Compound Partnership & Working Partner

🎯 Learning Objective

Handle capitals that change mid-year, partners joining late, and a working partner's salary.

💡 Concept

• Capital changes mid-year → effective capital = sum of (capital × months) piece by piece
• Partner joins after k months → his time = (12 − k) months
• Compare partners on capital-months, then split the profit in that ratio
• Working partner gets salary/commission FIRST; the remaining profit splits in capital ratio
• Withdrawals work the same way — reduce the capital from that month onward

🧮 Key Formulas

Effective capital = Σ(capital × months)

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Late joiner: time = 12 − k months

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Divisible profit = Total profit − salary

✏️ Easy Example

Q. A invests ₹10,000. After 4 months, B joins with ₹15,000. Find their profit ratio at the end of the year.

1. A = 10000 × 12 = 120000
2. B = 15000 × 8 = 120000
3. Ratio = 1:1

Answer: 1 : 1

🇮🇳 Real-Life Example

In a kirana shop one partner sits at the counter every day — he draws a monthly salary first, and only then are profits split. Exams copy this exact model.

📝 Exam-Level Example

Q. A invests ₹30,000 and adds ₹10,000 more after 4 months. B invests ₹40,000 for the whole year. The annual profit is ₹11,500. Find B's share.

1. A = 30000×4 + 40000×8 = 440000
2. B = 40000×12 = 480000
3. Ratio = 44:48 = 11:12; one part = 11500/23 = 500
4. B = 12 × 500

Answer: ₹6,000

📝 Exam-Level Example

Q. A and B invest ₹40,000 and ₹60,000. A is the working partner and gets ₹1,000 per month as salary; the rest of the ₹32,000 annual profit is divided in capital ratio. Find A's total earnings.

1. Salary = 12 × 1000 = 12000
2. Divisible profit = 32000 − 12000 = 20000
3. Ratio 2:3 → A's share = 8000
4. A's total = 12000 + 8000

Answer: ₹20,000

🪄 Memory Trick

Draw a month timeline per partner, write capital × months on each piece, add. Salary always leaves the profit before any ratio touches it.

⚠️ Common Mistakes

• ❌ Using 12 months for a partner who joined mid-year
• ❌ Splitting the FULL profit in capital ratio and adding salary afterwards
• ❌ Adding fresh capital but still counting the old amount for the whole year

🏆 Exam Tips

• ✅ Effective capitals often come out equal by design — expect clean ratios like 1:1 or 11:12
• ✅ Answer check: shares plus salary must total the full profit

📌 Summary

• Changed capital → add capital × months piecewise
• Late joiner gets fewer months
• Salary first, ratio later
• All shares + salary = total profit