A and B together invested Rs. 12000 is a business. At the end of the year, out of a total profit of Rs. 1800. A's share was Rs.750. Find the investment of A.

Answer

To solve the problem step by step, let's follow the reasoning laid out in the video transcript.

### Step 1: Understand the total investment and profit distribution
A and B together invested Rs. 12,000 in a business. The total profit at the end of the year is Rs. 1,800, and A's share of the profit is Rs. 750.

### Step 2: Calculate B's share of the profit
To find B's share of the profit, we subtract A's share from the total profit:
\[ \text{B's share} = \text{Total profit} - \text{A's share} = 1800 - 750 = 1050 \]

### Step 3: Set up the ratio of profits
The profit ratio of A to B is:
\[ \text{Profit ratio} = \frac{\text{A's share}}{\text{B's share}} = \frac{750}{1050} \]
This simplifies to:
\[ \frac{750 \div 150}{1050 \div 150} = \frac{5}{7} \]

### Step 4: Set up the ratio of investments
Since the profit ratio is the same as the investment ratio (assuming they invested for the same time period), we can express the investments of A and B as:
\[ \frac{\text{Investment of A}}{\text{Investment of B}} = \frac{5}{7} \]
Let the investment of A be \( x \). Then the investment of B would be:
\[ \text{Investment of B} = 12000 - x \]

### Step 5: Set up the equation
From the investment ratio, we can write:
\[ \frac{x}{12000 - x} = \frac{5}{7} \]

### Step 6: Cross-multiply to solve for x
Cross-multiplying gives us:
\[ 7x = 5(12000 - x) \]
Expanding this:
\[ 7x = 60000 - 5x \]

### Step 7: Combine like terms
Adding \( 5x \) to both sides:
\[ 7x + 5x = 60000 \]
\[ 12x = 60000 \]

### Step 8: Solve for x
Dividing both sides by 12:
\[ x = \frac{60000}{12} = 5000 \]

### Conclusion
Thus, the investment of A is Rs. 5,000.

---