Hello! Welcome to the fourth lesson in our Foundations of Arithmetic module. In our previous session, we mastered how to solve problems using direct and inverse proportion, a concept that describes how quantities change in relation to one another.
Today, we will apply those very principles to a new topic: Partnership. Your learning outcome is to solve partnership problems involving time-based and investment-based profit sharing. This is a classic topic in the SSC CGL exam and a direct extension of your knowledge of ratios and proportions. We'll start with the fundamental formula and then work through the various question patterns you'll encounter, focusing on both the logic and the time-saving methods you need for the exam.
1. The Fundamental Principle of Partnership
At its core, a partnership is a business arrangement where profits are shared among partners. The guiding principle for this profit distribution is a direct application of joint proportion, which we discussed in the last lesson. The profit a partner receives is directly proportional to two factors:
- The capital (investment) they put in.
- The time for which they invested their capital.
This gives us the single most important formula in this chapter:
Profit ∝ Capital × Time
Let's watch a short video from Aditya Ranjan Sir, a popular mentor for SSC aspirants, who explains this core idea very clearly.
Partnership (साझेदारी) के सभी Concepts + Questions 🔥 by Aditya Ranjan Sir Maths | Rankers Gurukul
This clip introduces the fundamental formula of partnership and the two main scenarios that arise from it.
Watch the initial segment from the beginning to the end of the explanation. Pay attention to the "IT Company" mnemonic (Investment × Time = Profit) and the distinction between two cases: When time is constant (equal for all partners). When time is different for partners.
To formalize what the video explained, we can categorize partnerships into two main types.
Partnership Formula Definition and Examples
This article from SSCAdda provides precise definitions and formulas for the two types of partnerships.
Please read the section titled "Types of Partnerships". Focus on the definitions and formulas for Simple Partnership (where time is the same) and Compound Partnership (where time is different). This formalizes the two cases discussed in the video.
So, the rule is simple:
- If partners invest for the same duration, the profit is shared in the ratio of their capitals:
- If partners invest for different durations, the profit is shared in the ratio of the product of their capital and time:
This second case, the Compound Partnership, is where most exam questions come from.
2. Solving Problems with Changing Investments
SSC CGL problems often involve dynamic situations where partners join late, leave early, or change their investment amount midway through the year. The logic remains the same, but the calculation requires careful attention to the timeline.
The effective investment for a partner is calculated by summing the (Capital × Time) products for each period they had a different investment.
Effective Investment = (C₁ × T₁) + (C₂ × T₂) + ...
Let's see how this works in practice with a typical exam question.
SSC CGL 2024 | MATHS | PARTNERSHIP: साझेदारी By Rakesh Yadav Sir
In this video, Rakesh Yadav Sir solves a classic problem where partners add or withdraw capital after a few months. This demonstrates the "sum of products" method clearly.
Watch the third problem discussed in the video, starting from this segment. Observe how the total investment for each partner is calculated: Partner A: (Initial Capital × 3 months) + (New Capital × 9 months) Partner B: (Initial Capital × 3 months) + (New Capital × 9 months) Partner C: (Capital × 6 months), as C joins late. The final ratio of these total investments is then used to divide the profit.
This method is foolproof, but the calculations can sometimes involve large numbers. A smart trick used by many toppers is to simplify the capital amounts at the very beginning by taking their ratio. This makes all subsequent multiplications much easier.
Let's see this shortcut in action.
Partnership (साझेदारी) के सभी Concepts + Questions 🔥 by Aditya Ranjan Sir Maths | Rankers Gurukul
This next clip from Aditya Ranjan Sir tackles a problem where partners join and leave at different times. The key technique to observe is how he simplifies the large investment amounts into a simple ratio before starting the calculation.
Watch from this example. Notice how the initial investments of ₹25,000, ₹50,000, and ₹30,000 are immediately converted to the ratio 5:10:6. The entire problem is then solved using these smaller, more manageable numbers. This is a crucial time-saving technique for your exam.
3. Working and Sleeping Partners
Another common variation in partnership problems involves distinguishing between different types of partners. The two main types you'll encounter are:
- Sleeping Partner: A partner who only invests capital and does not participate in running the business. They are entitled only to a share of the profit based on their investment.
- Working Partner: A partner who invests capital and actively manages the business. Because of their extra contribution of labour, they are often compensated with a salary or a fixed percentage of the total profit, which is paid out before the remaining profit is distributed among all partners.
Let's read a quick definition to solidify these terms.
Partnership Question Types, Formulas, Short Tricks and Tips
This section from an Oliveboard article provides a clear glossary of terms used in partnership.
In the table under the heading "Partnership Quantitative Aptitude Short Notes", focus on the descriptions for Working Partner and Sleeping Partner.
Here’s how you handle a problem with a working partner:
Example: A and B start a business with investments of ₹50,000 and ₹70,000. A is a working partner and is to receive 20% of the total profit as a salary. If the total profit at the end of the year is ₹48,000, what is A's total share?
Step 1: Calculate the working partner's salary.
Total Profit = ₹48,000
A's Salary = 20% of ₹48,000 = ₹9,600
Step 2: Calculate the remaining profit to be distributed.
Remaining Profit = Total Profit - A's Salary
Remaining Profit = ₹48,000 - ₹9,600 = ₹38,400
Step 3: Distribute the remaining profit according to the investment ratio.
Investment Ratio (A : B) = 50,000 : 70,000 = 5 : 7
Sum of ratios = 5 + 7 = 12
A's share of remaining profit = (5/12) × ₹38,400 = ₹16,000
B's share of remaining profit = (7/12) × ₹38,400 = ₹22,400
Step 4: Calculate the total earnings for the working partner.
A's Total Share = A's Salary + A's share of profit
A's Total Share = ₹9,600 + ₹16,000 = ₹25,600
This multi-step process is crucial to get right. Always deduct the salary or commission for the working partner first, and only then distribute the rest.
Conclusion
In this lesson, we've broken down the topic of Partnership, a direct and practical application of ratios and proportions. We established the unshakable foundation of this chapter and then built upon it to handle the complex scenarios you will face in the SSC CGL exam.
Key Takeaways:
- The fundamental rule is Profit ∝ Capital × Time. The ratio of profits is the ratio of the (Capital × Time) products.
- For problems with changing investments (partners joining, leaving, or changing capital), calculate each partner's total effective investment by summing up the (Capital × Time) products for each distinct period.
- A crucial shortcut is to simplify the initial investment amounts into a ratio before you begin your calculations.
- When dealing with a working partner, always calculate and subtract their salary/commission from the total profit before distributing the remaining amount based on the investment ratio.
In our next lesson, we will shift gears to another cornerstone of arithmetic: Averages. You will learn how to calculate simple and weighted averages, skills that are fundamental to solving a wide range of problems, especially those involving data interpretation.