## 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 ₹ I_{A} and ₹ I_{B} 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 = I_{A} : I_{B} |

**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 ₹ I_{A} and ₹ I_{B} for the same period and earned total profit ₹ P, then

A's share = | _{A} |
× p | ||

I_{A} + I_{B} |

B's share = | _{B} |
× p | ||

I_{A} + I_{B} |

**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 = | _{A} |
× p | ||

I_{A} + I_{B} |

= | × 44000 | |||

1600 + 6000 |

= | × 44000 | |||

22000 |

= ₹ 32000

∴ B's share = | _{B} |
× p | ||

I_{A} + I_{B} |

= | × 44000 | |||

16000 + 6000 |

= | × 44000 | |||

22000 |

= ₹ 12000

**Rule**- If three partners A, B and C invest ₹ I_{A}, ₹ I_{B}, I_{C},…… for the same period, then the ratio of their profit will be

Profit of A : Profit of B : Profit of C : …… = I_{A} : I_{B} : I_{C} : .......... |

**Exampl**e- 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, I_{A} = ₹ 5000

I_{B} = ₹ 8000

I_{C} = ₹ = ₹ 10000

Ratio of Profit = I_{A} : I_{B} : I_{C} |

= 5000 : 8000 : 10000

= 5 : 8 : 10

**Rule**- If three partners A, B and C invest ₹ I_{A}, ₹ I_{A} and ₹ I_{C} for the same period and earned total profit ₹ P, then

A's share = | _{A} |
× p | ||

I_{A} + I_{B} + I_{C} |

B's share = | _{B} |
× p | ||

I_{A} + I_{B} + I_{C} |

C's share = | _{C} |
× p | ||

I_{A} + I_{B} + I_{C} |

**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 = | _{A} |
× p | ||

I_{A} + I_{B} + I_{C} |

= | × 60000 | |||

5000 + 7000 + 8000 |

= | × 60000 | |||

20000 |

= 5000 × 3

= ₹ 15000

B's share = | _{B} |
× p | ||

I_{A} + I_{B} + I_{C} |

= | × 60000 | |||

5000 + 7000 + 8000 |

= | × 60000 | |||

20000 |

= 7000 × 3

= ₹ 21000

C's share = | _{C} |
× p | ||

I_{A} + I_{B} + I_{C} |

= | × 60000 | |||

5000 + 7000 + 8000 |

= | × 60000 | |||

20000 |

= 8000 × 3

= ₹24000

**Rule**- If two partners A and B invest ₹ I_{A} and ₹ I_{B} for the different time period t_{1} and t_{2}, then the ratio of their profit will be

Profit of A : Profit of B = I_{A} × t_{1} : I_{B}× t_{1} |

**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 = I_{A} × t_{1} : I_{B} × t_{2} |

= 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 = I_{A} × t_{1} : I_{B} × t_{2} |

= 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

** **