Understanding Partnership Investment Ratios and Profit Shares

In a business partnership, the profit earned is typically distributed among partners in the ratio of their investments and the time period for which the investment is made. In this specific question, the time period is assumed to be the same for all partners since it's not mentioned otherwise. Therefore, the ratio of their profit shares will be the same as the ratio of their investments.

Analyzing the Investment Relationships

We are given two key relationships between the investments of partners A, B, and C:

1. Three times the investment of A equals four times the Investment of B. Mathematically, this can be written as \(3 \times \text{Investment of A} = 4 \times \text{Investment of B}\).
2. The Capital (Investment) of B is twice that of C. Mathematically, this is \(\text{Investment of B} = 2 \times \text{Investment of C}\).

Calculating the Investment Ratio

Our goal is to find the ratio of the investments of A, B, and C, i.e., Investment of A : Investment of B : Investment of C. We can use the given relationships to express all investments in terms of a single variable, say, the investment of C.

• From the second relationship, we know that the Investment of B is \(2 \times \text{Investment of C}\). Let's represent Investment of C as \(I_C\). Then, Investment of B, \(I_B\), is \(2 I_C\).
• Now, use the first relationship: \(3 \times \text{Investment of A} = 4 \times \text{Investment of B}\). Substitute the value of \(I_B\) in terms of \(I_C\): \(3 \times I_A = 4 \times (2 I_C)\) \(3 I_A = 8 I_C\) To find \(I_A\) in terms of \(I_C\), divide both sides by 3: \(I_A = \frac{8}{3} I_C\)

Now we have the investments of A and B in terms of the investment of C:

• Investment of A, \(I_A = \frac{8}{3} I_C\)
• Investment of B, \(I_B = 2 I_C\)
• Investment of C, \(I_C = I_C\)

The ratio of their investments A : B : C is:

\(I_A : I_B : I_C = \frac{8}{3} I_C : 2 I_C : I_C\)

To simplify this ratio and express it using whole numbers, we can multiply each part of the ratio by the least common multiple of the denominators, which is 3.

\(\left(\frac{8}{3} I_C\right) \times 3 : (2 I_C) \times 3 : (I_C) \times 3\)

\(= 8 I_C : 6 I_C : 3 I_C\)

We can cancel out \(I_C\) from all parts of the ratio (assuming \(I_C \neq 0\)), which gives us the investment ratio:

\(8 : 6 : 3\)

Determining the Profit Share Ratio

Since the profit share ratio is the same as the investment ratio when the time period is constant, the ratio of the share of each partner in the profit is A : B : C = 8 : 6 : 3.

Revision Table: Key Concepts in Partnership Profit Sharing

Additional Information on Partnership Calculations

Understanding the relationship between investment and profit distribution is crucial in partnership problems. Here are some related points:

• Time Factor: If partners invest for different durations, the profit share ratio is calculated based on the product of their investment amount and the time duration. For example, if A invests Rs. 100 for 12 months and B invests Rs. 200 for 6 months, their effective investment ratio for profit sharing is \((100 \times 12) : (200 \times 6) = 1200 : 1200 = 1 : 1\).
• Variable Investments: Sometimes partners change their investment amount during the year. In such cases, the investment is calculated based on the sum of (investment amount \(\times\) time duration) for each period the investment was held.
• Working Partners: Some partnership agreements might include a provision for salary or commission for a working partner before the remaining profit is distributed based on the investment ratio.
• Loss Sharing: Losses are typically shared among partners in the same ratio as profits, unless the partnership deed specifies otherwise.

In this specific problem, since no time periods or special conditions are mentioned, we assume that all investments were made for the same duration, making the profit share ratio directly proportional to the investment ratio.