🤝 Partnership - Complete Theory

Master Partnership - profit sharing made easy!

🎯 What is Partnership?

Partnership is a business arrangement where two or more people agree to share profits and losses.

Key Concept: Profit/Loss is shared in the ratio of their investments (capital × time).

📐 Types of Partnership

1. Simple Partnership

When all partners invest for the SAME time period.

Profit Ratio = Capital Ratio

If A invests ₹x and B invests ₹y:
Profit sharing ratio = x : y

2. Compound Partnership

When partners invest for DIFFERENT time periods.

Profit Ratio = (Capital × Time) Ratio

If A invests ₹x for t₁ months and B invests ₹y for t₂ months:
Profit ratio = (x × t₁) : (y × t₂)

📊 Basic Formulas

Formula 1: Profit Sharing (Simple Partnership)

A's Profit : B's Profit = A's Capital : B's Capital

If total profit = P
A's share = [A's capital / Total capital] × P

Example: A invests ₹30,000, B invests ₹20,000. Profit = ₹5,000. Find A’s share.

Ratio = 30,000 : 20,000 = 3:2
Total parts = 5

A's share = (3/5) × 5,000 = ₹3,000

Formula 2: Profit Sharing (Compound Partnership)

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

A's share = [(C₁ × T₁) / (C₁×T₁ + C₂×T₂)] × Total Profit

Example: A invests ₹40,000 for 6 months, B invests ₹30,000 for 8 months. Profit = ₹10,000. Find each share.

Ratio = (40,000 × 6) : (30,000 × 8)
      = 240,000 : 240,000
      = 1 : 1

A's share = ₹5,000
B's share = ₹5,000

💼 Working & Sleeping Partners

1. Working Partner (Active Partner)

Partner who actively manages the business.

• Gets extra commission/salary for managing
• Receives: Share of profit + Management fees

2. Sleeping Partner (Silent Partner)

Partner who only invests money, doesn’t work.

• Only gets profit share
• No management fees

Example with Working Partner

Q: A and B invest ₹50,000 each. A is working partner and gets 10% of profit as salary. Remaining profit (₹9,000) is shared equally. Find A’s total income.

Solution:

Total profit before A's salary = P

A's salary = 10% of P = 0.1P
Remaining = P - 0.1P = 0.9P

This remaining = ₹9,000 (shared equally)
0.9P = 9,000
P = 10,000

A's salary = 10% of 10,000 = ₹1,000
A's profit share = 9,000 / 2 = ₹4,500

A's total income = 1,000 + 4,500 = ₹5,500

Answer: ₹5,500

💡 Solved Examples

Example 1: Simple Partnership

Q: A, B, C invest ₹20,000, ₹30,000, ₹50,000. Profit is ₹40,000. Find each partner’s share.

Solution:

Ratio = 20,000 : 30,000 : 50,000
      = 2 : 3 : 5
Total parts = 10

A's share = (2/10) × 40,000 = ₹8,000
B's share = (3/10) × 40,000 = ₹12,000
C's share = (5/10) × 40,000 = ₹20,000

Answer: A = ₹8,000, B = ₹12,000, C = ₹20,000

Example 2: Compound Partnership

Q: A invests ₹36,000 for 4 months, B invests ₹48,000 for 6 months. Profit = ₹18,000. Find shares.

Solution:

Ratio = (36,000 × 4) : (48,000 × 6)
      = 144,000 : 288,000
      = 1 : 2
Total parts = 3

A's share = (1/3) × 18,000 = ₹6,000
B's share = (2/3) × 18,000 = ₹12,000

Answer: A = ₹6,000, B = ₹12,000

Example 3: Finding Investment

Q: A and B are partners. A’s share of profit is ₹12,000 and B’s is ₹8,000. If A invested ₹60,000, find B’s investment.

Solution:

Profit ratio = 12,000 : 8,000 = 3:2
Capital ratio = 3:2 (same time period)

A's capital : B's capital = 3:2
60,000 : B's capital = 3:2

B's capital = (60,000 × 2) / 3 = ₹40,000

Answer: ₹40,000

Example 4: Finding Time Period

Q: A invests ₹30,000 for 12 months. B invests ₹40,000. They share profit equally. For how long did B invest?

Solution:

For equal profit sharing, ratio = 1:1

(A's capital × A's time) : (B's capital × B's time) = 1:1
(30,000 × 12) : (40,000 × t) = 1:1

360,000 = 40,000 × t
t = 9 months

Answer: 9 months

Example 5: Variable Investment

Q: A starts business with ₹50,000. After 3 months, B joins with ₹70,000. After 3 more months, C joins with ₹80,000. Profit at year-end = ₹1,26,000. Find shares.

Solution:

A's investment = 50,000 for 12 months
B's investment = 70,000 for 9 months (joined after 3)
C's investment = 80,000 for 6 months (joined after 6)

Ratio = (50,000 × 12) : (70,000 × 9) : (80,000 × 6)
      = 600,000 : 630,000 : 480,000
      = 60 : 63 : 48
      = 20 : 21 : 16
Total parts = 57

A's share = (20/57) × 126,000 = ₹44,211 (approx)
B's share = (21/57) × 126,000 = ₹46,421 (approx)
C's share = (16/57) × 126,000 = ₹35,368 (approx)

Answer: A = ₹44,211, B = ₹46,421, C = ₹35,368

Example 6: Adding/Withdrawing Capital

Q: A invests ₹40,000. After 6 months, he adds ₹10,000 more. B invests ₹50,000 whole year. Profit = ₹25,000. Find shares.

Solution:

A's investment:
First 6 months: 40,000 × 6 = 240,000
Next 6 months: 50,000 × 6 = 300,000
Total = 540,000

B's investment:
12 months: 50,000 × 12 = 600,000

Ratio = 540,000 : 600,000 = 9:10
Total parts = 19

A's share = (9/19) × 25,000 = ₹11,842 (approx)
B's share = (10/19) × 25,000 = ₹13,158 (approx)

Answer: A = ₹11,842, B = ₹13,158

🔄 Special Cases

Case 1: Partner Joins/Leaves Mid-year

Calculate capital × time for each period separately
Then add them up

Example: A invests ₹100 for 3 months, then ₹200 for 9 months
A's effective = (100 × 3) + (200 × 9) = 300 + 1800 = 2100

Case 2: Loss Sharing

Loss is shared in the SAME ratio as profit

If profit ratio = 3:2
Loss ratio = 3:2 (same!)

Case 3: Finding Total Profit

If one partner's share is known:

Total Profit = (Partner's share × Total parts) / Partner's parts

Example: A's share (3 parts) = ₹6,000, total parts = 8
Total profit = (6,000 × 8) / 3 = ₹16,000

⚡ Quick Shortcuts

Shortcut 1: Same Capital

If capitals are equal, profit ratio = Time ratio

A for 6 months, B for 9 months
Profit ratio = 6:9 = 2:3

Shortcut 2: Same Time

If time is equal, profit ratio = Capital ratio

A invests ₹40,000, B invests ₹60,000 (both for 1 year)
Profit ratio = 40,000:60,000 = 2:3

Shortcut 3: Proportional Share

Partner's share = (Partner's ratio / Sum of ratios) × Total profit

If ratio = 3:5, total = 8
First partner = 3/8 of profit
Second partner = 5/8 of profit

Shortcut 4: Reverse Calculation

If profit shares are known, capital ratio is the same (if time is same)

A gets ₹3,000, B gets ₹5,000
Capital ratio = 3:5

📊 Important Patterns

Pattern 1: Yearly Investment with Mid-year Change

Break into two periods:
Period 1: Jan-June (6 months)
Period 2: July-Dec (6 months)

Calculate each separately, then add

Pattern 2: Working Partner Commission

Total profit = P

Working partner's commission = c% of P
Remaining profit = P - (c% of P) = P(1 - c/100)

This remaining is shared in capital ratio

Pattern 3: Multiple Partners, One Leaves

Calculate capital × time for all periods

Partner who leaves: Calculate only till leaving date
Partner who stays: Calculate for full period

⚠️ Common Mistakes

❌ Mistake 1: Ignoring Time

Wrong: If capitals are different, profit ratio = capital ratio ✗
Right: Check if time is also same! If different time, use capital × time ✓

❌ Mistake 2: Adding Times Wrong

Wrong: A for 6 months + B for 3 months = 9 months together ✗
Right: Calculate (capital × time) for each separately ✓

❌ Mistake 3: Working Partner Commission

Wrong: Sharing total profit in capital ratio ✗
Right: First deduct working partner's commission, then share remaining ✓

❌ Mistake 4: Withdrawal/Addition

Wrong: Treating it as single investment for whole year ✗
Right: Break into periods before and after change ✓

📝 Practice Problems

Level 1:

1. A invests ₹40,000, B invests ₹60,000. Profit = ₹20,000. Find each share.
2. A invests ₹30,000 for 8 months, B invests ₹40,000 for 6 months. Find ratio.
3. Profit ratio is 3:5. A gets ₹6,000. Find total profit.

Level 2:

4. A invests ₹50,000. After 4 months B joins with ₹70,000. After year, profit = ₹42,000. Find shares.
5. A is working partner getting 20% commission. Remaining ₹16,000 shared with B equally. Find A’s total income.
6. A invests for 6 months, B for 12 months. Profit ratio = 3:4. If B invested ₹40,000, find A’s investment.

Level 3:

7. A starts with ₹60,000. After 4 months adds ₹20,000. B invests ₹80,000 whole year. Profit = ₹50,000. Find shares.
8. Three partners: A for whole year, B joins after 3 months, C joins after 6 months. All invest ₹100,000. Find profit ratio.
9. A and B’s capital ratio = 3:4, time ratio = 2:3. If total profit = ₹52,000, find each share.

🔗 Related Topics

Prerequisites:

• Ratio & Proportion - Foundation for partnership
• Percentage - For commission calculations

Related:

• Time & Work - Similar capital × time concept
• Profit & Loss - Business profit concepts
• Simple Interest - Investment returns

Practice:

• Partnership Questions
• Formula Sheet

Master Partnership - It’s all about (Capital × Time) ratio! 🤝