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 $x$.

Correct Answer: The correct option is $ 400.

Step-by-step Explanation:

1. Understand the Investments and Time Periods:

In a partnership, the profit is divided among partners in the ratio of their equivalent investments (Investment × Time period).

Clara's investment = $ 2000

Clara stayed in the venture for the whole year (12 months).

Damien's investment = $ ($2000+x$)

Damien withdrew after 8 months, so his investment period is 8 months.

2. Calculate the Ratio of Equivalent Investments:

Equivalent investment of Clara = $2000\times 12=24000$

Equivalent investment of Damien = $(2000+x)\times 8=16000+8x$

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

$\frac{\text{Damien's Profit}}{\text{Clara's Profit}}=\frac{16000+8x}{24000}$

3. Use the Profit Condition:

It is given that Damien's profit share is 20% less than Clara's profit share.

This means Damien's share is 80% of Clara's share:

$\text{Damien's Profit}=(1-0.20)\times \text{Clara's Profit}=0.80\times \text{Clara's Profit}$

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

$\frac{\text{Damien's Profit}}{\text{Clara's Profit}}=0.80=\frac{4}{5}$

4. Equate and Solve for $x$:

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

Cross-multiplying to solve for $x$:

$5\times (16000+8x)=4\times 24000$

$80000+40x=96000$

$40x=96000-80000$

$40x=16000$

$x=\frac{16000}{40}=400$

Thus, the value of $x$ is $ 400.