P, Q and R start a business jointly. Twice the capital of P is equal to thrice the capital of Q and the capital of Q is four times the capital of R. Find the share of Q in an annual profit of Rs 148500.

Options

A. Rs54000
B. Rs 64000
C. Rs56000
D. Rs 55000

Answer

To solve the problem step by step, we will denote the capitals of P, Q, and R as \( P \), \( Q \), and \( R \) respectively.

### Step 1: Set up the equations based on the given information
From the problem, we have two relationships:
1. Twice the capital of P is equal to thrice the capital of Q:
\[ 2P = 3Q \quad \text{(Equation 1)} \]
2. The capital of Q is four times the capital of R:
\[ Q = 4R \quad \text{(Equation 2)} \]

### Step 2: Express P in terms of Q
From Equation 1, we can express \( P \) in terms of \( Q \):
\[ P = \frac{3Q}{2} \]

### Step 3: Express R in terms of Q
From Equation 2, we can express \( R \) in terms of \( Q \):
\[ R = \frac{Q}{4} \]

### Step 4: Find the total capital in terms of Q
Now, we can express the total capital \( T \) as the sum of the capitals of P, Q, and R:
\[ T = P + Q + R \]
Substituting the expressions for \( P \) and \( R \) in terms of \( Q \):
\[ T = \frac{3Q}{2} + Q + \frac{Q}{4} \]

### Step 5: Simplify the total capital
To add these fractions, we need a common denominator. The least common multiple of 2 and 4 is 4:
\[ T = \frac{3Q}{2} + \frac{4Q}{4} + \frac{Q}{4} = \frac{6Q}{4} + \frac{4Q}{4} + \frac{Q}{4} = \frac{6Q + 4Q + Q}{4} = \frac{11Q}{4} \]

### Step 6: Calculate the share of Q in the profit
The share of Q in the total profit is proportional to its capital. The profit is Rs 148,500. Therefore, the share of Q can be calculated as follows:
\[ \text{Share of Q} = \frac{Q}{T} \times \text{Total Profit} = \frac{Q}{\frac{11Q}{4}} \times 148500 \]
This simplifies to:
\[ \text{Share of Q} = \frac{4}{11} \times 148500 \]

### Step 7: Calculate the final amount
Now, we calculate:
\[ \text{Share of Q} = \frac{4 \times 148500}{11} = \frac{594000}{11} = 54000 \]

Thus, the share of Q in the annual profit is Rs 54,000.

### Final Answer
The share of Q in an annual profit of Rs 148,500 is Rs 54,000.

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