Partnership

Overview

Partnership is a fundamental arithmetic topic in IBPS PO Prelims that tests your ability to divide profits (or losses) among business partners based on their capital contributions and the duration of investment. This topic directly builds on your understanding of ratios and proportions, making it a natural extension of basic arithmetic concepts.

In IBPS PO Prelims, partnership questions typically appear as 1-2 questions in the Quantitative Aptitude section. They range from straightforward profit-sharing calculations to more complex scenarios involving multiple partners, changing capitals, and working vs. sleeping partners. Mastering this topic gives you quick, reliable marks because the underlying logic remains consistent across all question variations.

Students must understand two core principles: profit shares are proportional to capital invested, and when time varies, you must calculate the "capital-time product" for each partner before finding ratios.

Key Concepts

• **Simple Partnership**: All partners invest for the same duration. Profit ratio equals the ratio of their capitals directly.

• **Compound Partnership**: Partners invest for different time periods. Profit ratio equals the ratio of (Capital × Time) for each partner.

• **Capital-Time Product**: The effective investment of a partner = Capital × Number of months invested. This is the basis for all compound partnership calculations.

• **Working Partner vs. Sleeping Partner**: A working partner manages the business and may receive a salary or extra share before profit distribution. A sleeping partner only contributes capital.

• **Changing Capital**: When a partner adds or withdraws capital mid-year, calculate separate capital-time products for each period and sum them.

• **Profit Distribution Formula**: Partner A's share = (A's capital-time product ÷ Total capital-time product) × Total Profit.

• **Loss Sharing**: Losses are shared in the same ratio as profits unless specifically stated otherwise in the partnership agreement.

Formulas / Key Facts

**Simple Partnership (same time period):** Profit Ratio = Capital of A : Capital of B : Capital of C

**Compound Partnership (different time periods):** Profit Ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃) Where C = Capital invested, T = Time in months

**Individual Share Calculation:** A's Profit = (A's contribution ÷ Total contribution) × Total Profit

**When capital changes during the year:** Effective Capital = (C₁ × T₁) + (C₂ × T₂) Where C₁ is initial capital for T₁ months, C₂ is changed capital for T₂ months

**Working Partner's Share:** First deduct salary/commission from total profit, then divide remaining profit in the investment ratio.

**If profit ratio and one partner's share are known:** Total Profit = (Given share × Sum of ratio parts) ÷ That partner's ratio part

Worked Examples

**Example 1: Simple Partnership** A and B start a business with investments of ₹40,000 and ₹60,000 respectively. If the annual profit is ₹25,000, find each partner's share.

*Solution:*

• Ratio of capitals = 40,000 : 60,000 = 2 : 3
• Total parts = 2 + 3 = 5
• A's share = (2/5) × 25,000 = ₹10,000
• B's share = (3/5) × 25,000 = ₹15,000

**Example 2: Compound Partnership** A invests ₹20,000 for 12 months, B invests ₹30,000 for 8 months. If total profit is ₹3,500, find B's share.

*Solution:*

• A's capital-time = 20,000 × 12 = 2,40,000
• B's capital-time = 30,000 × 8 = 2,40,000
• Ratio = 2,40,000 : 2,40,000 = 1 : 1
• B's share = (1/2) × 3,500 = ₹1,750

**Example 3: Changing Capital** A starts a business with ₹50,000. After 4 months, B joins with ₹40,000. After 2 more months, A withdraws ₹10,000. Find the profit ratio at year end.

*Solution:*

• A's capital-time = (50,000 × 6) + (40,000 × 6) = 3,00,000 + 2,40,000 = 5,40,000
• B's capital-time = 40,000 × 8 = 3,20,000
• Ratio = 5,40,000 : 3,20,000 = 54 : 32 = 27 : 16

**Example 4: Working Partner** A and B invest ₹30,000 and ₹45,000. A manages the business and receives 10% of profit as salary. If remaining profit is ₹13,500, find total profit.

*Solution:*

• Remaining profit = 90% of total (since 10% is A's salary)
• Total profit = 13,500 × (100/90) = ₹15,000
• A's salary = 10% of 15,000 = ₹1,500
• Investment ratio = 30,000 : 45,000 = 2 : 3
• A's share from remaining = (2/5) × 13,500 = ₹5,400
• A's total = 1,500 + 5,400 = ₹6,900

Common Mistakes

• **Ignoring time differences** → Students apply capital ratio directly even when partners join at different times. Always check if investment periods are equal; if not, use capital-time products.

• **Wrong time calculation for late joiners** → If B joins after 4 months in a 12-month period, B's time is 8 months, not 4. Count from joining date to year end.

• **Forgetting to handle capital withdrawal correctly** → When A withdraws ₹10,000 from ₹50,000, the new capital is ₹40,000, not ₹10,000. Calculate separate products for before and after withdrawal.

• **Mixing up salary and profit share** → Working partner's salary is deducted first from total profit. The remaining profit is then divided in the investment ratio. Don't add salary to the ratio calculation.

• **Ratio simplification errors** → After calculating capital-time products like 2,40,000 : 1,60,000, students sometimes simplify incorrectly. Divide both by HCF carefully (here, 80,000 gives 3 : 2).

Quick Reference

• Same time period → Profit ratio = Capital ratio directly

• Different time periods → Profit ratio = (Capital × Time) ratio

• Late joiner's time = Total period − Months before joining

• Working partner: Deduct salary first, then divide remaining profit

• Capital changes mid-year: Add separate (Capital × Time) products for each phase

• Losses are shared in the same ratio as profits unless stated otherwise