Liam and Noah started a joint business venture by investing $30,000 and $(30,000 + x), respectively. After six months, Noah withdrew 20% of his initial capital contribution. At the end of one year, the total profit earned by the business was $22,000, and Noah's share of the profit was $12,000. Find the value of 2x.

The correct option is 20000.

Step 1: Calculate Liam's effective investment for the year.
Liam invested $30,000 for the full 12 months.
Liam's total equivalent investment for 1 month:

$\text{Liam's Investment}=30000\times 12=360000$

Step 2: Calculate Noah's effective investment for the year.
Noah initially invested $(30,000 + x) for the first 6 months.
After 6 months, Noah withdrew 20% of his initial investment.
The remaining capital invested by Noah for the next 6 months is:

$(100-20)\%\text{ of }(30000+x)=0.8(30000+x)$

Noah's total equivalent investment for 1 month:

Step 3: Determine the profit sharing ratio.
Total profit earned at the end of 1 year = $22,000.
Noah's share of the profit = $12,000.
Liam's share of the profit:

$\text{Liam's Profit}=22000-12000=10000$

The ratio of Liam's profit share to Noah's profit share is:

$\frac{\text{Liam's Share}}{\text{Noah's Share}}=\frac{10000}{12000}=\frac{5}{6}$

Step 4: Solve for x by equating the ratio of investments to the ratio of profits.
Since profit is distributed in the ratio of equivalent investments:

$\frac{360000}{10.8(30000+x)}=\frac{5}{6}$

Cross-multiplying to solve for $x$:

$30000+x=\frac{2160000}{54}$

$30000+x=40000$

$x=40000-30000=10000$

Step 5: Calculate the value of 2x.

$2x=2\times 10000=20000$

Thus, the value of 2x is 20000.