Three partners A, B, C start a business. Twice A's capital is equal to thrice B's capital and B's capital is four times C's capital. Out of a total profit of Rs. 16,500 at the end of the year, B's share is :

Options

A. Rs. 4000
B. Rs. 8000
C. Rs. 7500
D. None of these

Answer

To solve the problem step by step, we will first establish the relationships between the capitals of partners A, B, and C based on the information provided.

### Step 1: Establish relationships based on the problem statement
We are given two relationships:
1. Twice A's capital is equal to thrice B's capital.
\[ 2A = 3B \quad \text{(1)} \]
2. B's capital is four times C's capital.
\[ B = 4C \quad \text{(2)} \]

### Step 2: Express A's capital in terms of B's capital
From equation (1), we can express A in terms of B:
\[ A = \frac{3B}{2} \quad \text{(3)} \]

### Step 3: Express C's capital in terms of B's capital
From equation (2), we can express C in terms of B:
\[ C = \frac{B}{4} \quad \text{(4)} \]

### Step 4: Find the ratio of A, B, and C
Now we can express A, B, and C in terms of B:
- From (3): \( A = \frac{3B}{2} \)
- From (2): \( B = B \)
- From (4): \( C = \frac{B}{4} \)

Now we can find the ratio of A, B, and C:
\[ \text{Ratio of A : B : C} = \frac{3B/2}{B} : 1 : \frac{B/4}{B} \]
This simplifies to:
\[ \text{Ratio of A : B : C} = \frac{3}{2} : 1 : \frac{1}{4} \]

To eliminate the fractions, we can multiply each term by 4:
\[ \text{Ratio of A : B : C} = 4 \times \frac{3}{2} : 4 \times 1 : 4 \times \frac{1}{4} = 6 : 4 : 1 \]

### Step 5: Calculate the total parts
The total parts in the ratio are:
\[ 6 + 4 + 1 = 11 \]

### Step 6: Calculate B's share of the profit
The total profit is Rs. 16,500. To find B's share, we take the part of B in the ratio, which is 4:
\[ \text{B's share} = \left(\frac{B's \, part}{Total \, parts}\right) \times Total \, profit \]
\[ \text{B's share} = \left(\frac{4}{11}\right) \times 16500 \]

### Step 7: Perform the calculation
Calculating B's share:
\[ \text{B's share} = \frac{4 \times 16500}{11} = \frac{66000}{11} = 6000 \]

### Final Answer
B's share of the profit is Rs. 6,000.

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