The following questions are accopanied by three statements (A) or (I), (B) or (II), and (C) or (III). You have to determine which statement(s) is/are sufficient/necessary to answer the questions.
A, B and Centered into a partnership. If the profit earned in the business is proportional to the investment and the period of investment then What is the profit of B if all of them invested the amount for one year and total profit is Rs. x.
A. A invested Rs. 1500 less than that of C.
B. B invested 2 times more than that of A. C invested 3 times more than that of A.
C. B invested 200 percent more than that of A and 25% less than that of C.

Options

A. Any two of them
B. Either B orCalone
C. Any of them
D. All statements are required

Answer

To determine the profit of B in the partnership of A, B, and C, we need to analyze the provided statements one by one. The profit earned is proportional to the investment made by each partner.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- Let the investments of A, B, and C be denoted as \( A \), \( B \), and \( C \) respectively.
- The total profit is given as Rs. \( x \).
- The profit of each partner is proportional to their investment.

2. **Analyzing Statement A:**
- Statement A says, "A invested Rs. 1500 less than that of C."
- This means \( A = C - 1500 \).
- However, we do not have any information about B's investment or its relation to A or C. Therefore, we cannot determine the exact amounts invested by A, B, and C from this statement alone.
- **Conclusion:** Statement A is **not sufficient**.

3. **Analyzing Statement B:**
- Statement B states, "B invested 2 times more than that of A. C invested 3 times more than that of A."
- Let \( A = a \). Then:
- \( B = 2a \)
- \( C = 3a \)
- The total investment becomes \( A + B + C = a + 2a + 3a = 6a \).
- The profit of B can be calculated as:
\[ \text{Profit of B} = \frac{B}{A + B + C} \times x = \frac{2a}{6a} \times x = \frac{1}{3}x \]
- **Conclusion:** Statement B is **sufficient**.

4. **Analyzing Statement C:**
- Statement C states, "B invested 200 percent more than that of A and 25% less than that of C."
- If \( A = a \), then:
- \( B = a + 2a = 3a \) (200% more than A)
- Let \( C = c \). Then \( B = c - 0.25c = 0.75c \).
- From \( B = 3a \) and \( B = 0.75c \), we can set up the equation:
\[ 3a = 0.75c \implies c = 4a \]
- Now we have:
- \( A = a \)
- \( B = 3a \)
- \( C = 4a \)
- The total investment becomes \( A + B + C = a + 3a + 4a = 8a \).
- The profit of B can be calculated as:
\[ \text{Profit of B} = \frac{B}{A + B + C} \times x = \frac{3a}{8a} \times x = \frac{3}{8}x \]
- **Conclusion:** Statement C is **sufficient**.

### Final Conclusion:
- Statement A is not sufficient.
- Statement B is sufficient.
- Statement C is sufficient.

Thus, the answer is that either Statement B or Statement C alone is sufficient to determine the profit of B.