Determine the correct value of the variable based on the partnership details provided below.

Clara and Damien invested $ 2000 and $ ($2000+x$) respectively. Damien withdrew from the venture after eight months. If the share of profit received by Damien at the end of a year is 20% less than the share received by Clara, then find the value of $x$.

The correct option is $ 400.

Step-by-Step Explanation:

1. Understand the concept of Partnership and Profit Sharing:
In a partnership, the profit is shared among the partners in direct proportion to the product of their investment amount and the duration of time for which the money remained invested.

$\text{Ratio of Profit}=({\text{Investment}}_1\times {\text{Time}}_1):({\text{Investment}}_2\times {\text{Time}}_2)$

2. Calculate the total equivalent investment for Clara and Damien:
Clara invested $ 2000 for the entire year (12 months).

$\text{Clara's equivalent investment}=2000\times 12=24000$

Damien invested $ ($2000+x$) for 8 months before withdrawing.

$\text{Damien's equivalent investment}=(2000+x)\times 8=16000+8x$

3. Determine the ratio of their profit shares:
The problem states that Damien's share of profit is 20% less than Clara's share.
If Clara's profit share is taken as 100%, then Damien's profit share is:

$100\%-20\%=80\%$

Thus, the ratio of Damien's profit share to Clara's profit share is:

$\frac{\text{Damien's Profit}}{\text{Clara's Profit}}=\frac{80}{100}=\frac{4}{5}$

4. Equate the ratios to solve for $x$:

$\frac{16000+8x}{24000}=\frac{4}{5}$

Cross-multiplying to solve for $x$:

$16000+8x=\frac{4\times 24000}{5}$

$16000+8x=4\times 4800$

$16000+8x=19200$

Subtract 16000 from both sides:

$8x=19200-16000$

$8x=3200$

Divide by 8:

$x=\frac{3200}{8}=400$

Therefore, the value of $x$ is 400.