RRB NTPC & Group D Maths: Master Partnership – Concepts, Formulas & Solved Questions

Introduction to Partnership for RRB Exams

Partnership is a fundamental arithmetic topic that frequently appears in various competitive examinations conducted by the Railway Recruitment Board (RRB), including RRB NTPC, RRB Group D, and RRB Technician Grade I & III. In real-world business scenarios, two or more individuals join hands and invest capital to run a business undertaking. At the end of a specific time period, the profit or loss generated by the business is divided among the partners based on specific rules and ratios.

Understanding Partnership concepts is essential for candidates as it bridges basic Ratio and Proportion with practical quantitative reasoning. Questions from this section test your ability to calculate individual profit shares, determine investment time durations, and find initial capital contributions. By mastering the core principles and shortcut tricks detailed in this guide, you can comfortably secure maximum marks from this high-yield topic.

Topic Weightage and Importance

In RRB examinations, quantitative aptitude plays a decisive role in clearing the cutoff scores. The topic of Partnership usually accounts for 1 to 2 direct questions in CBT-1 and CBT-2 of RRB NTPC and RRB Group D exams. Although the frequency may seem modest, these questions are universally scoring because they rely on straightforward algebraic ratios without complex formula variations.

Given the strict time constraints in RRB online exams, solving a Partnership question in less than 45 seconds can give you a crucial edge over millions of competitors.

Key Concepts and Formulas

The mathematical foundation of Partnership is directly derived from Ratio and Proportion. Profit or loss shared among partners is directly proportional to the product of the capital invested and the duration for which it remains in the business.

1. Fundamental Formula of Profit Sharing

If a person invests capital \(C\) for a time period \(T\), then the effective investment equivalent is given by \(C \times T\). Profit (\(P\)) distributed is proportional to this equivalent value:

\(P \propto C \times T\)

For two partners, A and B, with investments \(C_A\) and \(C_B\) for time periods \(T_A\) and \(T_B\):

\(\frac{P_A}{P_B} = \frac{C_A \times T_A}{C_B \times T_B}\)

2. Classification of Partnerships

• Simple Partnership: All partners invest their capital for the same duration of time (\(T_A = T_B = T\)). In this case, profit is distributed purely in the ratio of their capitals: \(P_A : P_B = C_A : C_B\).
• Compound Partnership: Partners invest different amounts for different time durations. Profit is shared according to the ratio of the product of capital and time: \(P_A : P_B : P_C = (C_A \times T_A) : (C_B \times T_B) : (C_C \times T_C)\).

3. Types of Partners

• Working Partner: A partner who manages the business actively along with investing capital. Working partners often receive a fixed salary or a fixed percentage of total profit before the remaining profit is distributed in the investment ratio.
• Sleeping (Silent) Partner: A partner who only invests money but does not take part in the daily operations of the business. Their share depends strictly on their investment ratio.

Solved Examples (Step-by-Step)

Example 1: Simple Partnership

Question: A and B start a business with investments of ₹40,000 and ₹60,000 respectively. At the end of one year, the total profit is ₹25,000. Find the profit share of A.

Solution:

Step 1: Identify capitals and duration. Here time duration is equal for both (1 year).

\(C_A = 40000\), \(C_B = 60000\)

Step 2: Find the ratio of profit shares:

\(P_A : P_B = C_A : C_B = 40000 : 60000 = 2 : 3\)

Step 3: Sum of ratio terms = \(2 + 3 = 5\)

Step 4: Calculate A's share from total profit (₹25,000):

\(P_A = \left(\frac{2}{5}\right) \times 25000 = 2 \times 5000 = ₹10,000\)

Answer: The profit share of A is ₹10,000.

Example 2: Compound Partnership with Variable Time

Question: X invested ₹12,000 for 12 months. Y joined the business after 3 months with an investment of ₹16,000. If the total annual profit earned is ₹18,000, what is Y's share?

Solution:

Step 1: Calculate effective time periods.

Time for X, \(T_X = 12\) months.

Since Y joined after 3 months, time for Y, \(T_Y = 12 - 3 = 9\) months.

Step 2: Find the equivalent capital-time product ratio:

\(P_X : P_Y = (C_X \times T_X) : (C_Y \times T_Y)\)

\(P_X : P_Y = (12000 \times 12) : (16000 \times 9) = 144000 : 144000 = 1 : 1\)

Step 3: Divide total profit equally since ratio is 1:1.

\(P_Y = \left(\frac{1}{2}\right) \times 18000 = ₹9,000\)

Answer: Y's profit share is ₹9,000.

Example 3: Working Partner Commission

Question: Ramesh and Suresh invest ₹50,000 and ₹30,000 in a firm. Ramesh is a working partner and receives 10% of total profit as management fee. The remaining profit is divided according to investment. If total profit is ₹40,000, find Ramesh's total income.

Solution:

Step 1: Calculate Ramesh's salary/commission from total profit.

\(\text{Commission} = 10\% \text{ of } 40000 = 0.10 \times 40000 = ₹4,000\)

Step 2: Calculate remaining profit to be distributed:

\(\text{Remaining Profit} = 40000 - 4000 = ₹36,000\)

Step 3: Profit distribution ratio = Investment ratio = \(50000 : 30000 = 5 : 3\)

Sum of ratio terms = \(5 + 3 = 8\)

Step 4: Calculate Ramesh's share from remaining profit:

\(P_{\text{Ramesh}} = \left(\frac{5}{8}\right) \times 36000 = 5 \times 4500 = ₹22,500\)

Step 5: Ramesh's total earnings = Commission + Profit share = \(4000 + 22500 = ₹26,500\)

Answer: Ramesh receives a total of ₹26,500.

Common Mistakes to Avoid

• Confusing Joining Time with Investment Duration: If a person joins "