A partnership is an arrangement in which two or more individuals invest money together to run a business and share the profits or losses. Partnership problems are common in aptitude and placement exams and are usually based on the ratio of capital invested and the time period of investment.
Types of Partnership
Partnerships are mainly of two types:
1. Simple Partnership:
When all partners invest their capital in the business at the same time, and the capital remains invested for the same period, it is called a simple partnership.
In this case, the profit is distributed among the partners in proportion to their invested capital.
Example:
Suppose P and Q invest Rs. a and Rs. b respectively in a business for the same period of time. Then the ratio of their profit or loss will be:
P’s share : Q’s share = a : b
2. Compound Partnership:
When the capital of the partners is invested in the business for different periods of time, the partnership is called a compound partnership.
In such cases, the profit-sharing ratio is calculated by multiplying the capital invested by the time period (usually measured in months).
Example:
If P invests Rs. a for t₁ months and Q invests Rs. b for t₂ months, then the profit or loss sharing ratio will be:
P’s share : Q’s share = a × t₁ : b × t₂
where
t₁ = duration of P’s investment
t₂ = duration of Q’s investment.
Relation Between Profit, Capital, and Time
In partnership problems, the distribution of profit depends on the capital invested and the time period for which the capital is invested.
1. Profit and Investment
Profit is directly proportional to the investment made by the partners.
Profit ∝ Investment
2. Profit and Time
Profit is also directly proportional to the time for which the capital remains invested.
Profit ∝ Time
3. Profit and Capital
If two partners invest money for the same time period, the profit shared between them will be proportional to their capital.
Profit (P) ∝ Capital (C)
4. Profit Sharing Ratio
If two partners invest C₁ and C₂ respectively, then their profit ratio will be:
P₁ : P₂ = C₁ : C₂
Important Formulas
1. When capital and time are both different:
Profit share = Capital × Time
2. Profit ratio between partners:
P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂)
where
C₁, C₂ = capital invested
T₁, T₂ = time duration
Solved Questions and Answers
Question 1: A and B invest Rs. 5000 and Rs. 7000 in a business. Find the ratio in which they share the profit.
Solution:
Profit ratio = Capital ratio
5000 : 7000
Divide by 1000
5 : 7
Question 2: P and Q invest Rs. 8000 and Rs. 6000 respectively for one year. If the total profit is Rs. 7000, find their shares.
Solution:
Capital ratio = 8000 : 6000
= 4 : 3Total parts = 7
P's share
= (4/7) × 7000
= 4000Q's share
= (3/7) × 7000
= 3000
Question 3: A invests Rs. 6000 for 12 months and B invests Rs. 8000 for 6 months. Find the profit-sharing ratio.
Solution:
A's share = 6000 × 12 = 72000
B's share = 8000 × 6 = 48000Ratio
72000 : 48000
Divide by 24000
3 : 2
Question 4: X and Y invest Rs. 10000 and Rs. 15000 respectively. X invests for 12 months and Y invests for 8 months. Find their profit ratio.
Solution:
X's share = 10000 × 12 = 120000
Y's share = 15000 × 8 = 120000Ratio
120000 : 120000
1 : 1
Question 5: A, B, and C invest Rs. 5000, Rs. 6000, and Rs. 9000 respectively for the same time. If the total profit is Rs. 10000, find their shares.
Solution:
Capital ratio
5000 : 6000 : 9000
Divide by 1000
5 : 6 : 9
Total parts
5 + 6 + 9 = 20
A's share
(5/20) × 10000 = 2500
B's share
(6/20) × 10000 = 3000
C's share
(9/20) × 10000 = 4500