Step 8: Compare the Two Quantities
- Quantity I: 100
- Quantity II: 75

### Conclusion
Since 100 (Quantity I) is greater than 75 (Quantity II), we conclude that:
\[ \text{Quantity I} > \text{Quantity II} \]

### Final Answer
The correct option is that Quantity I is greater than Quantity II.

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