Partnership

Module 09 — Lesson 0009 · In Depth · 35 min

Partnership profit is not about who put in more money or for longer — it is about money × time. Every rupee earns for every month it stays in the business; the product C×T is your share of the work done by capital.

1. The Fundamental Principle

Profit ∝ Investment × Time Period

If A invests C₁ for T₁ and B invests C₂ for T₂:

Ratio of Profit (A : B) = (C₁ × T₁) : (C₂ × T₂)

Monthly equivalent: C×T = capital if it were invested for exactly 1 month. 1000 for 12 months = 12000 for 1 month — same work.

2. Types of Partnership Problems

Type A: Simple Partnership (Same Time)

All partners invest for same duration → T cancels.

P₁ : P₂ = C₁ : C₂

Worked: A 3000, B 5000, both for 12 months → ratio 3:5. Profit 8000 → A gets 3/8×8000=3000, B gets 5000.

Type B: Compound Partnership (Different Times)

  1. Calculate monthly equivalent for each: C × months.
  2. Ratio of those products.
Worked: A 1000 for 12m, B 2000 for 6m → 1000×12=12000 vs 2000×6=12000 → 1:1 equal profit. Twice the money for half the time is equal work.

Type C: Changing Investments (Advanced)

A partner changes capital mid-year — break into time blocks and sum:

Total Contribution = (Inv₁×T₁) + (Inv₂×T₂) + …

Worked: A invests 5000 for 4m, then 4000 for 8m (withdrew 1000).
Block1: 5000×4=20000, Block2: 4000×8=32000 → total 52000.
If B invested 6000 for 12m (72000), ratio A:B = 52000:72000=13:18. Then distribute profit on 31 parts.
Key: always convert to same time unit (months). If given in days/months mix, normalize first.

3. Step-by-Step Solving Strategy

  1. Identify C and T for every partner (months).
  2. Handle joiners/leavers:
  3. Calculate ratios: C×T for each, simplify to smallest integers (e.g., 50000:75000 → 2:3).
  4. Distribute profit: total parts = sum of ratios, value of 1 part = total profit / total parts, individual = ratio part × value.

Joiners pitfall: "A starts business, B joins after 3 months" — don't give B 3 months; give B total − 3. Read whether the number given is waiting time or active time.

4. Working vs. Sleeping Partners

Sometimes the working partner (manager) gets a salary or x% commission before the rest is split by C×T.

  1. Subtract salary/commission from total profit.
  2. Divide remainder by C×T ratio.
Worked: Total profit 10000, A (working) gets 10% commission, remaining split A:B = 2:3 (on investment).
Step1: A commission = 10%×10000=1000, remaining =9000.
Step2: A gets 2/5×9000=3600, B gets 5400 → A total =1000+3600=4600, B=5400.
Order matters: commission first, then ratio.

5. Summary of Variables

Symbol Meaning
P Profit share
C / I Capital / investment
T Time period (months)
Total Profit Sum of all shares
Monthly Equivalent C × months — capital for 1 month

Profit share ratio is proportional to?

A:1000 for 12m, B:2000 for 6m → ratio?

Same time for all → ratio is?

A:5000×4m +4000×8m → total monthly equiv?

B joins after 4 months in a 12m year → B's time?

Working partner gets 10% of 10000, remainder 2:3 → working partner total?

Notes

Partnership :- Principle :-

Profit ∝ C×T ; monthly equiv ⇒ C×months (1-month work)

Ratio ⇒ (C₁T₁):(C₂T₂) ; same time → C₁:C₂

Types :-

a) Simple (same T) ⇒ C₁:C₂

b) Compound (diff T) ⇒ C×T products (eg 1000×12=2000×6→1:1)

c) Changing ⇒ sum blocks: Σ(Invᵢ×Tᵢ) (eg 5000×4+4000×8=52000)

Strategy :-

1) C,T per partner (months) ; 2) joins: active = total−wait ; 3) C×T → simplify ; 4) profit/parts

Working/Sleeping :-

Salary/commission first → remainder by C×T

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