Understanding Business Partnership Profit Sharing

This question deals with the concept of business partnerships where profits are shared among partners in proportion to the capital invested by each partner. We are given the total capital, the total profit, and the profit shares of two partners (A and B), and we need to find the capital contributed by the third partner (C).

The fundamental principle here is that the ratio of the profits received by the partners is equal to the ratio of their respective capitals invested in the business.

Let:

• \( C_A, C_B, C_C \) be the capitals of A, B, and C respectively.
• \( P_A, P_B, P_C \) be the profits received by A, B, and C respectively.

We are given the following information:

• Total Capital \( C_{total} = C_A + C_B + C_C = \text{Rs. } 33,000 \)
• Total Profit \( P_{total} = P_A + P_B + P_C = \text{Rs. } 15,000 \)
• A's Profit Share \( P_A = \text{Rs. } 4,500 \)
• B's Profit Share \( P_B = \text{Rs. } 5,500 \)

Calculating C's Profit Share

First, let's find the profit share of C. The total profit is the sum of the individual profit shares of A, B, and C.

\( P_{total} = P_A + P_B + P_C \)

We can find \( P_C \) by subtracting the known profits of A and B from the total profit:

\( P_C = P_{total} - P_A - P_B \)

\( P_C = 15,000 - 4,500 - 5,500 \)

\( P_C = 15,000 - (4,500 + 5,500) \)

\( P_C = 15,000 - 10,000 \)

\( P_C = \text{Rs. } 5,000 \)

So, C receives Rs. 5,000 as their share of the profit.

Determining the Ratio of Profits and Capitals

The profits of A, B, and C are Rs. 4,500, Rs. 5,500, and Rs. 5,000 respectively. The ratio of their profits is:

\( P_A : P_B : P_C = 4500 : 5500 : 5000 \)

We can simplify this ratio by dividing all numbers by a common factor. Dividing by 100:

\( 45 : 55 : 50 \)

Now, divide by 5:

\( 9 : 11 : 10 \)

So, the ratio of profits \( P_A : P_B : P_C \) is \( 9 : 11 : 10 \).

Since profits are divided in proportion to capitals, the ratio of capitals \( C_A : C_B : C_C \) is also \( 9 : 11 : 10 \).

Calculating C's Capital

The total capital is Rs. 33,000, and it is divided among A, B, and C in the ratio \( 9 : 11 : 10 \).

The sum of the ratio parts is \( 9 + 11 + 10 = 30 \).

C's capital \( C_C \) is the total capital multiplied by C's share of the ratio divided by the total ratio parts.

\( C_C = \frac{\text{C's ratio part}}{\text{Total ratio parts}} \times \text{Total Capital} \)

\( C_C = \frac{10}{30} \times 33,000 \)

\( C_C = \frac{1}{3} \times 33,000 \)

\( C_C = 11,000 \)

Therefore, C's capital is Rs. 11,000.

Alternative Method: Using Profit-to-Capital Ratio

The ratio of total profit to total capital is constant for all partners if profits are shared according to capital.

Ratio \( = \frac{\text{Total Profit}}{\text{Total Capital}} = \frac{15000}{33000} = \frac{15}{33} = \frac{5}{11} \)

This means that for every Rs. 11 of capital, there is a profit of Rs. 5.

For partner C, we know their profit share \( P_C = \text{Rs. } 5,000 \). We can use the constant ratio to find C's capital \( C_C \).

\( \frac{P_C}{C_C} = \text{Ratio} \)

\( \frac{5000}{C_C} = \frac{5}{11} \)

To solve for \( C_C \), we can cross-multiply:

\( 5000 \times 11 = 5 \times C_C \)

\( 55000 = 5 \times C_C \)

\( C_C = \frac{55000}{5} \)

\( C_C = 11000 \)

Again, we find that C's capital is Rs. 11,000.

The calculated capitals for A, B, and C are Rs. 9,900, Rs. 12,100, and Rs. 11,000 respectively. Let's verify if these sum up to the total capital:

\( 9900 + 12100 + 11000 = 22000 + 11000 = 33000 \)

The sum matches the total capital, confirming our calculation for C's capital is correct.

Conclusion on C's Capital

Based on the profit shares and the principle that profits are divided in proportion to capital, C's profit share is Rs. 5,000. This profit share corresponds to a capital investment of Rs. 11,000, maintaining the same profit-to-capital ratio as the overall business.

Revision Table: Business Partnership Key Concepts

Additional Information: Partnership Profit Distribution

In a business partnership, the method of distributing profits (and losses) is usually decided upon by the partners through a partnership deed. Common methods include:

• Equal Distribution: Regardless of capital, profits are shared equally among partners.
• Capital Ratio: Profits are shared in the exact proportion of the capital invested by each partner. This is the method used in the given problem.
• Agreed Ratio: Partners might agree on a specific ratio for profit sharing, which may or may not be directly based on their initial capital investment, but might consider factors like active participation, expertise, etc.
• Salary/Commission + Remaining Profit: Partners might receive a fixed salary or commission for their work, and the remaining profit is then distributed based on capital ratio or another agreed ratio.

Understanding the specific profit-sharing agreement is crucial for calculating individual shares of profit or determining capital contributions when profit shares are known.