## Partnership

When two or more persons invest some money for running a certain business together for a certain time and share the total profit at an agreed proportion.

#### Simple partnership

If all partners invest their money for the same time period.

#### Compound Partnership

If all partners invest their money for the different time period.

#### Sleeping Partner

A partner who only invests money and does not look after the business activities is called sleeping partner.

#### Active Partner

A partner who not only invests money but also takes part in the business activities is called an active partner.

### Some Useful Shortcut Method

Rule- If two partners A and B invest ₹ IA and ₹ IB for the same period, then the ratio of their profit will be

 ⇒ Profit of A = Investment of A Profit of B Investment of B

 ⇒ Profit of A : Profit of B = IA : IB

Example- A and B start a business by investing ₹ 5000 and ₹ 8000 respectively. Find the ratio of their profit after 1 year.
Solution:- given, Investment of A = ₹
Investment of B = ₹ 8000
Profit will be distributed between A and B in the ratio of their investment.

 ⇒ Profit of A = Investment of A Profit of B Investment of B
 ∴ Ratio of their profit = 5000 = 5 : 8 8000

Example– A and B jointly start a business. The investment of A is equal to four times the investment of B. Find the share of A in the annual profit of ₹ 55000.
Solution:- Let the investment of B = ₹ x
Then, investment of B = ₹ 4x

 Profit of A = Investment of A Profit of B Investment of B
 = 4x x

According to the question,
x + 4x = 55000
or, 5x = 55000
or, x = 55000/5
∴ x = 11000
∴ A’s share = 4x = 4 × 11000 = ₹ 44000

Rule- If two partners A and B invest ₹ IA and ₹ IB for the same period and earned total profit ₹ P, then

 A's share = IA × p IA + IB

 B's share = IB × p IA + IB

Example- A and B invested ₹16000 and ₹ 6000 for a period of 2 years. After 2 years, they earned ₹ 44000. What will be the share of A and B of this earning?
Solution:- given, Investment of A = ₹ 16000
Investment of B = ₹ 6000
Total Profit = ₹ 44000

 ∴ A's share = IA × p IA + IB
 = 16000 × 44000 1600 + 6000
 = 16000 × 44000 22000

= ₹ 32000
 ∴ B's share = IB × p IA + IB
 = 6000 × 44000 16000 + 6000
 = 6000 × 44000 22000

= ₹ 12000

Rule- If three partners A, B and C invest ₹ IA, ₹ IB, IC,…… for the same period, then the ratio of their profit will be

 Profit of A : Profit of B : Profit of C : …… = IA : IB : IC : ..........

Example- A, B and C start a business by investing ₹ 5000, ₹ 8000 and ₹ 10000 respectively. Find the ratio of their profit after 2 years.
Solution:- given, IA = ₹ 5000
IB = ₹ 8000
IC = ₹ = ₹ 10000

 Ratio of Profit = IA : IB : IC

= 5000 : 8000 : 10000
= 5 : 8 : 10

Rule- If three partners A, B and C invest ₹ IA, ₹ IA and ₹ IC for the same period and earned total profit ₹ P, then

 A's share = IA × p IA + IB + IC

 B's share = IB × p IA + IB + IC

 C's share = IC × p IA + IB + IC

Example- A, B and C invested ₹5000 and ₹ 7000 and ₹ 8000 for a period of 2 years. After 2 years, they earned ₹ 60000. What will be the share of A, B and C of this earning?
Solution:-
given, Investment of A = ₹ 5000
Investment of B = ₹ 7000
Investment of C = ₹ 8000
Total Profit = ₹ 60000

 A's share = IA × p IA + IB + IC
 = 5000 × 60000 5000 + 7000 + 8000
 = 5000 × 60000 20000

= 5000 × 3
= ₹ 15000
 B's share = IB × p IA + IB + IC
 = 7000 × 60000 5000 + 7000 + 8000
 = 7000 × 60000 20000

= 7000 × 3
= ₹ 21000
 C's share = IC × p IA + IB + IC
 = 8000 × 60000 5000 + 7000 + 8000
 = 8000 × 60000 20000

= 8000 × 3
= ₹24000

Rule- If two partners A and B invest ₹ IA and ₹ IB for the different time period t1 and t2, then the ratio of their profit will be

 Profit of A : Profit of B =  IA × t1 : IB× t1

Example- A starts a business with ₹ 4000 and B joins him after 4 months with ₹ 7000. Find the ratio of their profits at the end of the year.
Solution:- Here, A invest for 1 year = 12 months
B invest after 3 months = ( 12 - 4 ) = 8 months

 Ratio of profit A and B = IA × t1 : IB × t2

= 4000 × 12 : 7000 × 8
= 6 : 7

Example-A Starts a business with ₹ 3000 and B joins the business 4 months later with an investment of ₹ 8000. After 1 year, they earn a profit of ₹ 14000. Find the share of A and B.
Solution:- A invest for 1 year = 12 months
B invest after 6 months = ( 12 - 6 ) = 6 months

 Ratio of Profit A : Profit B = IA × t1 : IB × t2

= 3000 × 12 : 8000 × 6
= 3 : 4
Let the share of A = 3x and share of B = 4x
According to the question,
3x + 4x = 14000
or, 7x = 14000
or, x = 14000/7 = 2000
∴ A's share = 3x = 3 × 2000 = ₹ 6000
∴ B's share = 4x = 4 × 2000 = ₹ 8000