P and Q started a business by investing Rs 15000 & Rs (15000 + x) respectively. After four months, Q withdrew 40% of his initial investment. After a year, the total profit was Rs 47000 and the profit share of Q was Rs 22000. Find the value of ‘2x’ .

The correct option is 6000.

Let's break down the solution step-by-step:

Step 1: Define the investments and time periods for P and Q.
P invests Rs 15000 for the entire 12 months (1 year).
The equivalent investment-months for P is:

$\text{Equivalent Investment of P}=15000\times 12=180000$

Q starts with an initial investment of Rs (15000 + x).
This initial amount remains invested for the first 4 months.
After 4 months, Q withdraws 40% of his initial investment, which means 60% of his initial investment remains for the next 8 months.
Remaining investment for Q = $0.6\times (15000+x)$.
The equivalent investment-months for Q is:

$\text{Equivalent Investment of Q}=(15000+x)\times 4+0.6(15000+x)\times 8$

Simplifying the terms:

$\text{Equivalent Investment of Q}=4(15000+x)+4.8(15000+x)=8.8(15000+x)$

Step 2: Determine the ratio of their profits.
The total profit after one year is Rs 47000.
The profit share of Q is Rs 22000.
Therefore, the profit share of P is:

$\text{Profit Share of P}=47000-22000=25000$

The ratio of the profit shares of P and Q is:

$\frac{\text{Profit of P}}{\text{Profit of Q}}=\frac{25000}{22000}=\frac{25}{22}$

Step 3: Relate the investment ratio to the profit ratio.
The ratio of profits is equal to the ratio of their equivalent investments:

$\frac{180000}{8.8(15000+x)}=\frac{25}{22}$

Cross-multiplying to solve for x:

$180000\times 22=25\times 8.8\times (15000+x)$

Since $25\times 8.8=220$, we have:

$180000\times 22=220\times (15000+x)$

Dividing both sides by 22:

$180000=10\times (15000+x)$

Dividing by 10:

$18000=15000+x$

Subtracting 15000 from both sides:

$x=3000$

Step 4: Calculate the value of ‘2x’.
Now, we find the required value of 2x:

$2x=2\times 3000=6000$