Master Partnership for RRB NTPC & Group D: Complete Guide with Concepts, Formulas & Solved Questions

Introduction to Partnership for RRB Exams

Welcome, future railway professionals! If you are gearing up for the highly competitive RRB NTPC, Group D, or Technician exams, you know that every single mark counts. The Quantitative Aptitude section is often the make-or-break part of these exams. One topic that consistently appears and can fetch you easy marks, if understood correctly, is 'Partnership'.

So, what exactly is Partnership? In the context of mathematics, a partnership is an agreement between two or more individuals who decide to run a business together. They invest their capital (money or assets) and agree to share the resulting profits or losses. Questions from this topic test your understanding of ratios, percentages, and basic arithmetic operations in a business scenario. This comprehensive guide will walk you through every concept, formula, and trick you need to master Partnership and ace the questions in your RRB exam.

Topic Weightage and Importance

In RRB exams like NTPC (CBT-1 & CBT-2) and Group D, the Quantitative Aptitude section holds significant weightage. Within this section, you can expect to find 1-2 questions directly from the Partnership topic. While this might seem like a small number, remember that in an exam where lakhs of candidates compete, even a single question can significantly impact your rank.

The beauty of Partnership questions is that they are formula-based and logical. Once you grasp the core concepts of how investments and time periods affect profit distribution, these questions become highly scoring. They don't involve complex theorems like Geometry or Trigonometry, making them easier to master with dedicated practice.

Key Concepts and Formulas

To solve any Partnership problem, you must be crystal clear about the fundamental concepts and the formulas derived from them. Let's break them down.

1. Basic Terminology

• Partner: An individual who invests in the business.
• Capital: The amount of money or value of assets invested by a partner in the business.
• Partnership: The relationship or agreement between partners to run a business and share its outcomes.

2. Types of Partnership

Partnerships are primarily categorized into two types based on the duration of investment.

a) Simple Partnership

This is the most basic form of partnership. Here, all partners invest their capital for the same period of time. In such cases, the profit or loss is distributed among the partners in the ratio of their investments.

If partners A, B, and C invest capitals C₁, C₂, and C₃ respectively for the same duration, then the ratio of their profits (P₁, P₂, P₃) will be:

P₁ : P₂ : P₃ = C₁ : C₂ : C₃

b) Compound Partnership

This type is more common in exams. Here, partners invest their capital for different periods of time. In a compound partnership, the profit or loss is distributed in the ratio of the product of their capital and the time period for which they invested.

If partners A, B, and C invest capitals C₁, C₂, and C₃ for time periods T₁, T₂, and T₃ respectively, then the ratio of their profits (P₁, P₂, P₃) will be:

P₁ : P₂ : P₃ = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃)

This product (Capital × Time) is often called the 'equivalent capital' for a unit of time.

3. Types of Partners

Partners can also be classified based on their role in the business.

• Working (or Active) Partner: A partner who not only invests capital but also actively manages the business. A working partner may be entitled to a salary or a commission from the profit before the general distribution.
• Sleeping (or Dormant) Partner: A partner who only invests capital but does not participate in the management of the business. They are only entitled to their share of the profit based on their investment.

How to handle Working Partner problems:

If a working partner receives a salary or commission, this amount is first deducted from the total profit. The remaining profit is then distributed among all partners (including the working partner) according to the ratio of their investments (or equivalent capital).

Total Share of Working Partner = Salary/Commission + Share in the remaining profit

Solved Examples (Step-by-Step)

Let's apply these concepts to solve some typical RRB exam-level questions.

Example 1: Simple Partnership

Question: Anjali, Beena, and Charu start a business by investing ₹25,000, ₹30,000, and ₹35,000 respectively. If they make an annual profit of ₹45,600, what is Charu's share in the profit?

Solution:

• Step 1: Identify the type of partnership. Since the time period is not mentioned differently for any partner, we assume it's a simple partnership where everyone invested for the same duration (one year).
• Step 2: Find the ratio of their investments. Ratio of Investments = Anjali : Beena : Charu
  = 25000 : 30000 : 35000
  To simplify, divide all parts by 5000:
  = 5 : 6 : 7
• Step 3: Understand that the profit will be shared in this ratio. Sum of the ratio terms = 5 + 6 + 7 = 18. This means the total profit is divided into 18 parts.
• Step 4: Calculate Charu's share. Charu's share corresponds to the ratio part '7'.
  Charu's Profit Share = (Charu's ratio part / Sum of ratio parts) × Total Profit
  = (7 / 18) × 45600
  = 7 × (45600 / 18)
  = 7 × 2533.33 (approx) --- Let's recheck the calculation. 45600 / 18 = 2533.33 is not a clean number. Let's assume the profit was ₹46,800 for a cleaner example. Let's redo with a better number like ₹54,000. Let's stick to the original and round if needed, or point out that exam numbers are usually clean. Let's find a cleaner profit number. 18 * 2500 = 45000. Let's use 45000 as profit. Let's recalculate with the original number. Ah, 45600/18 is not integer. Let's adjust the profit amount for the example to make it cleaner for students. Let's say the profit is ₹46,800. 46800/18 = 2600. Perfect. Let's re-write the question. Revised Question: Anjali, Beena, and Charu start a business by investing ₹25,000, ₹30,000, and ₹35,000 respectively. If they make an annual profit of ₹46,800, what is Charu's share in the profit?

Solution:

• Step 1: This is a simple partnership.
• Step 2: Ratio of Investments = 25000 : 30000 : 35000 = 5 : 6 : 7
• Step 3: Sum of the ratio terms = 5 + 6 + 7 = 18.
• Step 4: Charu's Profit Share = (7 / 18) × 46800
  = 7 × (46800 / 18)
  = 7 × 2600
  = ₹18,200

Example 2: Compound Partnership

Question: Ram starts a business with ₹40,000. After 3 months, Shyam joins him with an investment of ₹60,000. At the end of one year, they earn a profit of ₹39,000. What is Shyam's share?

Solution:

• Step 1: Identify the type of partnership. This is a compound partnership as partners invested for different durations.
• Step 2: Calculate the equivalent capital for each partner. The business ran for one year (12 months). Ram's investment period (T₁) = 12 months. Ram's capital (C₁) = ₹40,000. Shyam joined after 3 months, so his investment period (T₂) = 12 - 3 = 9 months. Shyam's capital (C₂) = ₹60,000.
• Step 3: Find the ratio of their effective investments (Profit Sharing Ratio). Ratio of Profits = (Ram's Capital × Ram's Time) : (Shyam's Capital × Shyam's Time)
  = (40000 × 12) : (60000 × 9)
  = 480000 : 540000
  To simplify, divide by 60000:
  = 48/6 : 54/6 = 8 : 9
• Step 4: Calculate Shyam's share. Sum of ratio terms = 8 + 9 = 17. Shyam's Profit Share = (Shyam's ratio part / Sum of ratio parts) × Total Profit
  = (9 / 17) × 39000
  = 9 × (39000 / 17)
  = 9 × 2294.11... let's use a cleaner number again. Total profit of ₹34,000. 34000/17 = 2000. Perfect. Revised Question: Ram starts a business with ₹40,000. After 3 months, Shyam joins him with an investment of ₹60,000. At the end of one year, they earn a profit of ₹34,000. What is Shyam's share?

Solution:

• Step 1 & 2: Ram's equivalent capital = 40000 x 12. Shyam's equivalent capital = 60000 x 9.
• Step 3: Ratio of Profits = (40000 × 12) : (60000 × 9) = 480000 : 540000 = 8 : 9
• Step 4: Sum of ratio terms = 8 + 9 = 17. Shyam's Profit Share = (9 / 17) × 34000
  = 9 × 2000
  = ₹18,000

Example 3: Working Partner

Question: Amar and Bimal enter into a partnership with capitals of ₹50,000 and ₹70,000 respectively. Amar, being the working partner, gets 20% of the total profit as his salary. The total profit in a year is ₹80,000. Find the total amount received by Amar.

Solution:

• Step 1: Calculate the working partner's salary. Total Profit = ₹80,000 Amar's Salary = 20% of 80,000 = (20/100) × 80000 = ₹16,000.
• Step 2: Calculate the remaining profit to be distributed. Remaining Profit = Total Profit - Amar's Salary
  = 80000 - 16000 = ₹64,000.
• Step 3: Find the ratio of their investments to distribute the remaining profit. Ratio of Investments = Amar : Bimal = 50000 : 70000 = 5 : 7. Sum of ratio terms = 5 + 7 = 12.
• Step 4: Calculate Amar's share from the remaining profit. Amar's profit share = (5 / 12) × 64000
  = (5 × 64000) / 12 = 320000 / 12 = ₹26,666.67
• Step 5: Calculate the total amount Amar receives. Total amount for Amar = Amar's Salary + Amar's profit share
  = 16000 + 26666.67 = ₹42,666.67

Common Mistakes to Avoid

• Ignoring the Time Factor: Many students apply the simple partnership formula (C₁:C₂) even when the time periods are different. Always check if it's a simple or compound partnership.
• Incorrect Time Calculation: When a partner joins 'after' X months, their investment period is (12 - X) months. When a partner leaves 'after' Y months, their period is Y months. Read the phrasing carefully.
• Unit Mismatch: Ensure all time periods are in the same unit (usually months) before calculating the product of capital and time.
• Working Partner Calculation Error: Forgetting to subtract the working partner's salary from the total profit before distributing the rest is a very common mistake.
• Ratio Simplification Errors: Making a mistake while simplifying large numbers in a ratio can lead to a completely wrong answer. Double-check your calculations.
• Misinterpreting the Final Question: Sometimes the question might ask for the 'difference' in shares or the 'total amount' received by a working partner. Make sure you answer exactly what is asked.

Practice Questions with Solutions

Now it's your turn! Try to solve these questions. The solutions are provided at the end.

Question 1: A, B, and C invested amounts of ₹12,000, ₹15,000 and ₹18,000 respectively to start a business. At the end of the year, the profit earned is ₹18,000. What is B's share of the profit?

Question 2: P starts a business with ₹60,000. Q joins him after 4 months with ₹80,000. R joins them after another 4 months with ₹1,00,000. Find the ratio of their shares in the annual profit.

Question 3: X and Y enter into a partnership with capitals in the ratio 5 : 6. At the end of 8 months, X withdraws his capital. If they receive profits in the ratio 5 : 9, find for how long Y's capital was used.

Question 4: Arun and Varun are partners in a business. Arun invests ₹35,000 for 8 months and Varun invests ₹42,000 for 10 months. Out of a profit of ₹31,570, Arun's share is?

Question 5: Seeta and Geeta start a business with investments of ₹45,000 and ₹55,000. Seeta is a working partner and gets a monthly salary of ₹1,000. If the annual profit is ₹64,000, what is Geeta's total share?

Question 6: Three partners A, B, and C invest ₹5000, ₹6000, and ₹7000 respectively. A gets 70% of the profit for managing the business and the rest of the profit is divided in the ratio of their investments. If A gets ₹5400 more than B and C together, what is the total profit?

---

Solutions to Practice Questions

Solution 1: Ratio of investments = 12000 : 15000 : 18000 = 12 : 15 : 18 = 4 : 5 : 6. Sum of ratios = 4 + 5 + 6 = 15. B's share = (5/15) × 18000 = (1/3) × 18000 = ₹6,000.

Solution 2: P invested for 12 months. Q invested for (12-4) = 8 months. R joined 4 months after Q, so R invested for (12-4-4) = 4 months. Ratio of profits = (P's Capital × P's Time) : (Q's Capital × Q's Time) : (R's Capital × R's Time) = (60000 × 12) : (80000 × 8) : (100000 × 4) = 720000 : 640000 : 400000 Divide by 80000: = 9 : 8 : 5. The ratio is 9 : 8 : 5.

Solution 3: Let capitals be 5x and 6x. Let Y's capital be used for 't' months. Ratio of profits = (5x × 8) : (6x × t) = 5 : 9 => (40x) / (6xt) = 5 / 9 => 40 / (6t) = 5 / 9 => 40 × 9 = 5 × 6t => 360 = 30t => t = 360 / 30 = 12 months. Y's capital was used for 12 months.

Solution 4: Ratio of profits = (Arun's investment) : (Varun's investment) = (35000 × 8) : (42000 × 10) = 280000 : 420000 = 28 : 42 = 2 : 3. Sum of ratios = 2 + 3 = 5. Arun's share = (2/5) × 31570 = 2 × 6314 = ₹12,628.

Solution 5: Total annual profit = ₹64,000. Seeta's monthly salary = ₹1,000. So, annual salary = 1000 × 12 = ₹12,000. Remaining profit = 64000 - 12000 = ₹52,000. Ratio of investments = 45000 : 55000 = 45 : 55 = 9 : 11. Sum of ratios = 9 + 11 = 20. Geeta's share = (11/20) × 52000 = 11 × 2600 = ₹28,600.

Solution 6: Let the total profit be P. A's management fee = 70% of P = 0.7P. Remaining profit = P - 0.7P = 0.3P. Ratio of investments = 5000 : 6000 : 7000 = 5 : 6 : 7. Sum of ratios = 18. This remaining profit (0.3P) is divided in this ratio. A's share from remaining profit = (5/18) × 0.3P B's share = (6/18) × 0.3P C's share = (7/18) × 0.3P Total share of A = 0.7P + (5/18) × 0.3P Total share of (B+C) = (6/18) × 0.3P + (7/18) × 0.3P = (13/18) × 0.3P Given: Total A's share - Total (B+C)'s share = 5400 [0.7P + (5/18) × 0.3P] - [(13/18) × 0.3P] = 5400 0.7P + (0.3P/18) × (5 - 13) = 5400 0.7P - (0.3P/18) × 8 = 5400 0.7P - (2.4P / 18) = 5400 (12.6P - 2.4P) / 18 = 5400 10.2P / 18 = 5400 P = (5400 × 18) / 10.2 = 97200 / 10.2 = ₹9529.41 (Let's recheck the numbers. Maybe a cleaner question is needed). Let's rephrase the question slightly. Let A get 30% for managing. Revised Question 6: Three partners A, B, and C invest ₹5000, ₹6000, and ₹7000 respectively. A gets 30% of the profit for managing the business and the rest of the profit is divided in the ratio of their investments. If A's total share is ₹3600, what is the total profit? Let Total Profit = P. A's salary = 0.3P. Remaining Profit = 0.7P. Investment Ratio = 5:6:7. Sum=18. A's share from remaining = (5/18)*0.7P. Total A share = 0.3P + (3.5P/18) = 3600. (5.4P + 3.5P)/18 = 3600. 8.9P = 3600 * 18. P = 64800 / 8.9. Still not clean. Let's make it much simpler. Final attempt at a good complex question: Question 6 (Revised): A and B are partners investing ₹10,000 and ₹15,000 respectively. A is a working partner and is to receive a salary of ₹12,500 from the annual profit. After paying A's salary, the remaining profit is distributed. If in a year, A totally receives ₹20,000, what was the total profit? Solution 6 (Revised): A's total income = A's salary + A's profit share = ₹20,000. A's salary = ₹12,500. So, A's profit share = 20000 - 12500 = ₹7,500. The remaining profit (after salary) is distributed in the ratio of investments. Ratio of investments = 10000 : 15000 = 2 : 3. A's share (2 parts) corresponds to ₹7,500. 1 part = 7500 / 2 = ₹3,750. B's share (3 parts) = 3 × 3750 = ₹11,250. Total distributed profit = A's share + B's share = 7500 + 11250 = ₹18,750. This is the profit remaining after paying the salary. Total Profit = Distributed Profit + A's Salary = 18750 + 12500 = ₹31,250.

Frequently Asked Questions (FAQs)

1. What is the main difference between Simple and Compound Partnership?

The key difference is the time period of investment. In a Simple Partnership, all partners invest for the same duration, so profits are shared in the ratio of their capitals. In a Compound Partnership, partners invest for different durations, so profits are shared in the ratio of the product of their capital and time period (Capital × Time).




2. What prerequisite topic should I master before starting Partnership?

A strong command over 'Ratio and Proportion' is essential. Partnership problems are fundamentally an application of ratio concepts. A good understanding of 'Percentages' is also helpful, especially for problems involving working partners' commissions or profit/loss percentages.




3. Can there be a loss in a partnership? How is it shared?

Yes, a business can incur a loss. The principle for sharing losses is exactly the same as for sharing profits. The loss is distributed among the partners in the same ratio used for profit distribution (i.e., ratio of investments for simple partnership or ratio of 'capital × time' for compound partnership).




4. How much time should I spend on a Partnership question in the RRB exam?

With good practice, you should be able to solve a typical partnership question in about 45 to 60 seconds. The calculations are usually straightforward. The key is to quickly identify the type of partnership and set up the correct ratio.

Conclusion and Final Tips

Mastering the topic of Partnership is a definite step towards boosting your score in the RRB exams. It is a topic that rewards conceptual clarity and systematic practice over complex calculations. Remember these final tips:

• Memorize the Formulas: Know the difference between P₁:P₂ = C₁:C₂ and P₁:P₂ = C₁T₁:C₂T₂ by heart.
• Read Carefully: Pay close attention to keywords like 'joins after', 'leaves after', 'withdraws', 'adds more', and 'working partner'.
• Practice Regularly: Solve at least 15-20 questions from previous years' papers and mock tests to get a feel for the patterns and difficulty level.
• Focus on Calculation Speed: Improve your speed in multiplication and division, and in simplifying ratios. This will save crucial time during the exam.

By following this guide and putting in consistent effort, you can confidently turn Partnership problems into a scoring opportunity. Stay focused, practice diligently, and march confidently towards your goal of securing a job in the Indian Railways. All the best!