A and B both start a business. The sum. invested by B is ₹200 more than that invested by A. Six months after the start of the business, B leaves the business. If the total annual profit earned is ₹2800, out of which A receives ₹1600, what is the amount invested by A?

Options

A. ₹ 250
B. ₹ 400
C. ₹ 800
D. ₹ 500

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Investments
Let the amount invested by A be \( X \). According to the problem, B invests ₹200 more than A. Therefore, the amount invested by B is:
\[ X + 200 \]

### Step 2: Calculate the Time of Investment
A invests for the entire year (12 months), while B invests for only 6 months.

### Step 3: Calculate the Total Investment in Terms of Time
The total investment by A over the year is:
\[ 12X \]
The total investment by B over the 6 months is:
\[ 6(X + 200) = 6X + 1200 \]

### Step 4: Set Up the Profit Sharing Ratio
The total profit earned is ₹2800, and A receives ₹1600. Therefore, B's share of the profit is:
\[ 2800 - 1600 = 1200 \]
The profit sharing ratio between A and B is:
\[ \frac{A's\ profit}{B's\ profit} = \frac{1600}{1200} = \frac{4}{3} \]

### Step 5: Set Up the Equation Based on Investments
Since profit is directly proportional to the amount invested multiplied by the time, we can set up the following ratio:
\[ \frac{12X}{6X + 1200} = \frac{4}{3} \]

### Step 6: Cross-Multiply to Solve for X
Cross-multiplying gives us:
\[ 3 \cdot 12X = 4(6X + 1200) \]
This simplifies to:
\[ 36X = 24X + 4800 \]

### Step 7: Solve for X
Rearranging the equation:
\[ 36X - 24X = 4800 \]
\[ 12X = 4800 \]
\[ X = \frac{4800}{12} = 400 \]

### Step 8: Conclusion
The amount invested by A is:
\[ \text{Amount invested by A} = X = ₹400 \]