A,B, and C started a business with their capitals in the ratio 2,3:5. A increased his capital by 50% after 4 months, B increased his capital by `33 1/3%` after 6 months and C withdrew 50% of his capital after 8 months, from the start of the business. If the total profit at the end of ayear was Rs. 86,800 then the difference between the shares of Aand C in the profit was:

Options

A. Rs. 9,800
B. Rs. 8,400
C. Rs. 7,000
D. Rs. 12,600

Answer

To solve the problem step by step, we will calculate the effective capital contributions of A, B, and C over the year, taking into account their respective changes in capital and the time for which they were invested.

### Step 1: Determine Initial Capitals
Let the initial capitals of A, B, and C be represented as:
- A's capital = 2x
- B's capital = 3x
- C's capital = 5x

### Step 2: Calculate A's Effective Capital Contribution
A increases his capital by 50% after 4 months.
- Initial capital for the first 4 months = 2x
- After 4 months, A's capital becomes:
\[ 2x + 50\% \text{ of } 2x = 2x + x = 3x \]
- A's effective capital contribution:
- For the first 4 months: \( 2x \times 4 = 8x \)
- For the next 8 months: \( 3x \times 8 = 24x \)
- Total contribution by A:
\[ 8x + 24x = 32x \]

### Step 3: Calculate B's Effective Capital Contribution
B increases his capital by 33.33% after 6 months.
- Initial capital for the first 6 months = 3x
- After 6 months, B's capital becomes:
\[ 3x + 33.33\% \text{ of } 3x = 3x + x = 4x \]
- B's effective capital contribution:
- For the first 6 months: \( 3x \times 6 = 18x \)
- For the next 6 months: \( 4x \times 6 = 24x \)
- Total contribution by B:
\[ 18x + 24x = 42x \]

### Step 4: Calculate C's Effective Capital Contribution
C withdraws 50% of his capital after 8 months.
- Initial capital for the first 8 months = 5x
- After 8 months, C's capital becomes:
\[ 5x - 50\% \text{ of } 5x = 5x - 2.5x = 2.5x \]
- C's effective capital contribution:
- For the first 8 months: \( 5x \times 8 = 40x \)
- For the last 4 months: \( 2.5x \times 4 = 10x \)
- Total contribution by C:
\[ 40x + 10x = 50x \]

### Step 5: Calculate Total Effective Capital Contribution
Now, we can sum up the total contributions:
- Total contribution = \( 32x + 42x + 50x = 124x \)

### Step 6: Determine Profit Shares
The total profit at the end of the year is Rs. 86,800. We need to find the share of A and C:
- A's share of profit:
\[ \text{A's share} = \left( \frac{32x}{124x} \right) \times 86800 = \frac{32}{124} \times 86800 \]
- C's share of profit:
\[ \text{C's share} = \left( \frac{50x}{124x} \right) \times 86800 = \frac{50}{124} \times 86800 \]

### Step 7: Calculate the Difference Between A's and C's Shares
- A's share:
\[ A's share = \frac{32 \times 86800}{124} = 22,400 \]
- C's share:
\[ C's share = \frac{50 \times 86800}{124} = 35,000 \]
- Difference between A's and C's shares:
\[ \text{Difference} = C's share - A's share = 35,000 - 22,400 = 12,600 \]

### Final Answer
The difference between the shares of A and C in the profit is **Rs. 12,600**.