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:
A and B started a business. After eight months from the start of the business, A left and C joined. The amount invested by B was twice of that invested by A. The amount invested by C was thrice of that invested by A. B got Rs252 as the share from the total annual profit earned.
Quantities:
I. Total annual profit
II. Rs 9000

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, we need to determine the total annual profit based on the investments made by A, B, and C in the business. Let's break it down step by step.

### Step 1: Define the Investments
Let the amount invested by A be \( X \).
- Then, the amount invested by B is \( 2X \) (twice that of A).
- The amount invested by C is \( 3X \) (thrice that of A).

### Step 2: Calculate the Time of Investment
- A invested for 12 months.
- B invested for 12 months.
- C joined after 8 months, so C invested for 4 months.

### Step 3: Calculate the Contribution of Each Partner
- A's contribution to the capital is \( X \times 12 = 12X \).
- B's contribution to the capital is \( 2X \times 12 = 24X \).
- C's contribution to the capital is \( 3X \times 4 = 12X \).

### Step 4: Total Contribution
Now, we can calculate the total contribution:
\[ \text{Total Contribution} = A's \ contribution + B's \ contribution + C's \ contribution = 12X + 24X + 12X = 48X \]

### Step 5: Calculate the Profit Share of B
We know that B received Rs 252 as his share of the total profit. The profit share is distributed in the ratio of their contributions.

### Step 6: Determine the Ratio of Contributions
The ratio of contributions is:
- A : B : C = \( 12X : 24X : 12X \)
- Simplifying this gives us \( 1 : 2 : 1 \).

### Step 7: Calculate Total Annual Profit
Let the total annual profit be \( P \).
Since B's share is Rs 252, which corresponds to his part of the total profit:
\[ \text{B's Share} = \frac{2}{4} \times P = 252 \]
This means:
\[ \frac{1}{2} \times P = 252 \implies P = 252 \times 2 = 504 \]

### Step 8: Compare with Quantity II
Now we have the total annual profit \( P = 504 \) and we need to compare it with Quantity II, which is Rs 9000.

### Conclusion
Since \( 504 < 9000 \), we can conclude that:

**Quantity I (Total annual profit) is less than Quantity II (Rs 9000).**

### Final Answer
The answer is: Quantity I is less than Quantity II.

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