When two or more persons start a business jointly and share the profit or loss thereof in an agreed proper portion, it is known as partnership business and the persons carrying on such business are called Partners. Generally partners share the profit or loss in the ratio of the capitals invested by them.
Partnership may be (1) simple; or
Simple Partnership - When the capitals of the partners are invested for the same time, then this type of partnership is called simple partnership. In such a case, the profit or loss is distributed in proportional to the capital invested.
Compound Partnership - When the capital, which is equal or unequal, of the partners, is invested for different times, this type of partnership is called compound partnership. In such a caste the profit or loss is distributed in proportional to the products of the capital and the periods of their investment.
An important formula for solving the problems of partnership is -
Working Rule -
1. If the ratio of investment by three persons is a: b: c and ratio of time invested in their capital is x: y: z then the ratio of their profit will be ax: by: cz.
2. If the ratio of investment by three persons is a:b:c and ratio of their profit is p:q:r then, the ratio of time invested in their capital will be
Example 1: A, B and C enter into partnership. A contributes one-third of the capital while B contributes as much as A and C together contribute. If the profit at the end of the year amounts to Rs. 840 what would each receive?
Sol: As A contributes one-third of the capital
Now as B contributes as much as A and C
So Profit of B = Profit of A + Profit of C = Rs. 280 + Profit of C
⇒ Profit of B – Profit of C = Rs. 280
And Profit of B + Profit of C = Rs. 840 – Rs. 280
Adding 2 Profit of B = Rs. 840
Profit of B = Rs. 420
Hence Profit of C = 840 – 420 - 280
= Rs. 140
Example 2: A is working and B is a sleeping partner in a business. A puts Rs. 5, 000 and B puts in Rs. 6, 000. A receives 12 ½ % of the profit for Managing the business and the rest is divided in proportion of their capitals. What does each get out of a profit of Rs. 880?
Sol: The amount, which A receives for managing
= 12 ½% of Rs.880
The amount left = 880 – 110 = Rs. 770
The amount left is to be divided in the ratio = 5,000 : 6,000 = 5: 6
Total share received by A = 110 + 350 = Rs. 460
Share received by B = Rs. 420
Example 3: A and B enter into a partnership. A contributes Rs. 5000 while B contributes Rs. 40000. After 1 month B withdraws 1/4 part of his contribution and after 3 months from the starting A puts Rs. 2000 more. When B withdraws his money at the same C also joins them with Rs. 7000. If at the end of 1 year there is a profit of Rs. 1218, what will be share of C in the profit?
Sol: Since the contributions of three partners are different and their times also differ. Therefore, their contributions should be converted for equal durations. For this, contribution is multiplied by time.
∴ Contribution of A = Rs. 5000 for 12 months + Rs. 2000 for 9 months
∴ Contribution of A for 1 month
= 5000 × 12 + 2000 × 9
= 60000 + 18000
= Rs. 78000
Contribution of B = Rs. 4000 for 1 month + of Rs. 4000 for 11 months
∴ Contribution of B for 1 month
= 4000 × 1 + 3000 × 11
= 4000 + 33000 = Rs. 37000
∴ Contribution of C = Rs. 7000 for 11 months
Contribution of C for 1 month = 7000 × 11
= Rs. 77000
∴ Ratio in their contributions = 78000:37000:77000
∴ Sum of their ratios = 78 + 37 + 77 = 192
= Rs. 488.47
Example 4: Alok started a business by investment of Rs.90000 after 3 months Pranav joined him with an investment of Rs. 120000. If they had a profit of Rs. 96000 after 2 years then what is the difference in the shares of two?
Sol: Alok’s investment for 1 month = 9000 × 24 = 2160000
Pranav’s investment for 1 month = 120000 × 21= 25200
Ratio of their investment = 6:7
Example 5: A, B and C started a business in partnership. A invested Rs. 25 lacks and after 1 year he invested Rs. 10 lacks more. B invested Rs. 35 lacks in the beginning and withdrew Rs. 10 lacks after 2 years. C invested Rs. 30 lacks. What is the ratio of their profit after 3 years?
Sol: A’s investment = 25 × 3 + 10 × 2
= Rs. 95 lacks
B’s investment = 35 × 2 + 25 × 1
= Rs. 95 lacks
C’s investment = 30 × 3
= Rs. 90 lacks
∴ Ratio of their investment = 19:19:18
∴ Ratio of their profit = 19:19:18 (because time period is same, i.e., for 3 years)
Example 6: A, B and C investment in a partnership in the ratio of 5:6:8. Ratio of their profit is 5:3:12. Find the ratio of time for their investment.
Example 7: Three people A, B and C invested money in a partnership in the ratio of 4:2:8 ratio of their time of investment is 3:3:2. What is the ratio of their profit?
Sol: Required ratio = 4 × 3 : 2 × 3 : 8 × 2
= 12 : 6 :16
= 6 : 3 : 8