Directions: Solve the following problem.

Two business partners, Elena and Marcus, established a commercial enterprise with initial capitals of ₹50,000 and ₹40,000, respectively. Six months after starting the enterprise, Elena withdrew her complete capital from the business, whereas Marcus maintained his investment active for the entire 12-month period. At the end of the year, in what ratio should their total profits be shared between Elena and Marcus?

The correct option is 5:8.

Step 1: Understand the principles of profit sharing in a partnership.
In a business partnership, the total profit earned at the end of a given period is divided among partners in proportion to the product of their invested capital and the duration (time) for which that capital remains invested.

Step 2: Identify the given investments and time periods.
For Elena:
Capital invested = ₹50,000
Investment duration = 6 months (since she withdrew her complete capital 6 months after starting)

For Marcus:
Capital invested = ₹40,000
Investment duration = 12 months (since he maintained his investment for the full 1-year period)

Step 3: Calculate the ratio of equivalent monthly capital investments.
The ratio of profits shared between Elena and Marcus is given by:
$\text{Profit Ratio}=(\text{Elena's Capital}\times \text{Elena's Time}):(\text{Marcus's Capital}\times \text{Marcus's Time})$

Substituting the values into the formula:
$\text{Profit Ratio}=(50000\times 6):(40000\times 12)$

Calculate the individual products:
$\text{Elena's total contribution factor}=50000\times 6=300000$
$\text{Marcus's total contribution factor}=40000\times 12=480000$

Step 4: Simplify the ratio to its lowest terms.
$\text{Ratio}=300000:480000$
Divide both terms of the ratio by their greatest common divisor, 60,000:
$\text{Ratio}=\frac{300000}{60000}:\frac{480000}{60000}$
$\text{Ratio}=5:8$

At the end of the year, the total profits should be shared between Elena and Marcus in the ratio of 5:8.