A and B started a business. B’s investment was ₹4000 more than that of A. At the end of 8 months from the start of the business, B left and C joined with an investment which was ₹6000 more than that of A. If the ratio of the total annual profit to B’s share in profit was 11:5, then what was the investment made by A?

Options

A. ₹ 6000
B. ₹ 12000
C. ₹ 8000
D. None of these

Answer

To solve the problem step by step, we will denote A's investment as \( X \).

### Step 1: Define Investments
- A's investment = \( X \)
- B's investment = \( X + 4000 \) (since B's investment is ₹4000 more than A's)
- C's investment = \( X + 6000 \) (since C's investment is ₹6000 more than A's)

### Step 2: Determine Time of Investment
- A invests for 12 months.
- B invests for 8 months (he leaves after 8 months).
- C invests for 4 months (he joins when B leaves).

### Step 3: Calculate Total Investment Contributions
- A's total contribution = \( 12X \)
- B's total contribution = \( (X + 4000) \times 8 = 8X + 32000 \)
- C's total contribution = \( (X + 6000) \times 4 = 4X + 24000 \)

### Step 4: Total Investment
The total investment can be expressed as:
\[ \text{Total Investment} = 12X + (8X + 32000) + (4X + 24000) \]
Combining the terms:
\[ \text{Total Investment} = 12X + 8X + 4X + 32000 + 24000 = 24X + 56000 \]

### Step 5: Profit Sharing Ratio
Given that the ratio of total annual profit to B's share in profit is \( 11:5 \), we can denote:
- Total profit = \( 11Y \)
- B's share in profit = \( 5Y \)

### Step 6: Relate B's Profit to Investment
Since profit is directly proportional to investment:
\[ B's \, share = \frac{B's \, investment}{Total \, investment} \times Total \, profit \]
This gives us:
\[ 5Y = \frac{(8X + 32000)}{(24X + 56000)} \times 11Y \]
Cancelling \( Y \) from both sides (assuming \( Y \neq 0 \)):
\[ 5 = \frac{(8X + 32000) \times 11}{(24X + 56000)} \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ 5(24X + 56000) = 11(8X + 32000) \]
Expanding both sides:
\[ 120X + 280000 = 88X + 352000 \]

### Step 8: Rearranging the Equation
Rearranging gives:
\[ 120X - 88X = 352000 - 280000 \]
\[ 32X = 72000 \]

### Step 9: Solve for X
Dividing both sides by 32:
\[ X = \frac{72000}{32} = 2250 \]

### Conclusion
Thus, the investment made by A is ₹2250.