Partnership Explained: Concepts, Formulas, and Solved Examples - Aptitude Questions & Answers

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Partnership Explained: Concepts, Formulas, and Solved Examples

Partnership Explained: Concepts, Formulas, and Solved Examples is one of the most important topics in Quantitative Aptitude. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.

Partnership: Complete Guide with Formulas & Examples

What is Partnership?

Partnership is a business arrangement where two or more individuals pool their resources (capital, skills, or labor) to start and operate a business. Partners share the profits or losses according to their agreed terms, typically based on their investment ratios and time periods.

Example: Ram invests ₹50,000 and Shyam invests ₹30,000 to start a grocery store. They agree to share profits in proportion to their investments. This arrangement forms a partnership.

Types of Partnerships

1. Simple Partnership

All partners invest for the same time period. Profit sharing ratio is based only on capital investment.

Profit Share ∝ Capital Investment

Example: A invests ₹40,000, B invests ₹60,000 for 1 year. Profit: ₹50,000. Ratio = 40:60 = 2:3. A\'s share = (2/5)×50,000 = ₹20,000

2. Compound Partnership

Partners invest for different time periods. Profit sharing ratio considers both capital and time.

Profit Share ∝ (Capital × Time)

Example: A invests ₹50,000 for 6 months, B invests ₹30,000 for 12 months. Ratio = (50,000×6):(30,000×12) = 3,00,000:3,60,000 = 5:6

3. Partnership with Salary/Commission

One partner receives fixed salary/commission before profit distribution.

Example: A manages business and gets ₹10,000 monthly salary. After paying salary, remaining profit is shared 2:3 between A and B.

Fundamental Partnership Formulas

Profit Share Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : (Capital₃ × Time₃)
Individual Share = (Individual Ratio / Total Ratio) × Total Profit

Key Points to Remember:

  1. Time should be in same units (months or years)
  2. Ratio should be reduced to simplest form
  3. When time is same, ratio depends only on capital
  4. When capital is same, ratio depends only on time

Types of Partnership Problems

Type 1: Finding Profit Ratio Given Investments and Time

Example Problem: Three partners A, B, and C invested ₹5,000 for 4 months, ₹6,000 for 7 months, and ₹7,000 for 6 months respectively. Find their profit sharing ratio.

  1. Calculate investment×time for each partner:
    • A: 5,000 × 4 = 20,000
    • B: 6,000 × 7 = 42,000
    • C: 7,000 × 6 = 42,000
  2. Ratio = 20,000 : 42,000 : 42,000
  3. Simplify: Divide by 2,000 → 10 : 21 : 21
Profit Sharing Ratio = 10 : 21 : 21

Example 2: A starts business with ₹80,000. After 3 months, B joins with ₹1,00,000. After another 4 months, C joins with ₹1,20,000. At year end, profit is ₹1,50,000. Find each partner\'s share.

  1. Calculate investment periods:
    • A: 12 months (whole year)
    • B: 9 months (after 3 months)
    • C: 5 months (after 7 months)
  2. Calculate capital×time:
    • A: 80,000 × 12 = 9,60,000
    • B: 1,00,000 × 9 = 9,00,000
    • C: 1,20,000 × 5 = 6,00,000
  3. Ratio = 9,60,000 : 9,00,000 : 6,00,000
  4. Simplify: Divide by 60,000 → 16 : 15 : 10
  5. Total ratio = 16 + 15 + 10 = 41
  6. A\'s share = (16/41) × 1,50,000 = ₹58,536.59
  7. B\'s share = (15/41) × 1,50,000 = ₹54,878.05
  8. C\'s share = (10/41) × 1,50,000 = ₹36,585.37

Type 2: Finding Investment Ratio Given Profit Ratio and Time

Example Problem: Three partners shared profit in ratio 3:6:5. They invested for 12, 9, and 6 months respectively. Find their investment ratio.

  1. Let investments be A, B, C
  2. Given: 12A : 9B : 6C = 3 : 6 : 5
  3. From 12A : 9B = 3 : 6
  4. 12A/9B = 3/6 = 1/2
  5. 4A/3B = 1/2 → 8A = 3B → B = 8A/3
  6. From 12A : 6C = 3 : 5
  7. 12A/6C = 3/5 → 2A/C = 3/5 → 10A = 3C → C = 10A/3
  8. A : B : C = A : (8A/3) : (10A/3)
  9. Multiply by 3: 3A : 8A : 10A
  10. Ratio = 3 : 8 : 10
Investment Ratio = 3 : 8 : 10

Example 2: A and B invested in ratio 4:5. After 3 months, A withdraws 1/4 of his capital and B withdraws 1/5 of his capital. At year end, profit is ₹76,000. Find B\'s share.

  1. Let initial investments: A = 4x, B = 5x
  2. First 3 months: A = 4x, B = 5x
  3. After 3 months: A withdraws 1/4 → Remaining = 3x (for 9 months)
  4. B withdraws 1/5 → Remaining = 4x (for 9 months)
  5. Calculate weighted investments:
    • A: (4x × 3) + (3x × 9) = 12x + 27x = 39x
    • B: (5x × 3) + (4x × 9) = 15x + 36x = 51x
  6. Ratio = 39x : 51x = 13 : 17
  7. Total ratio = 13 + 17 = 30
  8. B\'s share = (17/30) × 76,000 = ₹43,066.67

Type 3: Finding Time Given Profit Ratio and Investments

Example Problem: Aman starts business with ₹70,000. Ajeet joins later with ₹35,000. At year end, profit is shared 8:2. For how many months did Ajeet invest?

  1. Aman\'s investment: ₹70,000 for 12 months
  2. Ajeet\'s investment: ₹35,000 for \'x\' months
  3. Profit ratio = 8:2 = 4:1
  4. (70,000 × 12) : (35,000 × x) = 4 : 1
  5. 8,40,000 : 35,000x = 4 : 1
  6. 8,40,000/35,000x = 4/1
  7. 24/x = 4/1
  8. x = 6 months
Ajeet invested for 6 months

Example 2: A starts business with ₹50,000. After 4 months, B joins. At year end, they share profit equally. How much did B invest?

  1. A\'s investment: ₹50,000 for 12 months
  2. B\'s investment: ₹x for 8 months (joins after 4 months)
  3. Profit share equal → Ratio = 1:1
  4. (50,000 × 12) : (x × 8) = 1 : 1
  5. 6,00,000 : 8x = 1 : 1
  6. 6,00,000 = 8x
  7. x = ₹75,000

Advanced Partnership Concepts

1. Partnership with Variable Capital

When partners add or withdraw capital during the partnership period.

Method: Product Method

Calculate capital×time for each period and sum them.

Example: A starts with ₹1,00,000. After 4 months, he adds ₹50,000. After 8 months, he withdraws ₹30,000. B starts with ₹1,20,000. After 6 months, he withdraws ₹40,000. Partnership for 1 year. Find profit sharing ratio.

  1. A\'s calculation:
    • Period 1: 1,00,000 × 4 = 4,00,000
    • Period 2: 1,50,000 × 4 = 6,00,000 (after adding ₹50,000)
    • Period 3: 1,20,000 × 4 = 4,80,000 (after withdrawing ₹30,000)
    • Total = 4,00,000 + 6,00,000 + 4,80,000 = 14,80,000
  2. B\'s calculation:
    • Period 1: 1,20,000 × 6 = 7,20,000
    • Period 2: 80,000 × 6 = 4,80,000 (after withdrawing ₹40,000)
    • Total = 7,20,000 + 4,80,000 = 12,00,000
  3. Ratio = 14,80,000 : 12,00,000 = 148 : 120 = 37 : 30

2. Partnership with Salary/Commission

When one partner receives fixed payment before profit distribution.

Step-by-Step Calculation:

  1. Deduct salary/commission from total profit
  2. Distribute remaining profit according to capital ratio
  3. Add salary to the partner\'s share

Example: A and B invest ₹80,000 and ₹1,20,000 respectively. A gets ₹2,000 monthly salary. After 1 year, profit is ₹1,80,000. Calculate their shares.

  1. A\'s annual salary = 2,000 × 12 = ₹24,000
  2. Remaining profit = 1,80,000 - 24,000 = ₹1,56,000
  3. Investment ratio = 80,000:1,20,000 = 2:3
  4. A\'s profit share = (2/5) × 1,56,000 = ₹62,400
  5. B\'s profit share = (3/5) × 1,56,000 = ₹93,600
  6. A\'s total = Salary + Profit = 24,000 + 62,400 = ₹86,400
  7. B\'s total = ₹93,600

3. Partnership with Interest on Capital

When partners receive interest on their capital before profit distribution.

Example: A and B invest ₹1,00,000 and ₹1,50,000. They get 10% annual interest on capital. Remaining profit shared 3:2. Total profit ₹80,000.

  1. Interest calculation:
    • A\'s interest = 1,00,000 × 10% = ₹10,000
    • B\'s interest = 1,50,000 × 10% = ₹15,000
    • Total interest = ₹25,000
  2. Remaining profit = 80,000 - 25,000 = ₹55,000
  3. A\'s profit share = (3/5) × 55,000 = ₹33,000
  4. B\'s profit share = (2/5) × 55,000 = ₹22,000
  5. A\'s total = 10,000 + 33,000 = ₹43,000
  6. B\'s total = 15,000 + 22,000 = ₹37,000

4. When Partners Join/Leave at Different Times

Example: A, B, C start with capitals ₹50,000, ₹60,000, ₹70,000. A leaves after 4 months, B leaves after 6 months. C remains for full year. Profit ₹1,50,000.

  1. Investment periods:
    • A: 4 months
    • B: 6 months
    • C: 12 months
  2. Capital×time:
    • A: 50,000 × 4 = 2,00,000
    • B: 60,000 × 6 = 3,60,000
    • C: 70,000 × 12 = 8,40,000
  3. Ratio = 2,00,000 : 3,60,000 : 8,40,000 = 20 : 36 : 84 = 5 : 9 : 21
  4. Total ratio = 5 + 9 + 21 = 35
  5. A\'s share = (5/35) × 1,50,000 = ₹21,428.57
  6. B\'s share = (9/35) × 1,50,000 = ₹38,571.43
  7. C\'s share = (21/35) × 1,50,000 = ₹90,000

5. Partnership with Guaranteed Minimum Profit

When one partner is guaranteed minimum profit regardless of actual profit.

Example: A and B invest ₹1,00,000 and ₹2,00,000. B guaranteed minimum ₹40,000 profit. Total profit ₹90,000.

  1. Normal distribution (ratio 1:2):
    • A\'s share = (1/3) × 90,000 = ₹30,000
    • B\'s share = (2/3) × 90,000 = ₹60,000
  2. Since B\'s normal share (₹60,000) > guaranteed minimum (₹40,000), normal distribution applies.
  3. Case 2: If total profit was ₹60,000:
    • Normal distribution: A = ₹20,000, B = ₹40,000
    • B gets exactly guaranteed minimum → No adjustment needed
  4. Case 3: If total profit was ₹45,000:
    • Normal distribution: A = ₹15,000, B = ₹30,000
    • But B guaranteed ₹40,000 → Shortfall = ₹10,000
    • This ₹10,000 comes from A\'s share
    • Final: A = 15,000 - 10,000 = ₹5,000, B = ₹40,000

Partnership: Frequently Asked Questions

What is the basic principle behind profit sharing in partnership?

Profit is shared in proportion to the product of capital invested and time period (Capital × Time). If all partners invest for same time, profit is shared in capital ratio only.

How to handle when partners join/leave at different times?

Calculate each partner\'s effective investment as (Capital × Time period they were active). Then find ratio of these products to determine profit share.

What happens when a partner receives salary or commission?

Salary/commission is deducted from total profit first. Remaining profit is then distributed according to capital ratio. The partner receiving salary gets both salary and their profit share.

How to calculate when capital changes during partnership?

Break the period into segments where capital remains constant. Calculate (Capital × Time) for each segment and sum for each partner. Use these sums to find ratio.

What is the difference between simple and compound partnership?

In simple partnership, all partners invest for same time period. In compound partnership, investment periods differ, so we multiply capital by time for each partner.

Practice Problems

Problem 1

A, B, C invest ₹20,000, ₹30,000, ₹40,000 respectively. B invests for 8 months, others for 12 months. Profit ₹56,000. Find C\'s share.

Solution: Ratio = (20,000×12) : (30,000×8) : (40,000×12) = 2,40,000 : 2,40,000 : 4,80,000 = 1:1:2. Total ratio = 4. C\'s share = (2/4)×56,000 = ₹28,000

Problem 2

A starts with ₹50,000. After 4 months, B joins with ₹75,000. After another 4 months, C joins with ₹1,00,000. At year end, profit ₹1,45,000. Find B\'s share.

Solution: A: 50,000×12 = 6,00,000. B: 75,000×8 = 6,00,000. C: 1,00,000×4 = 4,00,000. Ratio = 6:6:4 = 3:3:2. Total = 8. B\'s share = (3/8)×1,45,000 = ₹54,375

Problem 3

Three partners share profit 4:8:9. They invested for 6 months, 8 months, and 1 year respectively. Find their investment ratio.

Solution: Let investments be A, B, C. 6A:8B:12C = 4:8:9. From 6A:8B = 4:8 → 6A/8B = 1/2 → A/B = 2/3. From 8B:12C = 8:9 → 8B/12C = 8/9 → B/C = 4/3. A:B:C = 2:3 and B:C = 4:3 → A:B:C = 8:12:9

Problem 4

A and B invest ₹60,000 and ₹90,000. A gets 15% of profit as salary. Remaining shared 2:3. Total profit ₹1,20,000. Find A\'s total earnings.

Solution: A\'s salary = 15% of 1,20,000 = ₹18,000. Remaining = ₹1,02,000. A\'s profit = (2/5)×1,02,000 = ₹40,800. Total = 18,000 + 40,800 = ₹58,800

Problem 5

X invests ₹80,000. After 6 months, Y joins. At year end, they share equally. How much did Y invest?

Solution: X: 80,000×12 = 9,60,000. Y: P×6. Equal share → 9,60,000 = 6P → P = ₹1,60,000. Y invested ₹1,60,000

Quick Calculation Formulas

When time same:
Profit Share = (Individual Capital / Total Capital) × Total Profit
When capital same:
Profit Share = (Individual Time / Total Time) × Total Profit
General Formula:
Individual Share = (Capitalᵢ × Timeᵢ) / Σ(Capitalᵢ × Timeᵢ) × Total Profit
When partner joins/leaves:
Consider only period when partner was active
Equivalent Capital = Original Capital × (Active Months / Total Months)

Related Topics

Profit and Loss Calculations Percentage and Ratios Simple and Compound Interest Business Mathematics

Frequently Asked Questions

What is Partnership Explained: Concepts, Formulas, and Solved Examples?

Partnership Explained: Concepts, Formulas, and Solved Examples is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.

Is Partnership Explained: Concepts, Formulas, and Solved Examples important for competitive exams?

Yes, Partnership Explained: Concepts, Formulas, and Solved Examples is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.

How to prepare Partnership Explained: Concepts, Formulas, and Solved Examples easily?

Practice solved examples, learn formulas and shortcuts, and attempt practice questions regularly to master Partnership Explained: Concepts, Formulas, and Solved Examples.

What are the important formulas in Partnership Explained: Concepts, Formulas, and Solved Examples?

Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Partnership Explained: Concepts, Formulas, and Solved Examples.

How many questions come from Partnership Explained: Concepts, Formulas, and Solved Examples?

Typically 5-10 questions come from Partnership Explained: Concepts, Formulas, and Solved Examples in most competitive exams, making it a high-scoring section.

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