Two entrepreneurs, P and Q, launched a venture with a total capital of Rs. 10,000. P's initial capital was Rs. 2,000 less than Q's. Q maintained the investment for a full 12 months, whereas P kept the capital invested for x months less than Q. If the overall profit generated at the year's end was Rs. 10,000 and P's portion of the profit was Rs. 2,000, what is the value of x?

The correct answer is 7.5 months.

Step 1: Determine the investment capital of each entrepreneur
Let the initial capital invested by Q be $C_Q$ and by P be $C_P$.
We are given that the total combined capital is Rs. 10,000 and P's capital was Rs. 2,000 less than Q's capital.

$C_P+C_Q=10000$

$C_P=C_Q-2000$

Substitute the expression for $C_P$ into the total capital equation:

$(C_Q-2000)+C_Q=10000$

$2C_Q-2000=10000$

$2C_Q=12000$

$C_Q=6000$

Therefore, Q's initial capital is Rs. 6,000, and P's initial capital is:

$C_P=6000-2000=4000$

Step 2: Determine the duration of investment for P and Q
Q kept the investment for a full 12 months.
P kept the investment for $x$ months less than Q, so P's investment period is $(12-x)$ months.

Step 3: Express equivalent investments
In a partnership, profit is shared in proportion to the product of capital and time period ($\text{Capital}\times \text{Time}$).

$\text{Equivalent investment of P}=4000\times (12-x)$

$\text{Equivalent investment of Q}=6000\times 12=72000$

Step 4: Calculate profit sharing and solve for x
Total profit at year-end = Rs. 10,000
P's profit portion = Rs. 2,000
So, Q's profit portion = Rs. 10,000 - Rs. 2,000 = Rs. 8,000.

The ratio of P's profit to Q's profit must equal the ratio of their equivalent investments:

$\frac{\text{P's Profit}}{\text{Q's Profit}}=\frac{4000\times (12-x)}{72000}$

Substitute the actual profit values into the equation:

$\frac{2000}{8000}=\frac{4000\times (12-x)}{72000}$

Simplify both sides:

$\frac{1}{4}=\frac{12-x}{18}$

Multiply both sides by 18 to isolate $12-x$:

$12-x=\frac{18}{4}$

$12-x=4.5$

$x=12-4.5=7.5$

Hence, the value of $x$ is 7.5 months.