P invested `60%` more than Q and R invested `20%` more than Q. If the ratio of investment of time-periods (P: Q: R) is 2:4:3 and the sum of profit shares of Q and R is ₹8550 then find the profit share of P.

Options

A. ₹ 3200
B. ₹ 4000
C. ₹ 2400
D. ₹ 3600

Answer

To solve the problem step by step, we will follow the information given in the question regarding the investments of P, Q, and R, their time periods, and the profit shares.

### Step 1: Define the Investments
Let the investment of Q be \( x \).
- Since P invested 60% more than Q, the investment of P will be:
\[ P = x + 0.6x = 1.6x \]
- R invested 20% more than Q, so the investment of R will be:
\[ R = x + 0.2x = 1.2x \]

### Step 2: Define the Time Periods
The ratio of the time periods for P, Q, and R is given as 2:4:3. We can denote the time periods as:
- Time period of P = \( 2k \)
- Time period of Q = \( 4k \)
- Time period of R = \( 3k \)

### Step 3: Calculate the Profit Shares
The profit share is directly proportional to the product of investment and time period. Therefore, we can express the profit shares as follows:
- Profit share of P:
\[ \text{Profit share of P} = P \times \text{Time period of P} = 1.6x \times 2k = 3.2xk \]
- Profit share of Q:
\[ \text{Profit share of Q} = Q \times \text{Time period of Q} = x \times 4k = 4xk \]
- Profit share of R:
\[ \text{Profit share of R} = R \times \text{Time period of R} = 1.2x \times 3k = 3.6xk \]

### Step 4: Set Up the Equation for Total Profit Shares
According to the problem, the sum of the profit shares of Q and R is ₹8550:
\[ 4xk + 3.6xk = 8550 \]
Combining the terms:
\[ (4 + 3.6)xk = 8550 \]
\[ 7.6xk = 8550 \]

### Step 5: Solve for \( xk \)
Now, we can find \( xk \):
\[ xk = \frac{8550}{7.6} = 1125 \]

### Step 6: Calculate Profit Share of P
Now that we have \( xk \), we can find the profit share of P:
\[ \text{Profit share of P} = 3.2xk = 3.2 \times 1125 = 3600 \]

### Final Answer
Thus, the profit share of P is ₹3600.