A and B invests Rs.8000 and Rs.9000 in a business. After 4 months, A withdraws half of his capital and 2 months later, B withdraws one-third of his capital. In what ratio should they share the profits at the end of the year ?

Options

A. A) 21:29
B. B) 41:54
C. C) 32:45
D. D) 37:51

Answer

To solve the problem, we need to calculate the effective capital contribution of both A and B over the year, considering their withdrawals.

### Step-by-Step Solution:

1. **Initial Investments**:
- A invests Rs. 8000.
- B invests Rs. 9000.

2. **A's Investment Calculation**:
- A invests Rs. 8000 for the first 4 months.
- After 4 months, A withdraws half of his capital, which is Rs. 4000. So, A now has Rs. 4000 remaining.
- A's Rs. 4000 remains invested for the next 8 months (from month 5 to month 12).
- Total contribution of A = (8000 * 4) + (4000 * 8) = 32000 + 32000 = Rs. 64000.

3. **B's Investment Calculation**:
- B invests Rs. 9000 for the first 6 months (from month 1 to month 6).
- After 6 months, B withdraws one-third of his capital, which is Rs. 3000. So, B now has Rs. 6000 remaining.
- B's Rs. 6000 remains invested for the next 6 months (from month 7 to month 12).
- Total contribution of B = (9000 * 6) + (6000 * 6) = 54000 + 36000 = Rs. 90000.

4. **Calculating the Profit Sharing Ratio**:
- A's total contribution = Rs. 64000.
- B's total contribution = Rs. 90000.
- The ratio of their contributions = 64000 : 90000.
- Simplifying this ratio:
- Divide both sides by 16000, we get:
- A : B = 4 : 5.625.
- To express it in whole numbers, multiply by 8:
- A : B = 32 : 45.

5. **Final Ratio**:
- Therefore, A and B should share the profits in the ratio of 32 : 45.