Three investors, Arthur, Brian, and Charles, launched a startup venture with initial capital contributions in the ratio of 1:2:4. Four months into the business, Charles pulled out 50% of his investment. Two months later, Brian withdrew 25% of his initial funds. The venture completed one full year of operation, generating a overall net profit of ₹2,40,000. Determine the profit-sharing ratio among Arthur, Brian, and Charles (Arthur : Brian : Charles).

Correct Option: The correct profit-sharing ratio among Arthur, Brian, and Charles is 12 : 21 : 32 (Option 1).

Step-by-Step Explanation:

Step 1: Understand the initial investment ratios and duration
Let the initial investments of Arthur, Brian, and Charles be based on the ratio 1 : 2 : 4.
Let Arthur's initial investment = $x$
Let Brian's initial investment = $2x$
Let Charles's initial investment = $4x$
The total duration of the business operation is 1 full year, which is equal to 12 months.

Step 2: Calculate the effective investment of each person over 12 months

1. Arthur's Investment:
Arthur made no changes to his investment throughout the entire 12 months.

$\text{Arthur's Total Investment}=x\times 12=12x$

2. Brian's Investment:
Brian maintained his full investment of $2x$ for the first 6 months (4 months + 2 months later).
After 6 months, Brian withdrew 25% of his initial investment.

$\text{Amount withdrawn by Brian}=25\%\text{of}2x=0.5x$
$\text{Brian's remaining investment}=2x-0.5x=1.5x$
Brian kept this remaining amount for the remaining 6 months (12 - 6 = 6 months).

$\text{Brian's Total Investment}=(2x\times 6)+(1.5x\times 6)=12x+9x=21x$

3. Charles's Investment:
Charles maintained his full investment of $4x$ for the first 4 months.
After 4 months, Charles withdrew 50% of his initial investment.

$\text{Amount withdrawn by Charles}=50\%\text{of}4x=2x$
$\text{Charles's remaining investment}=4x-2x=2x$
Charles kept this remaining amount for the remaining 8 months (12 - 4 = 8 months).

$\text{Charles's Total Investment}=(4x\times 4)+(2x\times 8)=16x+16x=32x$

Step 3: Determine the profit-sharing ratio
The ratio of profit sharing is equal to the ratio of their equivalent total investments over the period: