Understanding Business Partnership and Profit Sharing

In a business partnership, the profit earned at the end of a period is typically shared among the partners based on the ratio of the product of their invested capital and the time period for which the capital was invested. This fundamental principle guides how profits are distributed when partners invest different amounts of capital for different durations.

Let's analyze the given problem involving partners A, B, and C.

• A started the business with a capital of Rs. 54,000.
• A's capital was invested for the entire year, which is 12 months.
• B was admitted after 4 months. This means B's capital was invested for \(12 - 4 = 8\) months.
• C was admitted after 6 months. This means C's capital was invested for \(12 - 6 = 6\) months.
• At the end of the year, the profit was divided among A, B, and C in the ratio \(1 : 4 : 5\).

Calculating the Product of Capital and Time for Each Partner

The ratio of profits is equal to the ratio of the product of capital and time for each partner. Let \(C_A\), \(C_B\), and \(C_C\) be the capitals invested by A, B, and C respectively, and \(T_A\), \(T_B\), and \(T_C\) be the time periods for which their capitals were invested (in months).

Given:

• \(C_A = 54,000\) Rs.
• \(T_A = 12\) months.
• \(T_B = 8\) months.
• \(T_C = 6\) months.

The product of Capital and Time for A is:

\(C_A \times T_A = 54,000 \times 12 = 648,000\)

Let the profit ratio be \(P_A : P_B : P_C\).

We are given \(P_A : P_B : P_C = 1 : 4 : 5\).

The ratio of (Capital × Time) is \(C_A T_A : C_B T_B : C_C T_C\). Therefore:

\(C_A T_A : C_B T_B : C_C T_C = P_A : P_B : P_C\)

\(648,000 : C_B \times 8 : C_C \times 6 = 1 : 4 : 5\)

Finding the Capital Invested by B (\(C_B\))

We can compare the ratios of A and B:

\(\frac{C_A T_A}{C_B T_B} = \frac{P_A}{P_B}\)

\(\frac{648,000}{C_B \times 8} = \frac{1}{4}\)

Cross-multiplying gives:

\(1 \times (C_B \times 8) = 4 \times 648,000\)

\(8C_B = 2,592,000\)

Now, divide by 8 to find \(C_B\):

\(C_B = \frac{2,592,000}{8}\)

\(C_B = 324,000\)

So, B invested Rs. 324,000.

Finding the Capital Invested by C (\(C_C\))

We can compare the ratios of A and C:

\(\frac{C_A T_A}{C_C T_C} = \frac{P_A}{P_C}\)

\(\frac{648,000}{C_C \times 6} = \frac{1}{5}\)

Cross-multiplying gives:

\(1 \times (C_C \times 6) = 5 \times 648,000\)

\(6C_C = 3,240,000\)

Now, divide by 6 to find \(C_C\):

\(C_C = \frac{3,240,000}{6}\)

\(C_C = 540,000\)

So, C invested Rs. 540,000.

Calculating the Sum of Capitals Invested by B and C

The question asks for the sum of the capitals invested by B and C.

Sum = \(C_B + C_C\)

Sum = \(324,000 + 540,000\)

Sum = \(864,000\)

The sum of the capitals invested by B and C is Rs. 864,000.

Let's summarize the investments and time periods in a table:

The ratio of Capital × Time is \(648,000 : 8C_B : 6C_C\). This ratio must be proportional to the profit ratio \(1 : 4 : 5\).

This confirms our calculations:

For B: \(\frac{648,000}{8C_B} = \frac{1}{4} \implies 8C_B = 648,000 \times 4 \implies C_B = 324,000\)

For C: \(\frac{648,000}{6C_C} = \frac{1}{5} \implies 6C_C = 648,000 \times 5 \implies C_C = 540,000\)

Sum of B and C's capital = \(324,000 + 540,000 = 864,000\)

Revision Table: Business Partnership Basics

Additional Information on Profit Calculation

In real-world scenarios, calculating profit share can involve more complexities such as salaries to active partners, interest on capital, and withdrawal of capital during the period. However, for typical aptitude problems, the principle of profit being proportional to the product of capital and time is the key.

When partners join or leave mid-way, it's crucial to correctly determine the effective time period for which each partner's capital was utilized in the business during the profit calculation cycle (usually a year). The capital of a partner who joins later is considered for the remaining part of the year, and the capital of a partner who leaves earlier is considered for the period they were part of the business.

This problem is a classic example of a compound partnership question, requiring careful calculation of the Capital × Time product for each partner.