To solve the problem step by step, we will denote the capital invested by B as \( x \). Therefore, the capital invested by A will be \( x + 6,00,000 \).
### Step 1: Define the investments
Let:
- Capital of B = \( x \)
- Capital of A = \( x + 6,00,000 \)
### Step 2: Define the time of investment
B invests his capital for \( 7.5 \) months, while A invests for \( 7.5 + 2.5 = 10 \) months.
### Step 3: Set up the profit sharing equation
The profit is shared in the ratio of their investments multiplied by the time for which they invested. The profit sharing ratio can be expressed as:
\[
\text{Profit of A} : \text{Profit of B} = (x + 6,00,000) \times 10 : x \times 7.5
\]
### Step 4: Express the profit in terms of B's share
From the problem, we know that B's share is Rs. 2,48,000 less than A's share. If we let A's share be \( A_s \) and B's share be \( B_s \), we can write:
\[
B_s = A_s - 2,48,000
\]
The total profit is Rs. 12,40,000, so:
\[
A_s + B_s = 12,40,000
\]
### Step 5: Substitute B's share into the total profit equation
Substituting \( B_s \) into the total profit equation gives us:
\[
A_s + (A_s - 2,48,000) = 12,40,000
\]
This simplifies to:
\[
2A_s - 2,48,000 = 12,40,000
\]
Adding \( 2,48,000 \) to both sides:
\[
2A_s = 12,40,000 + 2,48,000 = 14,88,000
\]
Dividing by 2:
\[
A_s = 7,44,000
\]
### Step 6: Calculate B's share
Now we can find B's share:
\[
B_s = A_s - 2,48,000 = 7,44,000 - 2,48,000 = 4,96,000
\]
### Step 7: Set up the ratio of profits
From the profit sharing ratio:
\[
\frac{(x + 6,00,000) \times 10}{x \times 7.5} = \frac{A_s}{B_s} = \frac{7,44,000}{4,96,000}
\]
This simplifies to:
\[
\frac{(x + 6,00,000) \times 10}{x \times 7.5} = \frac{3}{2}
\]
### Step 8: Cross-multiply and solve for x
Cross-multiplying gives:
\[
2 \times (x + 6,00,000) \times 10 = 3 \times x \times 7.5
\]
This simplifies to:
\[
20(x + 6,00,000) = 22.5x
\]
Expanding:
\[
20x + 1,20,00,000 = 22.5x
\]
Rearranging gives:
\[
1,20,00,000 = 22.5x - 20x
\]
\[
1,20,00,000 = 2.5x
\]
Dividing by 2.5:
\[
x = \frac{1,20,00,000}{2.5} = 48,00,000
\]
### Step 9: Conclusion
Thus, the capital of B is \( \text{Rs. } 48,00,000 \).
