In a partnership, A invests `1/6` of the capital for `1/6` of the time, B invests `1/3` of the capital and `1/3` of the time and C, the rest of the capital for the whole time. Out of a profit of Rs. 4600, B's share is :

Options

A. Rs. 650
B. Rs 800
C. Rs. 960
D. Rs. 1000

Answer

To solve the problem step by step, we will first determine the effective capital contribution of each partner based on their investment and the time for which they invested.

### Step 1: Define the total capital
Let's assume the total capital is Rs. 6 (this is a convenient choice since it allows us to work with fractions easily).

### Step 2: Calculate A's investment
A invests \( \frac{1}{6} \) of the capital:
\[ \text{A's investment} = \frac{1}{6} \times 6 = 1 \text{ Rs.} \]
A invests for \( \frac{1}{6} \) of the time.

### Step 3: Calculate B's investment
B invests \( \frac{1}{3} \) of the capital:
\[ \text{B's investment} = \frac{1}{3} \times 6 = 2 \text{ Rs.} \]
B invests for \( \frac{1}{3} \) of the time.

### Step 4: Calculate C's investment
C invests the rest of the capital. The total capital is Rs. 6, and A and B together have invested:
\[ \text{Total investment by A and B} = 1 + 2 = 3 \text{ Rs.} \]
Thus, C's investment is:
\[ \text{C's investment} = 6 - 3 = 3 \text{ Rs.} \]
C invests for the whole time.

### Step 5: Calculate the effective capital contribution
The effective capital contribution is calculated as:
\[ \text{Effective capital} = \text{Investment} \times \text{Time} \]
- For A:
\[ \text{A's effective capital} = 1 \times \frac{1}{6} = \frac{1}{6} \]
- For B:
\[ \text{B's effective capital} = 2 \times \frac{1}{3} = \frac{2}{3} \]
- For C:
\[ \text{C's effective capital} = 3 \times 1 = 3 \]

### Step 6: Calculate total effective capital
Now, we sum up the effective capital contributions:
\[ \text{Total effective capital} = \frac{1}{6} + \frac{2}{3} + 3 \]
To add these, we convert everything to a common denominator (which is 6):
\[ \text{Total effective capital} = \frac{1}{6} + \frac{4}{6} + \frac{18}{6} = \frac{1 + 4 + 18}{6} = \frac{23}{6} \]

### Step 7: Calculate B's share of the profit
The total profit is Rs. 4600. The share of each partner is proportional to their effective capital contribution:
\[ \text{B's share} = \frac{\text{B's effective capital}}{\text{Total effective capital}} \times \text{Total profit} \]
Substituting the values:
\[ \text{B's share} = \frac{\frac{2}{3}}{\frac{23}{6}} \times 4600 \]
Calculating the fraction:
\[ \text{B's share} = \frac{2}{3} \times \frac{6}{23} \times 4600 = \frac{12}{69} \times 4600 \]
Now simplifying:
\[ \text{B's share} = \frac{12 \times 4600}{69} = \frac{55200}{69} \approx 800 \]

### Final Answer
B's share of the profit is Rs. 800.

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