Partnership: Complete Master Guide for UPSC/MPSC CSAT (English)

Introduction to Partnership

In the realm of competitive exams like UPSC and MPSC CSAT, Partnership is a direct extension of the ‘Ratio and Proportion’ concept. It represents a practical application where two or more individuals pool their resources (usually money) to start a business or venture and subsequently share the profits or losses generated. Mastering Partnership questions is highly rewarding because the core logic remains consistent, and these questions are typically scoring if approached methodically. They test your ability to simultaneously handle proportional investments and time durations.

When multiple people invest their money, the fundamental principle is that the profit (or loss) should be distributed in proportion to the amount they invested and the time duration for which their money was active in the business. Understanding this delicate balance between capital and time is the key to solving any Partnership problem seamlessly.

Core Concepts, Formulas, and Tricks

Partnerships are generally classified into two main types based on the uniformity of the time period.

1. Simple Partnership:
In a simple partnership, all partners invest their capital for the exact same time period. In this scenario, time becomes a constant, and the profit or loss is distributed solely based on the ratio of their invested capital.

• Formula: If Partner A invests capital $C_A$ and Partner B invests capital $C_B$ for the same time duration, then:
• Ratio of Profit of A : Profit of B = $C_A$ : $C_B$

2. Compound Partnership:
In a compound partnership, partners invest their capital for different time periods. Here, both the capital amount and the time duration play a role in deciding the profit share. To find the profit sharing ratio, you must multiply the capital of each partner by the respective time period for which it was invested.

• Formula: If Partner A invests capital $C_A$ for time period $T_A$, and Partner B invests capital $C_B$ for time period $T_B$, then:
• Ratio of Profit of A : Profit of B = ($C_A \times T_A$) : ($C_B \times T_B$)
• This product (Capital × Time) is often referred to as the ‘Effective Capital’.

3. Working vs. Sleeping Partners:

• Working Partner: A partner who not only invests money but also manages the business. They usually receive a fixed salary or a certain percentage of the profit as compensation for their management efforts, before the remaining profit is distributed according to investments.
• Sleeping (or Dormant) Partner: A partner who merely invests the money and does not participate in managing the business. Their share of profit depends entirely on their investment and time.

Trick/Formula for Capital or Time:
From the basic formula $P \propto C \times T$ (Profit is proportional to Capital × Time), we can derive:
Ratio of Investments (Capital) = $P_A/T_A$ : $P_B/T_B$
Ratio of Time Periods = $P_A/C_A$ : $P_B/C_B$

Solved Examples with Step-by-Step Explanations

Example 1: Simple Partnership
A, B, and C started a business by investing ₹1,20,000, ₹1,35,000, and ₹1,50,000 respectively. Find the share of B out of an annual profit of ₹56,700.

Step-by-Step Solution:
Step 1: Notice that the time period is not mentioned for individuals, but the total profit is “annual”. This means all invested for 1 year (Simple Partnership).
Step 2: Find the ratio of their investments.
Ratio = 1,20,000 : 1,35,000 : 1,50,000
Step 3: Simplify the ratio. Divide by 1000.
= 120 : 135 : 150
Step 4: Divide by 15.
= 8 : 9 : 10
Step 5: The profit sharing ratio is A:B:C = 8:9:10. Total parts = 8 + 9 + 10 = 27.
Step 6: Calculate B’s share.
B’s share = (9 / 27) × 56,700 = (1 / 3) × 56,700 = ₹18,900.

Example 2: Compound Partnership
X started a business with ₹21,000. After a few months, Y joined him with ₹36,000. At the end of the year, the profit was divided equally. After how many months did Y join?

Step-by-Step Solution:
Step 1: Identify capitals and time periods. X’s capital = ₹21,000; X’s time = 12 months.
Let Y’s money be invested for ‘t’ months. Y’s capital = ₹36,000.
Step 2: Since profits are equal, their Effective Capitals must be equal.
(Capital of X × Time of X) = (Capital of Y × Time of Y)
21,000 × 12 = 36,000 × t
Step 3: Solve for t.
252,000 = 36,000t
t = 252,000 / 36,000 = 7 months.
Step 4: The question asks “after how many months did Y join?”, not how long he invested.
Since Y’s money was in the business for 7 months in a year (12 months), Y joined after (12 – 7) = 5 months.

Example 3: Adding/Withdrawing Capital
A and B started a partnership with ₹40,000 and ₹50,000. After 4 months, A withdrew ₹10,000 and B invested an additional ₹10,000. Find the ratio of their profit at the end of the year.

Step-by-Step Solution:
Step 1: Calculate Effective Capital for A.
A invested ₹40,000 for the first 4 months.
For the remaining 8 months, A’s capital became (40,000 – 10,000) = ₹30,000.
A’s Effective Capital = (40,000 × 4) + (30,000 × 8) = 1,60,000 + 2,40,000 = 4,00,000.
Step 2: Calculate Effective Capital for B.
B invested ₹50,000 for the first 4 months.
For the remaining 8 months, B’s capital became (50,000 + 10,000) = ₹60,000.
B’s Effective Capital = (50,000 × 4) + (60,000 × 8) = 2,00,000 + 4,80,000 = 6,80,000.
Step 3: Find the ratio.
Profit Ratio A:B = 4,00,000 : 6,80,000 = 40 : 68 = 10 : 17.

Example 4: Working Partner
A and B invest ₹30,000 and ₹40,000 in a business. A is a working partner and receives 20% of the total profit as his salary. If the total profit is ₹7,000, find the total share of A.

Step-by-Step Solution:
Step 1: Calculate A’s salary from the profit.
Salary of A = 20% of 7000 = ₹1,400.
Step 2: Calculate the remaining profit to be distributed.
Remaining Profit = 7000 – 1400 = ₹5,600.
Step 3: Find the ratio of their investments.
Investment Ratio A:B = 30,000 : 40,000 = 3:4.
Step 4: Distribute the remaining profit based on the ratio.
A’s share from remaining profit = (3/7) × 5600 = 3 × 800 = ₹2,400.
Step 5: Calculate A’s total share.
Total Share of A = Salary + Share of Profit = 1400 + 2400 = ₹3,800.

Pro-Tips to Avoid Common Mistakes

• “Invested For” vs “Joined After”: Read the time condition very carefully. If a business runs for 12 months and B joins after 3 months, B’s money is invested for (12 – 3) = 9 months. Do not multiply B’s capital by 3.
• Simplify the Zeros Early: In partnership problems, capitals usually have many trailing zeros (e.g., 50,000 and 75,000). To save time and avoid calculation errors, mentally cancel the common zeros right at the start and just write 50 and 75, or even simplify to 2 and 3.
• Be Careful with Withdrawals and Additions: When a partner adds or withdraws money halfway, break their investment into separate time blocks and add the products (Capital_1 × Time_1 + Capital_2 × Time_2) rather than trying to do it in one complex equation.
• Working Partner’s Dual Share: Never forget that a working partner gets two things: a salary/commission AND a share of the remaining profit based on their capital. Do not distribute the full profit before deducting the salary.

Practice Questions

1. P and Q started a business investing ₹85,000 and ₹15,000 respectively. In what ratio the profit earned after 2 years be divided between P and Q respectively?
2. A, B, and C enter into a partnership. A invests 3 times as much as B and B invests two-third of what C invests. At the end of the year, the profit earned is ₹6,600. What is the share of B?
3. A started a business with ₹4500 and another person B joined after some period with ₹3000. Determine this period after which B joined the business if the profit at the end of the year is divided in the ratio 2:1.
4. A and B start a business. A invests ₹6000 for 9 months and B invests ₹8000 for some time. If B’s profit is half of the total profit, for how many months did B invest his money?
5. Anand and Deepak started a business investing ₹22,500 and ₹35,000 respectively. Out of a total profit of ₹13,800, Deepak’s share is?