To solve the problem step by step, we will analyze the capital contributions of A, B, and C, how they change over time, and then calculate the total profit based on B's share.
### Step 1: Determine Initial Capitals
The initial capital contributions of A, B, and C are in the ratio 1:4:4. Let's assume the common multiple is \( x \).
- A's capital = \( 1x \)
- B's capital = \( 4x \)
- C's capital = \( 4x \)
### Step 2: Calculate Capitals After Each Quarter
The business operates in quarters (every 3 months). We will track the capitals over four quarters (one year).
**End of 1st Quarter:**
- A doubles his capital: \( 1x \rightarrow 2x \)
- B halves his capital: \( 4x \rightarrow 2x \)
- C's capital remains unchanged: \( 4x \)
**Capitals after 1st Quarter:**
- A = \( 2x \)
- B = \( 2x \)
- C = \( 4x \)
**End of 2nd Quarter:**
- A doubles his capital: \( 2x \rightarrow 4x \)
- B halves his capital: \( 2x \rightarrow x \)
- C's capital remains unchanged: \( 4x \)
**Capitals after 2nd Quarter:**
- A = \( 4x \)
- B = \( x \)
- C = \( 4x \)
**End of 3rd Quarter:**
- A doubles his capital: \( 4x \rightarrow 8x \)
- B halves his capital: \( x \rightarrow 0.5x \)
- C's capital remains unchanged: \( 4x \)
**Capitals after 3rd Quarter:**
- A = \( 8x \)
- B = \( 0.5x \)
- C = \( 4x \)
**End of 4th Quarter:**
- A doubles his capital: \( 8x \rightarrow 16x \)
- B halves his capital: \( 0.5x \rightarrow 0.25x \)
- C's capital remains unchanged: \( 4x \)
**Capitals after 4th Quarter:**
- A = \( 16x \)
- B = \( 0.25x \)
- C = \( 4x \)
### Step 3: Calculate Total Capital After One Year
Now, we sum up the final capitals:
- Total capital = \( 16x + 0.25x + 4x = 20.25x \)
### Step 4: Calculate B's Share in the Profit
Given that B's share in the profit is ₹4,50,000, we need to determine B's share of the total capital to find the total profit.
**B's Capital Contribution:**
B's final capital = \( 0.25x \)
**B's Share of Total Capital:**
B's share = \( \frac{0.25x}{20.25x} = \frac{0.25}{20.25} \)
### Step 5: Calculate Total Profit
Let the total profit be \( P \).
From B's share, we have:
\[
\frac{0.25}{20.25} \times P = 4,50,000
\]
Now, solving for \( P \):
\[
P = 4,50,000 \times \frac{20.25}{0.25}
\]
Calculating:
\[
P = 4,50,000 \times 81 = 3,64,50,000
\]
### Step 6: Convert to Lakhs
To convert to lakhs:
\[
P = 3,64,50,000 \div 1,00,000 = 36.45 \text{ lakhs}
\]
### Final Answer
Thus, the total profit in ₹ lakhs is approximately **36.45 lakhs**.
