P and Q started a business by investing Rs. 50000 and Rs. 40000 respectively. After 4 months, P withdraws Rs. 20000 while Q invests Rs. 70000 more after 6 months of starting the business. At the end of the year P's share in the profit is Rs. 11000. What will be the total profit (In Rs.) at the end of year?

Options

A. 33500
B. 27600
C. 42800
D. 38800

Answer

To solve the problem step-by-step, let's break it down:

### Step 1: Determine the initial investments
P invests Rs. 50,000 and Q invests Rs. 40,000.

### Step 2: Calculate the effective capital contribution of P
- P's investment of Rs. 50,000 is maintained for 4 months.
- After 4 months, P withdraws Rs. 20,000, leaving P with Rs. 30,000 for the remaining 8 months.

**Effective capital contribution of P:**
- For the first 4 months: \( 50,000 \times 4 = 200,000 \)
- For the next 8 months: \( 30,000 \times 8 = 240,000 \)

Total contribution by P = \( 200,000 + 240,000 = 440,000 \)

### Step 3: Calculate the effective capital contribution of Q
- Q's investment of Rs. 40,000 is maintained for 6 months.
- After 6 months, Q invests an additional Rs. 70,000, making Q's total investment Rs. 110,000 for the remaining 6 months.

**Effective capital contribution of Q:**
- For the first 6 months: \( 40,000 \times 6 = 240,000 \)
- For the next 6 months: \( 110,000 \times 6 = 660,000 \)

Total contribution by Q = \( 240,000 + 660,000 = 900,000 \)

### Step 4: Calculate the ratio of their contributions
Now, we have:
- P's total contribution = Rs. 440,000
- Q's total contribution = Rs. 900,000

To find the ratio of their contributions:
\[ \text{Ratio of P to Q} = \frac{440,000}{900,000} = \frac{44}{90} = \frac{22}{45} \]

### Step 5: Calculate the total profit using P's share
Given that P's share in the profit is Rs. 11,000, we can set up the following proportion based on their contribution ratio:
- P's share corresponds to 22 parts.
- Total parts = 22 (P) + 45 (Q) = 67 parts.

Let the total profit be \( x \). Then:
\[ \frac{22}{67} \times x = 11,000 \]

### Step 6: Solve for total profit \( x \)
To find \( x \):
\[ x = 11,000 \times \frac{67}{22} \]
Calculating \( \frac{67}{22} \):
\[ \frac{67}{22} = 3.0454545 \quad (\text{approximately}) \]
Now, multiply:
\[ x \approx 11,000 \times 3.0454545 \approx 33,500 \]

### Final Answer
The total profit at the end of the year is approximately Rs. 33,500.

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