A and B starts a business with their initial investments in the respective ratio of `3:2`. x months after the starts of the business, B leaves and C joins. Respective ratio between initial investments to B and C is 1:2. If B’s share out of the total annual profit of Rs5700 is Rs600, for how many months did C invest ?

Options

A. 4
B. 8
C. 3
D. 2

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Investment Ratios**:
- A and B start a business with their investments in the ratio of 3:2.
- Let A's investment be 3x and B's investment be 2x.

2. **Investment Ratio of B and C**:
- The ratio of B's investment to C's investment is 1:2.
- Since B's investment is 2x, C's investment will be 4x (because 2x is to 4x as 1 is to 2).

3. **Combining the Ratios**:
- We now have the investments as follows:
- A = 3x
- B = 2x
- C = 4x
- The combined investment ratio of A, B, and C is therefore 3:2:4.

4. **Time of Investment**:
- A invests for the entire year, which is 12 months.
- B invests for x months.
- C joins when B leaves, so C invests for (12 - x) months.

5. **Profit Sharing**:
- The profit is shared in the ratio of (Investment × Time).
- Therefore, the profit ratio will be:
- A's profit = 3x * 12 = 36x
- B's profit = 2x * x = 2x²
- C's profit = 4x * (12 - x) = 48x - 4x²

6. **Total Profit**:
- The total profit is given as Rs. 5700.
- B's share of the profit is Rs. 600.
- We can set up the equation for B's profit:
\[ \frac{2x^2}{36x + 2x^2 + 48x - 4x^2} = \frac{600}{5700} \]

7. **Simplifying the Equation**:
- The denominator simplifies to:
\[ 36x + 2x^2 + 48x - 4x^2 = 84x - 2x^2 \]
- Thus, we can rewrite the equation:
\[ \frac{2x^2}{84x - 2x^2} = \frac{600}{5700} \]
- Cross-multiplying gives:
\[ 5700 \cdot 2x^2 = 600(84x - 2x^2) \]

8. **Expanding and Rearranging**:
- Expanding the equation:
\[ 11400x^2 = 50400x - 1200x^2 \]
- Rearranging gives:
\[ 11400x^2 + 1200x^2 - 50400x = 0 \]
\[ 12600x^2 - 50400x = 0 \]
- Factoring out common terms:
\[ 12600x(x - 4) = 0 \]

9. **Solving for x**:
- This gives us two solutions: \(x = 0\) or \(x = 4\).
- Since \(x = 0\) does not make sense in this context, we have \(x = 4\).

10. **Finding C's Investment Duration**:
- Since C invests for \(12 - x\) months:
\[ C's \text{ investment duration} = 12 - 4 = 8 \text{ months} \]

### Final Answer:
C invested for **8 months**.