Today, we will discuss the basics of Partnership and types of problem asked in previous year exams and how to approach them. This will help you in clearing your basics and solving these questions.

Tips for Partnership Problems

When two or more people invest their money in a business, persons are called Partners, their relationship is Partnership and money is Capital.

• If they invest money for the same time, it is called simple partnership.
• If they invest money for different time, it is called compound partnership.

This topic is basically based on ratio and percentage. We gave basics of Ratio in Time and Work. To learn the basics of ratio and percentage click the link given below: 

Concepts of Percentages and its Important Shortcuts

How to solve Percentages

Partnership Problems:

Profit is directly proportional to Time and Investments.

Profit ∝ Time 
Profit ∝ Investments
=> Profit ∝ (Time × Investments)

Example 1:

Three partners A, B and C invest Rs.1500, Rs.1200 and Rs.1800 respectively in a company. How should they divide a profit of Rs.900 ?

Solution: Given, there is no time given, we can say profit is proportional to investment.
Ratio of profit = ratio of investment
Profit ratio  of A:B:C = 1500:1200:1800 =5:4:6
so, total profit is 5+4+6 = 15 i.e. equal to 900
profit of A = (5/15)× 900 = 300
profit of B = (4/15)× 900 = 240
profit of C = (6/15)× 900 = 360

Example 2:

In a company, A invested Rs.1500 for 4 months and B invested Rs.1200 for 6 months and C invested Rs.3600 for 2 months. If company has a profit of Rs.680. What will be the share of A,B and C ?

Solution:
Ratio of profit A:B:C = (1500 × 4):(1200 × 6):(3600 × 2)
                                     = 60:72:72
                                     = 5:6:6
total profit is 5+6+6 = 17 i.e. equal to 680.
we can say, 17 = 680
                   1 = 40
profit of A is 5, so 5× 40 = 200
profit of B is 6, so 6× 40 = 240
profit of C is 6, so 6 × 40 = 240
 

Note: Read questions carefully. If we can calculate capital invested and time for which capital invested. We can easily calculate share in profit.

Example 3:

A and B enter into a partnership with Rs.50000 and Rs.75000 respectively in a company for a year. After 7 months, C get into partnership with them with Rs.30000 and A withdraws his contribution after 9 months. How would they share their profit of Rs.2600 at the end of the year?

Solution: A, B and C do business for 1 year but, A contributed Rs.50000 for 9 months, B contributed 75000 for 12 months and C invested Rs.30000 for 5 months, not for 7 months.
So the ratio of profit  A:B:C = 50×9: 75×12: 30×5
                                          = 15 : 30 : 5
Hence total profit is (15+30+5) = 50 which is equal to 2600
So share of A = (15/50)× 2600 = 780
 share of B = (30/50)× 2600 = 1560
 share of C = (5/50) × 2600 = 260

Example 4:

A, B and C started a company in which A invested (1/3)^rd of the capital for (1/4)^th of the time, B invested (1/2)^nd of the capital for (1/6)th of the time and C invested the remaining capital for the whole of the time. If the profit at the end of the year is Rs.1200. How would they share it?

Solution:  A invested (1/3)^rd of the capital and B invested (1/2)^nd of the capital
So, remaining capital invested by C = 1-((1/3)+(1/2)) = 1/6
The ratio of profit A: B:C = (1/3)× (1/4): (1/2)× (1/6): (1/6)× 1
                                      =  (1/12):(1/12):(1/6)
                                      = 1 : 1 : 2
A’s share = (1/4)× 1200 = 300
B’s share = (1/4)× 1200 = 300
C’s share = (1/2)× 1200 = 600

Example 5:

A and B rent a field for 11 months. A puts 100 bags for 9 months. How many bags can be put by B for 3 months if the ratio of their rent is 2:3?

Solution: Let B puts X bags.
the ratio of rent of A: B is 2: 3
so, (100×9) : (X × 3 ) = 2 : 3
        X = 450 bags

Example 6:

If A and B entered into a partnership and invested their capital in the ratio of 19:15. At the end of 19 months, B withdraws his capital. If they share profit in the ratio of 3:2, then for how many months A invested his ratio?

Solution: Let A invested for X months.
Ratio of profit A : B = X × 19 : 19 × 15
So, 19X : 19×15 = 3:2
 X = 22(1/2) months

Example 7:

Sandeep, Vineet and Shekhar are three partners. Sandeep receives 1/5 of the profit and Vineet and Shekhar share the remaining profit equally. If Vineet’s income is increased by Rs.650 when the profit rises from 10% to 15%. Find the capitals invested by Sandeep, Vineet and Shekhar and total capital invested.

Solution: As given, the profit share of Sandeep is 1/5, remaining profit (1-1/5) = 4/5 is shared between Vineet and Shekar equally.
So, the profit share of Vineet = 2/5 and profit share of Shekhar = 2/5
when profit % increases, Vineet’s income increase by Rs.650
 (15%-10%) = 5% = 650
100% = 13000
So, Vineet’s capital = 13000
i.e (2/5) of total capital = 13000
total capital = 32500
and Shekhar’s capital = 13000
Sandeep’s capital i.e (1/5) of total capital or ½ of (Vineet or Shekhar’s Capital) = 6500

Example 8:

A, B and C start a business. Twice the capital of A is equal to thrice the capital of B and Capital of B is four times of the capital of C. What will be A’s share if the profit earned is Rs. 2,75,000 . 

Solution: Let the capital of C is C.
Given, 2A=3B and B = 4C
So, 2A = 3× 4C = 12 C
      A = 6C
Hence the ratio of capital A: B: C = 6: 4: 1
so, Share of A = (6/11)×2,75,000 = 1,50,000

Example 9:

A and B are partners in a business. They invest in the ratio 5: 6, at the end of 8 months B withdraws. If they receive profits at the end of the year in the ratio of 5: 9, find how long A’s investment was used? 

Solution:  Let A’s investment used for X months.
Given, the ratio of invest (A: B) = 5: 6
 ratio of time =  X : 8
the ratio of profit = 5X: 6×8 and given ratio of profit = 5: 9
so 5X/48 = 5/9
X = 48/9
X = 16/3 months

Example 10:

A, B and C started a business with their investments in the ratio 1: 2: 4. After 6 months A invested the half amount more as before and B invested same the amount as before while C withdrew (1/4)^th of his investment after the 9 months. Find the ratio of their profits at the end of the year. 

Solution: Ratio of investments A:B:C = 1:2:4, there are no changes in the investment of A and B up to 6 months and in the investment of C up to 9 months.
At the end of 6 months, A invested half the amount more as before so A’s investment = 1 +(1/2)
Similarly B invest the same amount more as before = 2 + 2 = 4
But, C withdraw the (1/4)^th of the amount after 9 months = 4 – 1 = 3
ratio of profit = (1×6 + (3/2)× 6) : (2× 6 + 4× 6) : (4× 9+3× 3)  
                         = 15 : 36 : 45
                         = 5 : 12 : 15

Example 11:

A sum of money is divided amongst P, Q and R in the ratio of 3: 4: 5. Another amount is divided amongst A and B in the respective ratio of 2: 1. If B got Rs. 1050 less than Q, what is the amount received by R?

Solution: Let the sum of money divided amongst P, Q and R is 3x, 4x and 5x respectively and the sum of money divided amongst A and B is 2y and y respectively.
4x – y = 1050
another relation between x and y cannot be established. So, it cannot be determined.

Directions (12-15): In the following table, the investments and profit of three persons is given for different years in a joint business.

Note:
1. Except for the year 2012, they invested the amounts for the same period.
2. Some values are missing. You have to calculate these values per given data.

Example 12:

If the total profit in 2011 is 45000, then find the ratio of the investment of B in 2010 to the investment of A in 2011.

Solution: profit of A in 2011 is 45000-(15000+17500) = 12500
B makes the profit of 15000 by investing 6000
So, investment of A in 2011 = (6000/15000)× 12500 = 5000
In 2010, 23000 investment of C makes the profit of Rs.115000
So, investment of B = (23000/115000)× 82500 = 16500
required ratio of (B:A) is 16500:5000 = 33:10

Example 13:

If the total investment in 2014 is 46000, then the ratio of profit in 2014 is?

Solution: investment of C is 46000 – (20000+11000) = 15000
The time period is the same, so the ratio of profit will be also the same as the ratio of investment = 11:20:15

Example 14:

In the year 2012 total investment of A and B is 30000, A and B invested their amount for 4 months and 6 months respectively then find the number of months that C invested his amount?

Solution:  ratio of profit (A:B) = 42000: 27000
              A× 4 : B× 6 = 42000 : 27000
                 A : B = 21 : 9 = 7 : 3
 So, investment of A is 21000 and investment of B is 9000.
let C invested 18000 for X months.
So, (18000× X) : (21000 × 4) = 24000 : 42000
 X = (8/3) months, Hence C invested for 8/3 months.

Example 15:

If the total profit in the year 2013 is 58800 then the investment of A is?

Solution: Rs.10000 investment of C gives a profit of Rs.14000
then, Rs.17000 investment of B will give the profit of Rs. (14000/10000)× 17000 = 23800
So, profit of A is 58800 – (14000+23800) = 21000
Investment of A is = (14000/10000)×21000 = 15000