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

Options

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

Answer

To solve the problem step by step, we will follow the information given and apply the concepts of ratios and proportions.

### Step 1: Define the investments
Let the investment of Q be \( 100x \).
Since P invested 60% more than Q, the investment of P will be:
\[ P = 100x + 60\% \text{ of } 100x = 100x + 60x = 160x \]

R invested 20% more than Q, so the investment of R will be:
\[ R = 100x + 20\% \text{ of } 100x = 100x + 20x = 120x \]

### Step 2: Write down the investments
Now we have:
- Investment of P = \( 160x \)
- Investment of Q = \( 100x \)
- Investment of R = \( 120x \)

### Step 3: Define the time periods
The ratio of time periods for P, Q, and R is given as \( 2:4:3 \).
Let the time periods be:
- Time period of P = \( 2t \)
- Time period of Q = \( 4t \)
- Time period of R = \( 3t \)

### Step 4: Calculate the profit shares
The profit share is calculated as the product of investment and time period.
Thus, the profit shares for P, Q, and R will be:
- Profit share of P = \( 160x \times 2t = 320xt \)
- Profit share of Q = \( 100x \times 4t = 400xt \)
- Profit share of R = \( 120x \times 3t = 360xt \)

### Step 5: Write the profit share ratio
The profit share ratio of P:Q:R will be:
\[ P:Q:R = 320xt : 400xt : 360xt \]
This simplifies to:
\[ P:Q:R = 32 : 40 : 36 \]
Dividing each term by 4 gives:
\[ P:Q:R = 8 : 10 : 9 \]

### Step 6: Calculate the total profit shares
The sum of the profit shares of Q and R is given as Rs. 8550.
Thus, we can express this as:
\[ Q + R = 10u + 9u = 19u = 8550 \]
Where \( u \) is the unit profit share.

### Step 7: Find the value of \( u \)
To find \( u \):
\[ u = \frac{8550}{19} = 450 \]

### Step 8: Calculate the profit share of P
Now, we can find the profit share of P:
\[ P = 8u = 8 \times 450 = 3600 \]

### Conclusion
The profit share of P is Rs. 3600.