Railway Exams Maths Arithmetic Partnership

Partnership is an important topic in Railway Exams Mathematics and is frequently asked in the Arithmetic section. It deals with the distribution of profit or loss among partners based on the amount of money they invest and the duration for which their investment remains in the business. 

Questions on the Partnership test a candidate's understanding of ratios, proportions, percentages, and time-based calculations. By learning the fundamental profit-sharing principles and practicing different types of problems, candidates can solve Partnership questions quickly and improve their overall score in competitive examinations. 

Understanding Partnership and Profit Distribution Principles

Partnership occurs in business when individuals combine resources. The core concept revolves around the distribution of Profit.

Key Principle:

Profit is directly proportional to the product of the Rupees (Investment) and Time for which the investment is made. 

Profit ∝ (Rupees × Time), often denoted as RT. Partnership is essentially an application of ratio and proportion concepts.

Comparative Structures in Profit Distribution:

Examples on Partnership

The following worked examples illustrate the most important Partnership concepts commonly asked in Railway Exams. These questions cover profit distribution based on investment and time, partners joining or leaving a business, determining investment and profit ratios, and solving real exam-style scenarios. Practicing these examples will help you understand the underlying concepts and improve your speed and accuracy in solving Partnership problems. 

Worked Example 1: Profit Distribution with Equal Time

Problem: A and B invest ₹42,000 and ₹56,000 respectively. At the end of the year, they make a profit of ₹87,220. Find B's share 

Solution:

1. Time for both partners is equal.

2. Profit Ratio: Distributed in the ratio of their investments: A : B = 42,000 : 56,000 = 3 : 4.

3. Total Ratio Parts: 3 + 4 = 7.

4. Value per Part: ₹87,220 / 7 = ₹12,460.

5. B's Share: 4 parts × ₹12,460 = ₹49,840.

Worked Example 2: Finding Difference in Shares (Equal Time)

Problem: P, Q, R invested sums in the ratio 34:75:6. If they earned a total profit of ₹3600 at the end of the year, what is the difference between the share of Q and R?

Solution:

1. Time for all partners is equal.

2. Profit Ratio: P : Q : R = 34 : 75 : 6.

3. Total Ratio Parts: 34 + 75 + 6 = 115.

4. Value per Part: ₹3600 / 115 = ₹720 / 23.

5. Difference in Shares: Difference in ratio parts for Q and R = 75 - 6 = 69. Difference in profit = 69 × (₹720 / 23) = 3 × ₹720 = ₹2160.

Worked Example 3: Profit Distribution with Different Times

Problem: P invests ₹40,000 for 9 months. Q invests ₹60,000 for 6 months. Total profit ₹45,000. Find Q's share.

Solution:

1. Time is different. Apply Profit ∝ (Investment × Time).

2. P's (R×T): ₹40,000 × 9 months = ₹360,000.

3. Q's (R×T): ₹60,000 × 6 months = ₹360,000.

4. Profit Ratio: P : Q = 360,000 : 360,000 = 1 : 1.

5. Total Ratio Parts: 1 + 1 = 2.

6. Value per Part: ₹45,000 / 2 = ₹22,500.

7. Q's Share: 1 part × ₹22,500 = ₹22,500.

Worked Example 4: Partner Joining Later (Time Calculation)

Problem: Ashish started a business with ₹60,000. Sachin joined him 8 months later with ₹35,000. What is the respective share of the two in profit after 2 years? 

Solution:

1. Total Duration: 2 years = 24 months.

2. Ashish's Time: Full 24 months.

3. Sachin's Time: Joined 8 months later, so 24 - 8 = 16 months.

4. Profit Ratio (Ashish : Sachin): (₹60,000 × 24) : (₹35,000 × 16).

5. Simplify Ratio: (60 × 24) : (35 × 16) → (12 × 24) : (7 × 16) → (12 × 3) : (7 × 2) = 18 : 7.

Worked Example 5: Finding Profit Ratio from Investment and Time Ratios

Problem: A, B, C invest in the ratio 5:6:7. Their time of investment is in the ratio 4:3:2. What is the ratio of their profits?

Solution:

1. Profit ∝ (Investment × Time).

2. Profit Ratio (A : B : C): (5 × 4) : (6 × 3) : (7 × 2) = 20 : 18 : 14.

3. Simplify Ratio: Divide by 2: 10 : 9 : 7.

Worked Example 6: Finding Investment Ratio from Time and Profit Ratios

Problem: A, B, C are partners. Their time of investment is in the ratio 6:12:14. Their profit is in the ratio 5:4:7. What is the ratio of their respective investments? 

Solution:

1. Investment = Profit / Time.

2. Initial Investment Ratio (A : B : C): (5/6) : (4/12) : (7/14) = (5/6) : (1/3) : (1/2).

3. Convert to Integer Ratio: LCM of denominators (6, 3, 2) is 6. Multiply each term by 6:
   (5/6 × 6) : (1/3 × 6) : (1/2 × 6) = 5 : 2 : 3.

Worked Example 7: Complex Investment Relationships and Profit Sharing

Problem: A, B, C, D are four partners. A's investment is 2/3 of D's. B's and C's investments are equal. C's investment is half of D's. Total profit ₹137.6 lakhs. Find combined profit of A and B.

Solution:

1. Define Relationships: A = (2/3)D, B = C, C = (1/2)D.

2. Assume D: Let D = 6 units (LCM of 3 and 2).

3. Calculate Investments: A = (2/3) × 6 = 4 units. C = (1/2) × 6 = 3 units. B = C = 3 units.

4. Investment Ratio (A:B:C:D): 4 : 3 : 3 : 6. Time is equal, so profit ratio is same.

5. Total Ratio Parts: 4 + 3 + 3 + 6 = 16.

6. Value per Part: ₹137.6 lakhs / 16 = ₹8.6 lakhs.

7. Combined Profit of A and B: (4 + 3) parts = 7 parts. 7 × ₹8.6 lakhs = ₹60.2 lakhs.

Worked Example 8: Profit Distribution with Charity Deduction

Problem: Pankaj, Meera, Ashok invested ₹41,000, ₹39,000, and ₹84,000. Total profit ₹49,200. 29% to charity. Find Ashok's share.

Solution:

1. Investment Ratio (Pankaj:Meera:Ashok): 41 : 39 : 84 (simplified by dividing by 1,000).

2. Time is equal, so profit ratio is same.

3. Percentage for Partners: 100% - 29% (charity) = 71%.

4. Value of 1 Ratio Part (from total profit): Total parts = 41+39+84 = 164. Value per part = ₹49,200 / 164 = ₹300.

5. Ashok's Share (before charity): 84 parts × ₹300 = ₹25,200.

6. Ashok's Actual Share: 71% of ₹25,200 = 0.71 × ₹25,200 = ₹17,892.

Worked Example 9: Mixed Units and Conversion in Partnership (Renting a Pasture)

Problem: A uses 18 cows for 3 months, 20 sheep for 4 months. B uses 35 sheep for 6 months. If 3 cows = 6 sheep, determine A's share of the rent. 

Solution:

1. Conversion Factor: 1 cow = 2 sheep.

2. A's Equivalent Sheep-Months: (18 cows × 2 sheep/cow × 3 months) + (20 sheep × 4 months) = 108 sheep-months + 80 sheep-months = 188 sheep-months.

3. B's Sheep-Months: 35 sheep × 6 months = 210 sheep-months.

4. Rent Ratio (A : B): 188 : 210 = 94 : 105.

5. A's Share: A's share = 94 / (94 + 105) = 94 / 199 of the total rent.

Worked Example 10: Multiple Partners with Different Joining Times

Problem: Sonu invested ₹40,000. 4 months later Prem joined with ₹80,000. 8 months later (from start) Alok joined with ₹120,000. Total profit ₹156,000 in one year. Find Alok's share. 

Solution:

1. Total Duration: 12 months.

2. Investment × Time (Sonu): ₹40,000 × 12 months.

3. Investment × Time (Prem): ₹80,000 × (12 - 4) = ₹80,000 × 8 months.

4. Investment × Time (Alok): ₹120,000 × (12 - 8) = ₹120,000 × 4 months.

5. Profit Ratio (Sonu : Prem : Alok): (40k × 12) : (80k × 8) : (120k × 4) = 480k : 640k : 480k.

6. Simplify Ratio: 48 : 64 : 48 (divide by 10k) = 3 : 4 : 3 (divide by 16).

7. Total Ratio Parts: 3 + 4 + 3 = 10.

8. Value per Part: ₹156,000 / 10 = ₹15,600.

9. Alok's Share: 3 parts × ₹15,600 = ₹46,800.