Partnership Profit Sharing Calculation

This problem involves calculating the duration of investment in a business partnership based on the profit-sharing ratio. In a partnership, profits are typically divided among partners in proportion to the product of their investment amount and the time period for which the investment was made.

Understanding Business Partnership Profit Share

When partners invest different amounts for different durations, the share of profit for each partner is determined by the ratio of their respective (Investment × Time) values. The formula used is:

$$ \frac{\text{Profit}_1}{\text{Profit}_2} = \frac{\text{Investment}_1 \times \text{Time}_1}{\text{Investment}_2 \times \text{Time}_2} $$

In this scenario, we have two partners, Mohan and Sohan.

Analyzing the Given Information

• Mohan's investment: Rs. 100,000
• Mohan stayed in the business for the entire year. A year has 12 months. So, Mohan's time in the business is 12 months.
• Sohan joined after a few months with an investment of Rs. 40,000.
• Let's assume Sohan stayed in the business for $x$ months.
• The total profit at the end of the year was divided in the ratio 3 : 1 (Mohan : Sohan).

We can summarize the information in a table:

Setting up the Profit Ratio Equation

Using the formula for profit sharing based on Investment × Time:

$$ \frac{\text{Mohan's Profit}}{\text{Sohan's Profit}} = \frac{\text{Mohan's Investment} \times \text{Mohan's Time}}{\text{Sohan's Investment} \times \text{Sohan's Time}} $$

Substitute the given values:

$$ \frac{3}{1} = \frac{100000 \times 12}{40000 \times x} $$

Solving for Sohan's Time in Business

Now, we solve the equation to find the value of $x$, which represents the number of months Sohan was in the business.

$$ 3 = \frac{100000 \times 12}{40000 \times x} $$

We can simplify the fraction on the right side by canceling out common factors.

$$ 3 = \frac{100 \times 12}{40 \times x} $$

$$ 3 = \frac{10 \times 12}{4 \times x} $$

$$ 3 = \frac{120}{4x} $$

Cross-multiply:

$$ 3 \times 4x = 120 $$

$$ 12x = 120 $$

Divide by 12:

$$ x = \frac{120}{12} $$

$$ x = 10 $$

This means Sohan's investment was in the business for 10 months.

Calculating When Sohan Joined

The question asks "After how many months did Sohan join the business?". The total duration is one year, which is 12 months. If Sohan was in the business for the last 10 months of the year, he must have joined after the first $12 - 10$ months.

Number of months after which Sohan joined = Total time - Sohan's time in business

Number of months after which Sohan joined = $12 - 10 = 2$ months.

Therefore, Sohan joined the business after 2 months.

Revision Table - Partnership Profit Sharing

Additional Information - Business Partnership Concepts

Understanding partnership dynamics is crucial for solving such problems. Here are some related concepts:

• Active vs. Sleeping Partner: An active partner participates in the daily management of the business, while a sleeping partner only contributes capital. Profit sharing principles usually apply regardless of this distinction, based on the partnership deed.
• Partnership Deed: A written agreement among partners outlining the terms and conditions of the partnership, including profit/loss sharing, investment details, duration, etc.
• Types of Partnerships: Includes General Partnership, Limited Partnership (LP), and Limited Liability Partnership (LLP), each with different structures regarding liability and management roles.

Problems like this help reinforce the understanding of how investment duration impacts profit distribution in a business partnership.