Aptitude Partnership Problems

Partnership Problems

Type I: Profit Share Of Partners When Investment Durations Are Same. (Why This Is The Easiest And Basic Type?)

This type is very easy to solve. You will understand why after reading the below example. In this type, you will find amounts invested by 2 or more partners forsame duration of time. You will be given the profit amount. You have to find the profit share of each member in the group.

Let us see an example.

Example Question 1: Ravi, Kiran and Raj start a business by investing Rs.1,00,000, Rs.1,50,000 and 2,00,000 respectively. Find the share of each out of an annual profit of Rs.50,000.

Solution:
To solve this question you have to remember the below formula:

If duration (months/years) of investment of partners is same, the profit will be shared in the ratio of their investments.

In other words, if duration of investment is same,
 

Ratio of profit between partners = Ratio of investment between partners

In our example, the ratio of investments by Ravi, Kiran and Raj is as follows

1,00,000 : 1,50,000 : 2,00,000

Therefore, based on our formula,

The ratio of their profits is also equal to 1,00,000 : 1,50,000 : 2,00,000

If we simplify the above ratio, we will get

Ratio of profits = 2 : 3 : 4

You know that the total profit earned = Rs. 50,000

Therefore, the profit of each person can be calculated as shown below. (If you don’t understand the below calculations, please refer to formula in example question 2 of ratio problems tutorial. )

Ravi’s profit share = 50,000 x 2/9 = Rs.11,111.11

Kiran’s profit share = 50,000 x 3/9 = Rs.16,666.67

Raj’s profit share = 50,000 x 4/9 = Rs.22,222.22

 

Type II: Profit Share Of Partners When Investment Durations Are Different (How This Is Different From Type 1?)

This type is same as type 1, but here the investment duration of partners will be different. To solve this you have to apply a slightly advanced formula as shown in the example below:

Example Question 2: Sam and Rita are partners in a business. Rita invests Rs.40,000 for 6 months and Sam invests Rs.45,000 for 3 months. Out of a profit of Rs.30,000, Rita’s share is ____ .

Solution:
You have to apply the below formula to solve this problem:

Assume that three partners P1, P2 and P3 have invested amounts A1, A2 and A3 (in Rupees) for durations T1, T2 and T3 (months or years) respectively. Then,
 

Ratio of profit between P1, P2 and P3 = A1 x T1 : A2 x T2 : A3 x T3 

Note: You have to extend the above formula if there are more than 3 partners. For example, for 4 partners the above formula will

become A1 x T1 : A2 x T2 : A3 x T3 : A4 X T4

You know that the total profit earned = Rs. 30,000

From the question, you know that Rita invests Rs. 40,000 for 6 months and Sam invests Rs. 45,000 for 3 months.

In this example, there only 2 partners. Therefore, you have to use the below formula:

Ratio of profit shared between Rita and Sam = Investment of Rita x Duration of investment of Rita : Investment of Sam x Duration of investment of Sam

You know the below values from question:

Investment of Rita = Rs. 40,000

Duration of investment of Rita = 6 months

Investment of Sam = Rs. 45,000

Duration of investment of Sam = 3 months

If you substitute the above values in the formula, you will get

Ratio of profit between Rita and Sam = 40,000 x 6 : 45,000 x 3

= 2,40,000 : 1,35,000

= 16 : 9

Therefore, you can find Rita’s profit as shown below:

Rita’s share = 30,000 x 16/25 = Rs. 19,200

 

Type III: Profit Between Partners When Amounts Are Invested/Withdrawn In Parts (How To Solve This Difficult Type?)

This is the most difficult type of partnership problems. But you can easily solve if you carefully read the below example.

Here is our example.

Example Question 3: Anu, Mahesh and Naren enter into a partnership and invest Rs. 30,000, Rs. 40,000 and Rs.50,000 respectively. Mahesh withdraws Rs.20,000 at the end of first year and Naren withdraws Rs.25,000 at the end of second year. At what ratio will they share their profit at the end of 3 years.

Solution:
In this case, you know Anu has invested Rs 30000 for 3 years (for years I, II and III).

Mahesh invests 40,000 at start but withdraws Rs 20,000 a the end of first year. Therefore, his remaining investment in business will be 40,000 – 20,000 = 20,000.

In other words, Mahesh’s investment is 40,000 for year I and 20,000 for years II & III

And, Naren invests 55,000 at start but withdraws 25,000 at the end of second year. 

Therefore, Naren’s investment is 50,000 for years I & II and 25,000 for year III.

Now, let us see the formula you should use to solve such problems. (This is actually an extension to formula in example question 2)

Let partner P1 has invested amount A1a for T1a months, A1b for T1b months and so on…

Let partner P2 has invested amount A2a for T2a months, A2b for T2b months and so on…

Let partner P3 has invested amount A3a for T3a months, A3b for T3b months and so on… 

Then, ratio of profits between P1, P2 and P3 = (A1a x T1a + A1b x T1b …) : (A2a x T2a + A2b x T2b…) … : (A3a x T3a + A3b x T3b …)

Is the above formula very difficult to understand? It will become easy after you finish reading this example.

To solve our example using the above formula, first let us convert years into months. So, you will get

Anu’s investment = 30000 for 36 months

Mahesh’s investment = 40000 for 12 months and 20000 for 24 months

Naren’s investment = 50000 for 24 months and 25000 for 12 months

Now, if you apply the formula, you will get

Ratio of their profits = (30,000 x 36) : (40,000 x 12 + 20,000 x 24) : (50,000 x 24 + 25,000 x 12)

= 10,80,000 : 96,000 : 15,00,000

= 54 : 48 : 75