Partnership Profit Sharing Calculation Explained

In a partnership, the profit is shared among the partners in a ratio that is proportional to the product of their individual capital investments and the duration for which the capital was invested.

Understanding the Given Information

We are given the following ratios for partners A, B, and C:

• Ratio of Capital Invested: A : B : C = 2 : 3 : 5
• Ratio of Time Period for Investment: A : B : C = 4 : 2 : 3

The difference between the profit shares of A and B is given as Rs. 1,86,000.

We need to find C's share of the total profit.

Calculating the Ratio of Profit Shares

The ratio of profit shares is calculated by multiplying the corresponding capital ratio and time ratio for each partner.

Profit Share Ratio = (Capital Ratio × Time Ratio)

• A's Profit Share ∝ Capital A × Time A
• B's Profit Share ∝ Capital B × Time B
• C's Profit Share ∝ Capital C × Time C

Let the capital ratio be $c_A : c_B : c_C = 2 : 3 : 5$ and the time ratio be $t_A : t_B : t_C = 4 : 2 : 3$.

The ratio of profit shares ($P_A : P_B : P_C$) is:

$\frac{P_A}{P_B} = \frac{c_A \times t_A}{c_B \times t_B}$ and $\frac{P_B}{P_C} = \frac{c_B \times t_B}{c_C \times t_C}$

So, $P_A : P_B : P_C = (c_A \times t_A) : (c_B \times t_B) : (c_C \times t_C)$

Let's calculate the products:

• For A: $2 \times 4 = 8$
• For B: $3 \times 2 = 6$
• For C: $5 \times 3 = 15$

Thus, the ratio of profit shares of A, B, and C is 8 : 6 : 15.

The profit sharing ratio is A : B : C = 8 : 6 : 15.

Using the Difference in Profit Shares to Find the Value of One Ratio Unit

Let the actual profit shares of A, B, and C be $8k$, $6k$, and $15k$, where $k$ is a constant representing the value of one unit in the profit ratio.

We are given that the difference between the profit shares of A and B is Rs. 1,86,000.

Difference = Profit Share of A - Profit Share of B

$1,86,000 = 8k - 6k$

$1,86,000 = 2k$

Now, we can find the value of $k$:

$k = \frac{1,86,000}{2}$

$k = 93,000$

So, one unit in the profit sharing ratio is equal to Rs. 93,000.

Calculating C's Share of Profit

C's share in the profit ratio is 15.

C's actual profit share = C's ratio unit × value of one ratio unit

C's actual profit share = $15 \times k$

C's actual profit share = $15 \times 93,000$

Let's perform the multiplication:

$15 \times 93,000 = 15 \times (90,000 + 3,000)$

$= 15 \times 90,000 + 15 \times 3,000$

$= 1,350,000 + 45,000$

$= 1,395,000$

C's share of profit is Rs. 13,95,000.

Summary of Profit Shares

• A's share: $8 \times 93,000 = 7,44,000$
• B's share: $6 \times 93,000 = 5,58,000$
• Difference between A and B: $7,44,000 - 5,58,000 = 1,86,000$ (Matches the given information)
• C's share: $15 \times 93,000 = 13,95,000$

Therefore, C's share of profit is Rs. 13,95,000.

Revision Table: Partnership Profit Ratios

Additional Information: Partnership Basics

A partnership is a business arrangement where two or more individuals agree to share in the profits or losses of their business.

• Capital: The amount of money or assets invested by each partner in the business.
• Time: The duration for which each partner's capital is invested. This is crucial if partners invest for different periods.
• Profit Sharing Agreement: The partners agree on how profits (and losses) will be divided. Often, this is based on the capital contributed and the time period. If no agreement exists, profits are usually shared equally, but problems like this specify a method based on investment and time.
• Types of Partners: Can include active partners, sleeping partners, nominal partners, etc., each with different roles and liabilities, which might affect profit sharing depending on the agreement. However, in typical ratio problems, the focus is purely on capital and time investment unless stated otherwise.

Understanding ratios is fundamental to solving partnership problems. A ratio represents a relationship between quantities. In this problem, ratios are used to represent the relative amounts of capital, time, and subsequently, profit shares.