A and B start a business with ₹3000 and ₹P respectively. At the end of six months from the start of the business, B leaves the business and C joins the business with a certain amount. The respective ratio between their profit shares, at the end of the year, was `5 : 4 : 3`. How much money did C invest in the business ?

Options

A. ₹3600
B. ₹4800
C. ₹4200
D. ₹3000

Answer

To solve the problem, we need to determine how much money C invested in the business given the profit-sharing ratios of A, B, and C.

### Step-by-Step Solution:

1. **Identify the Investments and Time Periods:**
- A invests ₹3000 for 12 months.
- B invests ₹P for 6 months (since he leaves after 6 months).
- C joins after 6 months and invests ₹X for 6 months.

2. **Calculate the Effective Investment:**
- The effective investment for A is:
\[ \text{A's investment} = 3000 \times 12 = 36000 \]
- The effective investment for B is:
\[ \text{B's investment} = P \times 6 = 6P \]
- The effective investment for C is:
\[ \text{C's investment} = X \times 6 = 6X \]

3. **Set Up the Ratio of Profits:**
- The profit-sharing ratio is given as 5:4:3 for A, B, and C respectively.
- This means:
\[ \frac{\text{A's effective investment}}{5} = \frac{\text{B's effective investment}}{4} = \frac{\text{C's effective investment}}{3} \]

4. **Express the Ratios in Terms of A's Investment:**
- From A's investment:
\[ 5 \text{ parts} = 36000 \implies 1 \text{ part} = \frac{36000}{5} = 7200 \]
- Therefore:
- A's investment corresponds to 5 parts: \( 5 \times 7200 = 36000 \)
- B's investment corresponds to 4 parts: \( 4 \times 7200 = 28800 \)
- C's investment corresponds to 3 parts: \( 3 \times 7200 = 21600 \)

5. **Relate C's Investment to the Effective Investment:**
- We know that \( 6X = 21600 \) (since C's effective investment is 6 months).
- To find X, we divide by 6:
\[ X = \frac{21600}{6} = 3600 \]

### Conclusion:
C invested ₹3600 in the business.