Sophia founded a tech startup with an initial investment of $40,000. Marcus joined the company 3 months later, contributing $60,000 in capital. If the total annual profits are divided at the end of 12 months, what is the ratio of Sophia's profit share to Marcus's profit share?

The correct option is 8:9.

To find the ratio of Sophia's profit share to Marcus's profit share, we calculate the product of each person's capital investment and the duration (in months) for which their investment was active during the 12-month period.

Step 1: Calculate Sophia's investment-time product
Sophia founded the tech startup, so her investment of $40,000 was invested for the entire 12 months.

$\text{Sophia's Product}=\mathrm{40,000}\times 12=\mathrm{480,000}$

Step 2: Calculate Marcus's investment-time product
Marcus joined 3 months later, which means his investment of $60,000 was active for 12 - 3 = 9 months.

$\text{Marcus's Product}=\mathrm{60,000}\times 9=\mathrm{540,000}$

Step 3: Determine the ratio of profit shares
The total profit is shared in the ratio of their respective investment-time products.

$\text{Ratio}=\mathrm{480,000}:\mathrm{540,000}$

Simplifying the ratio by dividing both terms by 60,000:

$\text{Ratio}=8:9$

Therefore, the ratio of Sophia's profit share to Marcus's profit share is 8:9.