Veer and Manyu entered into a business by investing Rs 6000 and Rs 8000 for x and `(x+4)` months respectively . At the end , profit share of Veer is 3900 Rs less than profit share of Manyu. Find the value of xif total profit is Rs . 12900 .(in months)

Options

A. A)12
B. B)8
C. C)6
D. D)10

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the investments and time periods
Veer invests Rs 6000 for x months, and Manyu invests Rs 8000 for (x + 4) months.

### Step 2: Calculate the ratio of investments
The ratio of their investments is:
- Veer's investment = 6000
- Manyu's investment = 8000

This can be simplified:
\[ \text{Investment Ratio} = \frac{6000}{8000} = \frac{3}{4} \]

### Step 3: Calculate the ratio of time
The time periods for their investments are:
- Veer's time = x months
- Manyu's time = (x + 4) months

The time ratio is:
\[ \text{Time Ratio} = \frac{x}{x + 4} \]

### Step 4: Calculate the profit ratio
The profit share is calculated based on the product of investment and time. Therefore, the profit ratio can be expressed as:
\[ \text{Profit Ratio} = \text{Investment Ratio} \times \text{Time Ratio} = \frac{3}{4} \times \frac{x}{x + 4} = \frac{3x}{4(x + 4)} \]

### Step 5: Set up the profit shares
Let Manyu's profit be \( y \). According to the problem, Veer's profit is Rs 3900 less than Manyu's profit:
\[ \text{Veer's Profit} = y - 3900 \]

### Step 6: Total profit equation
The total profit is given as Rs 12900:
\[ y + (y - 3900) = 12900 \]
Simplifying this gives:
\[ 2y - 3900 = 12900 \]
\[ 2y = 16800 \]
\[ y = 8400 \]

### Step 7: Calculate Veer's profit
Now we can find Veer's profit:
\[ \text{Veer's Profit} = y - 3900 = 8400 - 3900 = 4500 \]

### Step 8: Set up the equation based on profit ratios
Now we can set up the equation using the profit ratio:
\[ \frac{3x}{4(x + 4)} = \frac{4500}{8400} \]

### Step 9: Simplify the right side
Simplifying the right side:
\[ \frac{4500}{8400} = \frac{15}{28} \]

### Step 10: Cross-multiply to solve for x
Cross-multiplying gives:
\[ 3x \cdot 28 = 4(x + 4) \cdot 15 \]
\[ 84x = 60x + 240 \]

### Step 11: Solve for x
Rearranging gives:
\[ 84x - 60x = 240 \]
\[ 24x = 240 \]
\[ x = 10 \]

### Conclusion
The value of \( x \) is 10 months.