Profit Distribution in a Business Partnership: Explanation and Calculation

In a business context, profit distribution among partners is a critical aspect of partnership agreements. This article explores how the profits are distributed based on the capital contribution and the duration of the investment. We will use a series of examples and logical reasoning to help clarify the concept through specific scenarios.

Understanding the Scenario

A and B are partners in a business. A contributes 1/4 of the capital for 12 months, and B receives 2/3 of the profit. The question is to determine for how long B's money was used.

First Approach: Direct Calculation

Let's assume the total capital is Rs. 100.

A invests 1/4 of the capital for 15 months, which is Rs. 25 for 15 months. The capital of A in respect of time (one month) can be calculated as Rs. 25 × 15 Rs. 375. A is to receive 1/3 of the profit, which implies that the capital of A needed on a monthly basis to get 1/3 of the profit is Rs. 375. Therefore, to get 2/3 of the profit, the needed capital is Rs. 2 × 375 Rs. 750. The time duration for which B's money was used can be calculated as 750 ÷ 75 10 months.

Based on the calculation, B's money was used for 10 months.

Answer: B’s money was invested for 10 months.

Second Approach: Using Profit Ratio

From the given information, B gains 2/3 of the profit. Hence, the profit ratio for A:B is 1:2. Therefore, if A invests 1/4 of the capital for 15 months, B will invest 3/4 of the capital for an unknown duration, say x months.

Using the profit ratio formula:

1/4 × 15 / 3/4 × x 1/2.

Solving for x:

1/4 × 15 / (3/4 × x) 1/2

15/(3x) 1/2

30 3x

x 10

Thus, B's money was used for 10 months.

Answer: B’s money was invested for 10 months.

Third Approach: Using Direct Proportionality

Profits in a partnership are directly proportional to the amount of capital invested and the duration of the investment.

Let A invest 5x and B invest 6x for 18 months. Since B's capital and investment period result in 9x in 18 months, the profit distribution is in the ratio of 5:9.

To find the duration for which B’s capital was used, the following proportion is used:

5/7 ÷ 15 (2/3) 2/7 ÷ x (1/3).

Where 5/7 represents A's profit share, 15 (2/3) is A's capital for 15 months, 2/7 represents B's profit share, and x (1/3) is the duration for which B’s capital was used.

Solving for x:

35x 420

x 12

Thus, B's capital was used for 12 months.

Answer: B contributed 1/3 of the capital for 12 months.

Conclusion

The different approaches to solving this problem demonstrate the importance of understanding the relationship between capital contribution, profit sharing, and the duration of investment. In this case, the correct answer can be derived using the second approach or the third approach, both leading to the conclusion that B's money was used for 12 months.

Answer: B’s money was invested for 12 months.