A, B and C are partners in a business B's capital is `1/6` of the total and A's capital is equal to that of B and C together. How much does C receive out of a total annual profit of Rs. 2400.

Answer

To solve the problem step by step, we will first define the capital contributions of A, B, and C based on the information given.

### Step 1: Define Total Capital
Let the total capital of A, B, and C be represented as \( T \).

### Step 2: Determine B's Capital
According to the problem, B's capital is \( \frac{1}{6} \) of the total capital. Therefore, we can express B's capital as:
\[ B = \frac{1}{6} T \]

### Step 3: Express A's Capital
The problem states that A's capital is equal to the combined capital of B and C. We can express this as:
\[ A = B + C \]
Substituting B's capital into this equation gives us:
\[ A = \frac{1}{6} T + C \]

### Step 4: Set Up the Equation
From the above equations, we know:
\[ A + B + C = T \]
Substituting the values of A and B, we get:
\[ \left(\frac{1}{6} T + C\right) + \frac{1}{6} T + C = T \]
This simplifies to:
\[ \frac{1}{6} T + \frac{1}{6} T + 2C = T \]
Combining the terms gives:
\[ \frac{1}{3} T + 2C = T \]

### Step 5: Solve for C
Rearranging the equation to isolate C:
\[ 2C = T - \frac{1}{3} T \]
This simplifies to:
\[ 2C = \frac{2}{3} T \]
Dividing both sides by 2 gives:
\[ C = \frac{1}{3} T \]

### Step 6: Determine A's Capital
Now substituting C back into the equation for A:
\[ A = \frac{1}{6} T + C = \frac{1}{6} T + \frac{1}{3} T = \frac{1}{6} T + \frac{2}{6} T = \frac{3}{6} T = \frac{1}{2} T \]

### Step 7: Summary of Capital Contributions
Now we have:
- \( A = \frac{1}{2} T \)
- \( B = \frac{1}{6} T \)
- \( C = \frac{1}{3} T \)

### Step 8: Calculate the Profit Share
The total profit is Rs. 2400. To find out how much C receives, we need to calculate C's share of the total profit:
\[ \text{C's share} = \frac{C}{T} \times \text{Total Profit} \]
Substituting C's capital:
\[ \text{C's share} = \frac{\frac{1}{3} T}{T} \times 2400 = \frac{1}{3} \times 2400 = 800 \]

### Final Answer
C receives Rs. 800 out of the total annual profit of Rs. 2400.

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