Partnership Business

When two or more individuals (called partners বা অংশীদার) combine their financial capital, labor, and skills to jointly operate a commercial enterprise and agree to distribute the resulting profit or share the loss according to a predetermined agreement, the business is called a partnership business (অংশীদারি কারবার).

• Principal / Capital (মূলধন): The initial sum of money invested by each partner into the business fund.
• Duration of Investment (সময়কাল): The length of time (in months or years) for which each partner's capital remains actively employed in the business.
• Gross Profit vs. Net Distributable Profit: Total revenue minus basic operational expenses gives the gross profit. If any managing salaries, reserve funds, or loan repayments are stipulated, these are subtracted first to yield the net distributable profit.
• Share of Profit or Loss: Distributed strictly in proportion to the agreed ratio of investments. If a business suffers a financial loss, partners must absorb the loss in the exact same ratio as profit!

A partnership business is classified as a simple partnership (সম অংশীদারি কারবার) when the capitals of all partners remain invested in the business for the exact same duration of time (e.g., each partner's capital is invested for exactly 1 full year or 6 months).

Let three partners $A, B, C$ invest capitals $C_1, C_2, C_3$ respectively for 1 year, and let the total net profit at year end be $P$:

1. Find the simplified, irreducible ratio of capitals: $$\text{Ratio} = C_1 : C_2 : C_3$$
2. Calculate the sum of the ratio terms: $$\text{Sum of Terms} = S = C_1 + C_2 + C_3$$
3. Individual profit shares are computed by multiplying total profit by each partner's proportional fraction:
   $$\text{Share of } A = P \times \frac{C_1}{C_1 + C_2 + C_3}$$
   $$\text{Share of } B = P \times \frac{C_2}{C_1 + C_2 + C_3}$$
   $$\text{Share of } C = P \times \frac{C_3}{C_1 + C_2 + C_3}$$
4. Sanity Check: The sum of individual profit shares must exactly equal the total net profit: $\text{Share}_A + \text{Share}_B + \text{Share}_C = P$.

In commercial reality, partners rarely invest funds for identical durations. One partner may invest at the beginning of the year, while another joins 4 months later, or a partner may withdraw part of their investment mid-year. This is a compound partnership (মিশ্র অংশীদারি কারবার).

It would be fundamentally unfair to compare Rs. 10,000 invested for 12 months with Rs. 10,000 invested for only 3 months! To equalize different durations, we convert every investment into its equivalent capital for 1 month (১ মাসের তুল্য মূলধন):

Therefore, in a compound partnership, profit or loss is distributed in the ratio of their equivalent monthly capitals: $$\mathbf{\text{Ratio of Profit} = (C_1 \times t_1) : (C_2 \times t_2) : (C_3 \times t_3)}$$

When a partner alters their capital investment during the course of the financial year, the equivalent monthly capital is calculated in separate time intervals:

• Case A (Additional Investment): Partner starts with capital $C$, and after $m$ months invests an additional amount $\Delta C$:
  For the first $m$ months, capital is $C$.
  For the remaining $(12 - m)$ months, capital is $(C + \Delta C)$. $$\mathbf{\text{Equivalent Monthly Capital} = C \times m + (C + \Delta C) \times (12 - m)}$$
• Case B (Partial Withdrawal): Partner starts with capital $C$, and after $m$ months withdraws an amount $\Delta C$:
  For the first $m$ months, capital is $C$.
  For the remaining $(12 - m)$ months, capital is $(C - \Delta C)$. $$\mathbf{\text{Equivalent Monthly Capital} = C \times m + (C - \Delta C) \times (12 - m)}$$

In advanced Madhyamik board problems, profit distribution follows a strict sequential protocol (waterfall) before remaining funds are divided according to capital ratios:

1. Gross Profit: Total annual earnings of the business.
2. Pre-Distribution Deduction 1 (Working Partner Remuneration):
   If partner $A$ manages the business and receives a monthly salary $S$, total annual salary $= 12 \times S$ is deducted.
   If partner $A$ is entitled to a management commission of $p\%$ of profit, $P \times \frac{p}{100}$ is deducted.
3. Pre-Distribution Deduction 2 (Bank Loans & Reserves):
   Any loan repayment installment or emergency reserve fund is deducted from the gross profit.
4. Net Distributable Profit: $$\text{Net Distributable Profit} = \text{Gross Profit} - (\text{Salaries} + \text{Commissions} + \text{Loan Installments})$$
5. Division in Capital Ratio:
   The remaining net profit is divided among partners according to their simple or compound capital ratio.
6. Total Income for Working Partner: $$\text{Total Income} = \text{Annual Salary / Commission} + \text{Share of Remaining Net Profit}$$

Another classic board problem format states: "A specified percentage (e.g., $50\%$) of the profit is divided equally among all partners, and the remaining $50\%$ is divided in the ratio of their capital investments."

• Total Profit $= P$.
• Part 1 ($50\% = 0.5P$): Divided equally among $n$ partners: each receives $\frac{0.5P}{n}$.
• Part 2 ($50\% = 0.5P$): Divided among partners in the ratio $C_1 : C_2 : C_3$.
• Total share of each partner $= \text{Equal Share} + \text{Capital Proportional Share}$.