Directions: Select the correct option.

Liam and Maya launched a tech startup by investing $25,000 and $40,000, respectively. Three months later, Noah joined the venture with a capital contribution of $50,000. Determine the ratio in which the annual profit should be distributed among Liam, Maya, and Noah.

The correct option is 10 : 16 : 15.

Step 1: Understand the Partnership Profit Sharing Rule
When partners invest different amounts of capital for different periods of time, the total profit is distributed among them in proportion to the product of their capital investment and the time duration for which the capital was invested.

Step 2: Calculate the Duration for Each Partner
The annual profit corresponds to a total duration of 12 months.
• Liam invested at the start: 12 months.
• Maya invested at the start: 12 months.
• Noah joined 3 months later, so his investment period is:
$12-3=9\text{months}$

Step 3: Calculate the Capital-Time Product for Each Partner
• Liam:
$\text{Capital}\times \text{Time}=25000\times 12=300000$
• Maya:
$\text{Capital}\times \text{Time}=40000\times 12=480000$
• Noah:
$\text{Capital}\times \text{Time}=50000\times 9=450000$

Step 4: Determine the Ratio of Profit Share
The ratio of profit distribution among Liam, Maya, and Noah is:
$\text{Liam}:\text{Maya}:\text{Noah}=300000:480000:450000$
Dividing each term by 10,000:
$=30:48:45$
Dividing each term by 3:
$=10:16:15$
Therefore, the annual profit should be distributed among Liam, Maya, and Noah in the ratio of 10 : 16 : 15.