Understanding the Business Partnership Problem

This question involves a business partnership among three individuals: Ram, Mohan, and Sohan. They invested a total amount of $\text{Rs. } 50000$ to start the business. The problem provides relationships between their individual investments and asks us to find Mohan's share of the total profit of $\text{Rs. } 20000$. In a typical business partnership, profits are shared in the ratio of the capital invested by each partner, provided the investment duration is the same for all.

Calculating Individual Investments

To find Mohan's share of the profit, we first need to determine the exact amount of capital each partner invested. Let $\text{R}$, $\text{M}$, and $\text{S}$ represent the investments of Ram, Mohan, and Sohan, respectively.

We are given the following information:

• The total capital invested is $\text{Rs. } 50000$. So, $\text{R} + \text{M} + \text{S} = 50000$.
• Mohan invested $\text{Rs. } 10000$ less than Ram. This can be written as $\text{M} = \text{R} - 10000$. Rearranging this, we get $\text{R} = \text{M} + 10000$.
• Mohan invested $\text{Rs. } 8000$ more than Sohan. This can be written as $\text{M} = \text{S} + 8000$. Rearranging this, we get $\text{S} = \text{M} - 8000$.

Now we can substitute the expressions for $\text{R}$ and $\text{S}$ from the second and third points into the total capital equation:

$(\text{M} + 10000) + \text{M} + (\text{M} - 8000) = 50000$

Combine the $\text{M}$ terms and the constant terms:

$3\text{M} + (10000 - 8000) = 50000$

$3\text{M} + 2000 = 50000$

Subtract $2000$ from both sides:

$3\text{M} = 50000 - 2000$

$3\text{M} = 48000$

Divide by $3$ to find Mohan's investment:

$\text{M} = \frac{48000}{3}$

$\text{M} = 16000$

So, Mohan invested $\text{Rs. } 16000$. Now we can find Ram's and Sohan's investments using the relationships given:

• Ram's investment: $\text{R} = \text{M} + 10000 = 16000 + 10000 = 26000$
• Sohan's investment: $\text{S} = \text{M} - 8000 = 16000 - 8000 = 8000$

To verify, check if the sum of their investments equals the total capital: $26000 + 16000 + 8000 = 50000$. This is correct.

Determining the Investment Ratio

The profit from the business will be shared among Ram, Mohan, and Sohan in the ratio of their capital investments. Their investments are $\text{R} = 26000$, $\text{M} = 16000$, and $\text{S} = 8000$.

The investment ratio $\text{R} : \text{M} : \text{S}$ is:

$26000 : 16000 : 8000$

To simplify the ratio, divide all numbers by their greatest common divisor. We can start by dividing by $1000$:

$26 : 16 : 8$

Now, divide by $2$:

$13 : 8 : 4$

The simplified investment ratio of Ram, Mohan, and Sohan is $13 : 8 : 4$.

Calculating Mohan's Share of Profit

The total profit earned is $\text{Rs. } 20000$. This profit needs to be distributed according to the investment ratio $13 : 8 : 4$.

The total number of parts in the ratio is the sum of the ratio parts: $13 + 8 + 4 = 25$.

Mohan's share corresponds to $8$ parts out of the total $25$ parts. Therefore, Mohan's share of the total profit can be calculated as:

$\text{Mohan's Share} = \left(\frac{\text{Mohan's Ratio Part}}{\text{Total Ratio Parts}}\right) \times \text{Total Profit}$

$\text{Mohan's Share} = \left(\frac{8}{25}\right) \times 20000$

To calculate this, we can first divide $20000$ by $25$:

$\frac{20000}{25} = 800$

Now, multiply the result by $8$:

$\text{Mohan's Share} = 8 \times 800$

$\text{Mohan's Share} = 6400$

So, Mohan's share of the total profit of $\text{Rs. } 20000$ is $\text{Rs. } 6400$.

Let's verify the shares for all partners:

• Ram's Share: $\frac{13}{25} \times 20000 = 13 \times 800 = 10400$
• Mohan's Share: $\frac{8}{25} \times 20000 = 8 \times 800 = 6400$
• Sohan's Share: $\frac{4}{25} \times 20000 = 4 \times 800 = 3200$

Total Profit Shares = $10400 + 6400 + 3200 = 20000$, which matches the total profit.

Here is a summary table:

The calculation shows that Mohan's share of the profit is $\text{Rs. } 6400$.

Revision Table: Business Partnership Key Concepts

Additional Information: Profit Distribution in Partnerships

Understanding how profits are distributed in a partnership is crucial. While the investment ratio is the standard basis, partnership agreements can introduce variations. Here are some common aspects:

• Default Rule: If there is no explicit agreement, profits and losses are generally shared equally among partners, regardless of investment. However, competitive exams often imply profit sharing based on investment unless otherwise stated. The problem's structure here strongly suggests using the investment ratio.
• Partnership Deed: A legal document outlining the terms of the partnership, including profit and loss sharing ratios, salaries, interest on capital, etc. This overrides the default rules.
• Time Factor: If investments are made for different periods, the profit sharing is based on the product of investment and time (Capital $\times$ Time). The current problem implies investments were for the same duration as time is not mentioned.
• Capital Contribution: The primary factor in determining profit share in simple partnerships. A higher investment typically leads to a larger share of the profit.

This problem is a classic example of profit distribution based on the ratio of capital investments in a simple partnership.