A & B entered into a business by investing total capital of Rs 17000. B withdraws Rs 1500 after 6 months and gets Rs 8100 as profit of Rs 19500 at the end of year . Find capital of B after 6 months from starting .

Options

A. A)Rs 7000
B. B)Rs 9500
C. C)Rs 7500
D. D)Rs 6000

Answer

To solve the problem step by step, we will follow the logical flow of the information provided in the question.

### Step 1: Define the Investments
Let the investment of A be \( k \). Since the total capital invested by A and B is Rs 17,000, the investment of B can be expressed as:
\[ \text{Investment of B} = 17000 - k \]

### Step 2: Calculate B's Investment After 6 Months
B withdraws Rs 1500 after 6 months. Therefore, the amount of B's capital after 6 months will be:
\[ \text{Capital of B after 6 months} = (17000 - k) - 1500 = 15500 - k \]

### Step 3: Determine the Time of Investment
- A's capital \( k \) is invested for 12 months.
- B's capital \( 15500 - k \) is invested for 6 months (after which he withdrew Rs 1500).

### Step 4: Calculate the Profit Sharing Ratio
The profit sharing ratio is based on the product of capital and time. Thus, we have:
- A's contribution to the profit: \( k \times 12 \)
- B's contribution to the profit: \( (15500 - k) \times 6 \)

### Step 5: Set Up the Profit Equation
The total profit at the end of the year is Rs 19,500. According to the problem, B receives Rs 8,100 as his share of the profit. Therefore, the profit ratio can be set up as:
\[ \frac{k \times 12}{(15500 - k) \times 6} = \frac{8100}{19500} \]

### Step 6: Simplify the Profit Ratio
First, simplify the right side:
\[ \frac{8100}{19500} = \frac{27}{65} \]
Now, we can rewrite the equation:
\[ \frac{12k}{6(15500 - k)} = \frac{27}{65} \]
This simplifies to:
\[ \frac{2k}{15500 - k} = \frac{27}{65} \]

### Step 7: Cross Multiply to Solve for k
Cross multiplying gives:
\[ 2k \times 65 = 27 \times (15500 - k) \]
This expands to:
\[ 130k = 418500 - 27k \]

### Step 8: Combine Like Terms
Bringing all terms involving \( k \) to one side:
\[ 130k + 27k = 418500 \]
\[ 157k = 418500 \]

### Step 9: Solve for k
Now, divide both sides by 157:
\[ k = \frac{418500}{157} \approx 2665.6 \]

### Step 10: Calculate B's Capital After 6 Months
Now, substitute \( k \) back into the equation for B's capital after 6 months:
\[ \text{Capital of B after 6 months} = 15500 - k \]
Substituting \( k \):
\[ \text{Capital of B after 6 months} = 15500 - 2665.6 \approx 12834.4 \]

### Final Answer
Thus, the capital of B after 6 months from the start is approximately Rs 12,834.4.