Three entrepreneurs, Alex, Brian, and Clara, launched a tech startup with initial investments of $30,000, $20,000, and $25,000 respectively. After 6 months, Alex added 20% more to his initial investment, while Brian withdrew $5,000. If Clara's share of the annual profit at the end of the year was $4,500, find the difference between the profit shares of Alex and Brian.

The correct answer is $2,790.

Step 1: Calculate the effective investment for each person over the 12-month period.
In a business partnership, profit is distributed based on the product of the capital invested and the duration of the investment (Investment × Time in months).

For Alex:
Initial investment for the first 6 months = $30,000.
After 6 months, Alex added 20% more to his initial investment:

$\text{New Investment}=30000+(0.20\times 30000)=30000+6000=36000$

Alex's total equivalent investment for 1 year (12 months):

$\text{Alex's Total Investment-Months}=(30000\times 6)+(36000\times 6)=180000+216000=396000$

For Brian:
Initial investment for the first 6 months = $20,000.
After 6 months, Brian withdrew $5,000:

$\text{New Investment}=20000-5000=15000$

Brian's total equivalent investment for 1 year (12 months):

$\text{Brian's Total Investment-Months}=(20000\times 6)+(15000\times 6)=120000+90000=210000$

For Clara:
Clara kept her initial investment of $25,000 unchanged for the entire 12-month period.

$\text{Clara's Total Investment-Months}=25000\times 12=300000$

Step 2: Find the simplified ratio of profit shares.
The ratio of their profit shares is equal to the ratio of their total investment-month values:

$\text{Alex}:\text{Brian}:\text{Clara}=396000:210000:300000$

Dividing each value by 6,000 to simplify:

$\text{Alex}:\text{Brian}:\text{Clara}=66:35:50$

Step 3: Calculate the monetary value of one ratio unit.
Clara's share of the annual profit is $4,500, which corresponds to 50 ratio units:

$50\text{units}=\$4500$

$1\text{unit}=\frac{4500}{50}=\$90$

Step 4: Find the difference between the profit shares of Alex and Brian.
The difference between Alex's and Brian's shares in ratio units is:

$\text{Difference in units}=66-35=31\text{units}$

Multiplying by the value per unit:

$\text{Difference in profit shares}=31\times \$90=\$2790$

Therefore, the difference between the profit shares of Alex and Brian is $2,790.