Partnership
Overview
Partnership is a fundamental arithmetic topic in SBI Clerk Prelims that tests your ability to distribute profits (or losses) among business partners based on their investments and time periods.
The concept mirrors real-world business scenarios where multiple people invest different amounts for varying periods, and profits must be divided fairly. Questions range from simple two-partner problems to slightly complex scenarios involving working and sleeping partners. Mastering partnership helps you tackle related topics like ratio-proportion and time-work more confidently.
Key Concepts
- Basic Principle: Profit sharing ratio = Ratio of (Capital × Time) for each partner. If A invests ₹x for t₁ months and B invests ₹y for t₂ months, their profit ratio is x×t₁ : y×t₂.
- 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. You must calculate Capital × Time for each partner before finding the ratio.
- 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.
- Entry/Exit of Partners: When a partner joins mid-way or leaves early, calculate their contribution only for the months they were invested.
- Capital Changes During Year: If a partner adds or withdraws capital, treat each phase separately and sum up (Capital × Time) for all phases.
- Ratio Simplification: Always reduce the final ratio to its simplest form before calculating actual profit shares.
Formulas / Key Facts
Profit Share Formula: A's Share : B's Share = (A's Capital × A's Time) : (B's Capital × B's Time)
Individual Profit Calculation: A's Profit = (A's ratio part / Total ratio parts) × Total Profit
When Salary is Involved: Step 1: Deduct working partner's salary from total profit Step 2: Distribute remaining profit according to capital-time ratio
Equal Profit Condition: If profits are equal, then Capital₁ × Time₁ = Capital₂ × Time₂
Monthly Investment Calculation: When capital changes, Total Contribution = (C₁ × t₁) + (C₂ × t₂) + ...
Ratio of Investments When Profit Ratio is Given: If profit ratio and time are known, work backwards: Capital Ratio = Profit Ratio / Time Ratio
Worked Examples
Example 1: Simple Partnership A and B start a business with investments of ₹15,000 and ₹20,000 respectively. At the end of the year, total profit is ₹7,000. Find each partner's share.
Solution: Since both invest for the same period (1 year): Ratio of profits = 15,000 : 20,000 = 3 : 4 Total parts = 3 + 4 = 7 A's share = (3/7) × 7,000 = ₹3,000 B's share = (4/7) × 7,000 = ₹4,000
Example 2: Compound Partnership A invests ₹12,000 for 8 months and B invests ₹9,000 for 12 months. If total profit is ₹6,300, find the difference between their shares.
Solution: A's contribution = 12,000 × 8 = 96,000 B's contribution = 9,000 × 12 = 1,08,000 Ratio = 96,000 : 1,08,000 = 8 : 9 Total parts = 17 A's share = (8/17) × 6,300 = ₹2,964.70 ≈ ₹2,965 B's share = (9/17) × 6,300 = ₹3,335.30 ≈ ₹3,335 Difference = 3,335 - 2,965 = ₹370
Example 3: Partner Joining Later A starts a business with ₹8,000. After 4 months, B joins with ₹12,000. At the end of the year, profit is ₹4,200. Find B's share.
Solution: A invests for 12 months, B invests for (12 - 4) = 8 months A's contribution = 8,000 × 12 = 96,000 B's contribution = 12,000 × 8 = 96,000 Ratio = 96,000 : 96,000 = 1 : 1 B's share = (1/2) × 4,200 = ₹2,100
Example 4: With Salary Component A and B invest ₹5,000 and ₹6,000 respectively. A manages the business and gets ₹500 per month as salary. Annual profit is ₹8,800. Find A's total earning.
Solution: A's annual salary = 500 × 12 = ₹6,000 Remaining profit = 8,800 - 6,000 = ₹2,800 Investment ratio = 5,000 : 6,000 = 5 : 6 A's share from remaining = (5/11) × 2,800 = ₹1,272.72 ≈ ₹1,273 A's total earning = 6,000 + 1,273 = ₹7,273
Common Mistakes
- Ignoring time factor in compound partnership → Always check if investment durations differ. Even if not explicitly stated as "compound partnership," different entry/exit dates mean you must use Capital × Time.
- Using investment amounts directly as profit ratio → This works ONLY when all partners invest for equal time periods. Otherwise, multiply each investment by its respective time period first.
- Forgetting to deduct salary before profit distribution → When a working partner receives salary, subtract it from total profit first, then distribute the remainder according to the investment ratio.
- Calculating time incorrectly when partner joins later → If B joins after 4 months in a 12-month period, B's time is 8 months, not 4 months. Count from joining month to year-end.
- Not simplifying ratios → Large numbers like 96,000 : 1,08,000 should be simplified (8:9) to make calculations easier and reduce errors.
Quick Reference
- Profit Ratio = Capital₁ × Time₁ : Capital₂ × Time₂
- Same time period → Profit ratio equals capital ratio directly
- Partner joins after 'n' months → Their investment time = (12 - n) months
- Working partner's salary: deduct first, then distribute remaining profit
- Equal profits means equal (Capital × Time) products
- Always simplify ratios before calculating actual amounts