Partnership

Business Arithmetic. Master the relation between Capital, Time, and Profit ($P = C \times T$) and handle Active/Sleeping partners.

1. The Fundamental Law

Profit Ratio = (Capital ├Ч\times Time) Ratio.

  • Formula: PA:PB=(CA├ЧTA):(CB├ЧTB)P_A : P_B = (C_A \times T_A) : (C_B \times T_B).
  • If Time is same: Profit тИЭ\propto Capital.
  • If Capital is same: Profit тИЭ\propto Time.

Example:

Q: A invests 20k for 1 year. B 30k for 1 year.
Solution: PA:PB=20:30=2:3P_A : P_B = 20 : 30 = 2 : 3.

2. The 'Month-Ratio' Method

When capitals change mid-year (e.g., adding/withdrawing money).

  • Method: Calculate 'Month-Equivalent Capital'.
  • TotalCapital=(C1├ЧM1)+(C2├ЧM2)+...Total Capital = (C_1 \times M_1) + (C_2 \times M_2) + ...

Example:

Q: A invests 1000. After 4 months, adds 500. Total year.
Solution: 1000├Ч4+1500├Ч8=4000+12000=160001000 \times 4 + 1500 \times 8 = 4000 + 12000 = 16000.

3. Active vs Sleeping Partner

Active partner gets salary/commission, Rest profit is shared.

  • Equation: Net Shared Profit = Total Profit - (Salary + Commission).
  • This Net Profit is then split by C├ЧTC \times T ratio.

Example:

Q: Total Profit 1000. A gets 10% for management. Ratio A:B is 1:1.
Solution: Management Fee: 100. Distributable: 900.
A gets 100+450=550100 + 450 = 550. B gets 450.

4. Working Backwards

Finding Capital or Time from Profit Ratio.

  • Capital Ratio: CA:CB=(PA/TA):(PB/TB)C_A : C_B = (P_A / T_A) : (P_B / T_B).
  • Time Ratio: TA:TB=(PA/CA):(PB/CB)T_A : T_B = (P_A / C_A) : (P_B / C_B).

Example:

Q: Profit 2:3. Capital 1:2. Find Time Ratio.
Solution: TA:TB=(2/1):(3/2)=2:1.5=4:3T_A : T_B = (2/1) : (3/2) = 2 : 1.5 = 4 : 3.

5. Donation & Reserve Fund

Often x% is donated or kept as reserve.

  • Logic: Distribute only (100тИТx)%(100-x)\% of the Total Profit amongst partners.

Example:

Q: Profit 2000. 10% to charity. A:B is 3:2.
Solution: Distributable = 90% of 2000 = 1800.
A = 3/5├Ч1800=10803/5 \times 1800 = 1080.