Calculating Profit Sharing Ratio in a Business

This problem involves calculating the profit sharing ratio among three partners (A, B, and C) in a business where their initial investments change over the course of a year (12 months).

The profit sharing ratio is directly proportional to the total investment amount contributed by each partner over the entire duration of the business period. The total investment contribution for each partner is calculated by summing up the product of their investment amount and the duration for which that amount was invested.

Understanding the Investment Changes

Let's break down the investment periods and amounts for each partner over the 12 months:

• Partner A: Started with Rs. 20000. After 5 months, invested Rs. 4000 more.
• Partner B: Started with Rs. 25000. After 4 months, withdrew Rs. 8000.
• Partner C: Started with Rs. 10000. After 6 months, invested Rs. 8000 more.

Calculating Total Investment Contribution for Each Partner

Partner A's Contribution:

• Initial investment: Rs. 20000 for the first 5 months. Contribution = $\text{Rs. } 20000 \times 5 \text{ months}$.
• New investment: Rs. 20000 + Rs. 4000 = Rs. 24000 for the remaining $12 - 5 = 7$ months. Contribution = $\text{Rs. } 24000 \times 7 \text{ months}$.

Total contribution for A = $(\text{Rs. } 20000 \times 5) + (\text{Rs. } 24000 \times 7)$

Total contribution for A = $\text{Rs. } 100000 + \text{Rs. } 168000$

Total contribution for A = $\text{Rs. } 268000$

Partner B's Contribution:

• Initial investment: Rs. 25000 for the first 4 months. Contribution = $\text{Rs. } 25000 \times 4 \text{ months}$.
• New investment: Rs. 25000 - Rs. 8000 = Rs. 17000 for the remaining $12 - 4 = 8$ months. Contribution = $\text{Rs. } 17000 \times 8 \text{ months}$.

Total contribution for B = $(\text{Rs. } 25000 \times 4) + (\text{Rs. } 17000 \times 8)$

Total contribution for B = $\text{Rs. } 100000 + \text{Rs. } 136000$

Total contribution for B = $\text{Rs. } 236000$

Partner C's Contribution:

• Initial investment: Rs. 10000 for the first 6 months. Contribution = $\text{Rs. } 10000 \times 6 \text{ months}$.
• New investment: Rs. 10000 + Rs. 8000 = Rs. 18000 for the remaining $12 - 6 = 6$ months. Contribution = $\text{Rs. } 18000 \times 6 \text{ months}$.

Total contribution for C = $(\text{Rs. } 10000 \times 6) + (\text{Rs. } 18000 \times 6)$

Total contribution for C = $\text{Rs. } 60000 + \text{Rs. } 108000$

Total contribution for C = $\text{Rs. } 168000$

Determining the Profit Sharing Ratio

The profit sharing ratio for A : B : C is the ratio of their total investment contributions:

Ratio = Total contribution of A : Total contribution of B : Total contribution of C

Ratio = Rs. 268000 : Rs. 236000 : Rs. 168000

To simplify the ratio, we can divide all terms by a common factor. Let's divide by 1000:

Ratio = 268 : 236 : 168

All numbers are divisible by 4. Let's divide by 4:

$268 \div 4 = 67$

$236 \div 4 = 59$

$168 \div 4 = 42$

The simplified ratio is 67 : 59 : 42.

This ratio represents the proportion in which the total profit of Rs. 'x' will be shared among A, B, and C at the end of the year.

Comparing with Options

Let's compare our calculated ratio with the given options:

Our calculated ratio matches Option 4.

Revision Table: Profit Sharing Calculation

Ratio A : B : C = 268000 : 236000 : 168000

Dividing by 1000: 268 : 236 : 168

Dividing by 4: 67 : 59 : 42

Additional Information: Partnership Basics

In a business partnership, profits or losses are typically shared among partners based on their agreed-upon ratio. When investments change over time, the ratio is calculated based on the effective capital contributed by each partner over the entire period.

• The effective capital is the sum of (investment amount × time period) for each different investment amount held during the partnership duration.
• If all partners invest for the same duration and do not change their investments, the profit sharing ratio is simply the ratio of their initial investments.
• If investments change, the time period for each investment amount is crucial. The total duration is usually the life of the business or the accounting period (like one year).
• The total profit is then distributed according to the calculated ratio. For example, if the ratio is $a:b:c$ and the total profit is $P$, Partner A's share would be $\frac{a}{a+b+c} \times P$.