*Thursday, March 18th, 2021*

To start a business there is always a requirement of an adequate amount of money which in business terms called “INVESTMENT” OR “CAPITAL”. It is not necessary that a person always has enough money to start a business. In this case, he tries to get partners who are interested in his business.

Two or more people can get together to do business by pooling their resources. The people who have invested money in the partnership are called PARTNERS.

While two or more partners would have invested money, it is not necessary that all of them should be involved in the day to day running of the business. The partners involved in the day-to-day activities are called “working partners” and the others are called “sleeping partners” or “dormant partners”.

The profits left after paying the working partners’ remuneration are shared amongst all the partners.

Sometimes, the partners also take interest in their investment and only the remaining profits are shared by the partners.

Sharing of profits among the partners also depends on the understanding between the partners. However, if no special scheme of sharing the profits is specified (in a problem), then the profits are shared based on the investments of the partners. There are three different possibilities that exist here.

- If the partners invest DIFFERENT amounts each for the SAME period of time, then the profits at the end of the year are shared in the ratio of their investments.
- If the partners invest the SAME amounts for a DIFFERENT period of time, then the profits at the end of the year are shared in the ratio of the time periods for which their respective investments have been in business.
- If the partners invest DIFFERENT amounts and the time periods for which their investments are in the business are also DIFFERENT, then the profits at the end of the year are shared in the ratio of the product (investment * time period) for each partner.

There can be problems that are modeled along with the sharing of profits in partnerships.

**Example 1:** X and Y invest 21,000 and 17,500 respectively in a business and at the year they make a profit of 26,400. Find their individual shares in the profit.

**Sol:** Since, both their investments are there in the business for the same duration (1 year), profits will be shared in the ratio of their investments i.e. 21,000: 17,500 = 6: 5.

∴ X share = 6/11 * 26,400= 14, 400

Y share= 5/11 * 26,400= 12,000

**Example 2: **Krishna starts a business with 45, 000. Three months later Arjun joins him with 30, 000. At the end of the year, in what should they share the profits?

**Sol:** Sharing of profits will be in the ratio of investments multiplied by the time period.

Hence the ratio is :

(45, 000 * 12 ) : (30, 000 * 9) = 2 : 1

**Example 3: **Nakul started a business with 25, 000 and after 4 months Sanjay joined him with 60, 000. Nakul received 58, 000 including 10% of the profits as a commission for managing the business. What amount did Sanjay receive?

**Sol:** Ratio of shares of profit is : (25, 000 * 12) : (60, 000 * 8) = 5: 8

Let the total profit be P, as Nakul receives 10% of this as commission, the remaining 90% of P is shared in the ratio of 5: 8.

Hence Nakul’s receipts will be 5/13^{th} of 90% of total profit plus his commission.

0.1P + 5/1/3(0.9P) = 58, 000

⇒ 5.8P/13 = 58,000

⇒ P= 1, 30, 000

Therefore, Sanjay share will be 8/13 *0.9 * 1,30, 000= 72, 000

**Example 4: **A started a business with 40, 000. After 2 months B joined him with 60, 000. C joined them after some more time with 1, 20,000. At the end of the year, out of a total profit of 3, 75, 000, C gets 1,50, 000 as his share. How many months after B joined the business did C join?

**Sol:** The ratio of the shares of profits is : (40, 000 *12) : (60, 000 * 10) : (1,20, 000 * T), here T is the number of months that C was with the business

= 24: 30 : 6T = 4: 5: T

Hence, C share = T/ (T+9)

This is equivalent to 1,50, 000 out of 3,75,000

⇒ T/ (T+9) = 0.4

⇒ T= 6

So, C was with the business for 6 months. Hence C joined the business 4 months after, B

joined the business.

**Example 5: **Peter started a business with 20, 000. John joined him 4 months later with 30, 000. After 2 months Peter withdrew 5,000 of his capital and 2 more months later, John brought in 20, 000 more. At the end of the year what should be the ratio in which they should share the profits?

**Sol:** Here, even for each individual, the capital was not the same for the entire period his money was in the business. So, the term of the ratio for a person will be the sum of products of investment multiplied by time period for different parts of the year.

Peter has 20, 000 for 6 months and then since he withdrew 5,000, he had only 15, 000 for the rest of the 6 months. His term of the ratio will be

(20, 000 * 6) + ( 15, 000 * 6) = 2,10,000

John joined with 30,000 which remained unchanged for 4 months and then he brought in 20, 000 more. So he had 50, 000 for 4 months only as he joined 4 months after the business began. His term of the ratio will be

(30, 000 * 4) + (50, 000 * 4) = 3, 20, 000

Hence, the ratio of shares of the profit will be 2,10, 000 : 3, 20, 000 = 21 : 32.

**Example 6: **Ram, Krishna, and Arjun start a business with 30K, 40K, and 50K respectively. Ram stays for the entire year. Krishna leaves the business after two months but rejoins after another 4 months but only with 3/4^{th} of his initial capital. Arjun leaves after 3 months and rejoins after another 5 months but with only 4/5 of his capital. If the year-end profit is 27,900, how much more than Krishna did Arjun get?

**Sol:** The ratio of the investments of Ram, Krishna, and Arjun is:

(30,000*12): (40,000*2+3/4 *40,000 *6): (50,000*3+ 4/5*50,000*4)

⇒ (360,000) : (2,60,000) : (3,10,000)

⇒ 36:26:31

Let’s Ram’s share is 36x, Arjun’s share is 31x and Krishna’s share is 26x, then the difference is 31x -26x = 5x.

Hence the amount, Arjun get more than Krishna is: 5x/93x * 27, 900 = 1500

**Example 7: **A and B starts a business with different capitals. A was to get 15% of the profit as salary and the rest was to be divided in the ratio of their investments. Had the entire profit been distributed in the ratio of their investments, B would have got 1350 more than what he actually got. What is B’s actual share of the profit?

**Sol:** Let the profit be P and the ratio of profits of A and B is : A: B

Therefore, the profit B would have got is: B/A+B * P……………………………………..(1)

However, A gets 15% of the profit as a salary, so the remaining profit is 0.85P.

Now, the profit gets by B is: B/A+B *0.85P………………………………………….(2)

Subtract (2) from (1)

0.15BP/A+B = 1,350

BP/A+B=9,000

Put this value in Eq. (2)

The profit get by B is: 9,000 * 0.85 = 7,650

**Example 8: **A, B, C starts a partnership. The capital of A, B, C are in the ratio of 10:9:6, and the time period of A: B is in the ratio of 2:3. B gets 10, 800 shares out of a total profit of 26, 000. If A’s capital was in the business for 8 months, for how many months was C’s capital there?

**Sol:** Given, A was in the business for 8 months, therefore, B must be in the business for 12

months. As A:B = 2:3

Let the months for C’s capital presence is x.

A:B:C

10:9:6

8:12:x

⇒ 80:108: 6x

B’s share in profit = 108/(188+6x )* 26,000

This is equivalent to 10,800

108/(188+6x )* 26,000= 10,800

260= 188+6x

6x=72

X=12

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Sharing of profits among the partners also depends on the understanding between the partners. However, if no special scheme of sharing the profits is specified (in a problem), then the profits are shared based on the investments of the partners. There are three different possibilities that exist here.