Partnership Problems: A Simple Approach
Core Concept
A partnership is a relationship between two or more persons who have agreed to share the profits of a business carried out by all or any of them. In CMAT Quantitative Ability, partnership problems test your ability to distribute total profit based on the capital invested and the duration of the investment.
The Fundamental Formula
The golden rule of partnership is that the profit of each partner is directly proportional to the product of their investment and the time for which that investment is held. If Partner A invests $I_1$ for time $T_1$, and Partner B invests $I_2$ for time $T_2$, the ratio of their profits ($P_1 : P_2$) is given by:
$$P_1 : P_2 = (I_1 \times T_1) : (I_2 \times T_2)$$
If the time period is the same for all partners, the profit ratio is simply equal to the investment ratio: $P_1 : P_2 = I_1 : I_2$.
Calculating Profit Shares
When a specific profit percentage or amount is involved, follow these steps to isolate the share:
- Calculate the total ratio units: Add the ratio parts (e.g., if ratio is $3:2$, total units = $3+2=5$).
- Determine the individual fraction: Partner A's share = $\frac{3}{5}$ of total profit, Partner B's share = $\frac{2}{5}$ of total profit.
- Apply percentages: If the total profit is given as a percentage of the total investment, find the value in currency first, then apply the ratio.
Worked Example
Ram invests Rs. 10,000 for 12 months, and Shyam invests Rs. 15,000 for 8 months. If the total profit at the end of the year is Rs. 10,000, find Ram's share.
- Calculate profit ratio:
- Ram's contribution factor = $10,000 \times 12 = 120,000$
- Shyam's contribution factor = $15,000 \times 8 = 120,000$
- The ratio is $120,000 : 120,000$, which simplifies to $1:1$.
- Ram's share = $\frac{1}{1+1} \times 10,000 = Rs. 5,000$.
Exam Tips for CMAT
- Standardize time: Always ensure the time $T$ is in the same unit (all in months or all in years) before multiplying.
- Ignore zeros: When dealing with large investment figures, cancel out common zeros first to simplify the ratio calculation (e.g., $10,000 : 15,000$ becomes $10 : 15$ or $2 : 3$).
- Working vs Sleeping Partners: Sometimes a partner may receive a salary for management (a 'working partner'). Always subtract this salary from the total profit before distributing the remainder among partners in the agreed ratio.
- Percentage Traps: Read carefully to see if the question asks for the share of the total profit or the share after a specific deduction (like tax or management salary).
Practise these as timed MCQs to lock the pattern in.