A Partnership Problems is primarily a business venture in which two or more individuals/parties known as “partners” invest money and other valuable resources and share ownership and profits and losses.

**Ratio of Division of Gains**:

1. Suppose A and B invest Rs. \(x\) and Rs. \(y\) respectively for a year in a business, then at the end of the year:

(A’s share of profit) : (B’s share of profit) = \(x\) : \(y\)

Here investment of all partners are for same time, and the gain or loss is distributed among them in the ratios of their investments.

2. Suppose A invests Rs. \(x\) for ‘p’ months and B invests Rs. \(y\) for ‘q’ months, then

(A’s share of profit) : (B’s share of profit) = \(x\)p : \(x\)q

Here investments are for different time periods, equivalent capitals are calculated for a unit of time by taking,

(capital x number of units of time).

Profit or loss is divided in the ratio of these capitals.

**Working partner**: The partner one who works for the business is called a working partner.

**Sleeping partner**: The partner who simply invests the money for the business and doesn’t work is called a sleeping partner.

Here first person invested amount A for \(t_{1} \) period, second persons invested amount B for \(t_{2} \) period and so on.

**Example 1**:

A starts a business with Rs 2,000, B joins him after 3 months with Rs 4,000. C puts a sum of Rs 10, 000 in the business for 2 months only. At the end of the year the business gave a profit of Rs 5600. How should the profit be divided among them?

**Solution**:

- Ratio of their profits (A’s : B’s : C’s) = 2 x 12 : 4 x 9 : 10 x 2 = 6 : 9 : 5

Now, 6 + 9 + 5 = 20

Then A’s share = \(\frac{5600}{20} \times 6\) = Rs 1680

B’s share = \(\frac{5600}{20} \times 9\) = Rs 2520

C’s share = \(\frac{5600}{20} \times 5\) = Rs 1400

**Example 2**:

A, B and C invested capital in the ratio 2 : 3 : 5, the timing of their investments being in the ratio 4 : 5 : 6. In what ratio would their profit be distributed?

**Solution**:

- We should know that If the duration for their investments be in the ratio x : y : z, and investments is in ratio a : b : c then the profit would be distributed in the ratio ax : by : cz.

Thus, following the same rule, the required ratio = 2 x 4 : 3 x 5 : 5 x 6 = 8 : 15 : 30

**Example 3**:

A, B and C invested capital in the ratio 5 : 6: 8. At the end of the business term, they received the profits in the ratio 5 : 3 : 12. Find the ratio of time for which they contributed their capital?

**Solution**:

- Using the above formula, we have the required ratio

= \(\frac{5}{5}\) : \(\frac{3}{6}\) : \(\frac{12}{8}\)

= 1 : \(\frac{1}{2}\) : \(\frac{3}{2}\) : 2 : 1 : 3

- Given that

Ratios of their investments = 6,00,000 : 8,00,000 : 14,00,000

⇒ 6 : 8 : 14 ⇒ 3 : 4 : 7

Sum of the ratios = 3 + 4 + 7 = 14

Now, Share of first person = 60,000 x \(\frac{3}{14}\) = 12857.1

Share of second person = 60,000 x \(\frac{4}{14}\) = 17142.9

Share of third person = 60,000 x \(\frac{7}{14}\) = 30000

Therefore, Share of each person is

Rs.12857.1, Rs.17142, Rs.30000 respectively.

**2. Joseph, Johnson and Tom started a business each investing Rs. 20,000. After 5 months Joseph withdrew Rs.5000, Johnson withdrew Rs. 4000 and Tom invests Rs. 6000 more. At the end of the year, a total profit of Rs. 69,900 was recorded. Find the share of each.**

**Solution**:

- Given that

Joseph, Johnson and Tom invested Rs.20,000 together

and Tom invested more of Rs.6000 after 5 months

After 5 months, Joseph drew Rs.5000 and Johnson drew Rs. 4000

Total profit of the year = Rs. 69,900

Therefore ratio of the capitals of Joseph, Johnson and Tom is

= 20000 x 5 months + 15000 x 7 months : 20000 x 5 months + 16000 x 7 months : 20000 x 5 + 26000 x 7 months

= 205000 : 212000 : 282000

= 205 : 212 : 282

Sum of the ratios of capitals = 205 + 212 + 282 = rs. 699

Now, Joseph’s share = Rs.( 69900 x \(\frac{205}{699}\)) = Rs. 20500;

Johnson’s share = Rs.( 69900 x \(\frac{212}{699}\)) = Rs. 21200;

Tom’s share = Rs.( 69900 x \(\frac{282}{699}\)) = Rs. 28200.

**3. P, Q and R enter into partnership. P invests 2 times as much as Q invests and Q invests two- third of what R invests. At the end of the year, the profit earned is Rs. 7000. What is the share of Q?**

**Solution**:

- Given that,

Profit earned by all of them at the end of the year = Rs. 7000

Let the R’s capital be Rs. \(x\)

Then, Q’s capital = Rs.\(\frac{2}{3}x\)

P’s share = 2 x \(\frac{2}{3}x\) = Rs. \(\frac{4}{3}x\)

Therefore, Ratios of their capitals = \(\frac{4}{3}x\) : \(\frac{2}{3}x\) : \(x\) = 4\(x\) : 2\(x\) : 3\(x\)

Sum of the ratios = 4 + 2 + 3 = 9

Then, Q’s share = Rs. (7000 x \(\frac{2}{9}\)) = Rs. 1555.5 ≅ Rs. 1556

**4. A, B, and C bought a plot for Rs. 2 lakh. A contributed Rs. 1,50,000 when they sold that from the profit, B got Rs. Rs. 5050 while C got Rs. 3000. What was the profit of A?**

**Solution**:

- Given that,

Capital of A + capital of B + capital of C = Rs. 2,00,000

Capital of A = Rs. 1,50,000

Then, Capital of (A + B + C) = Rs. 2,00,000

⇒ Capital of (B + C) = Rs. (2,00,000 – 1,50,000)

⇒ Capital of (B + C) = Rs. 50,000

Also given, Profit of (B + C) = Rs. (5050 + 3000) = Rs. 8050

Therefore, A’s share = Rs. (8050 x \(\frac{150000}{50000}\))

⇒ Rs. 8050 x 3

⇒ RS. 24150

Therefore, profit of A = Rs. 24150

**5. P and Q started a business with capitals in the ratio of 5 : 8. After 2 months, Q took back his money. If they got profit in the ratio 3 : 6, for how many months P’s capital continued in the business?**

**Solution**:

- Given that,

Let P continued the business for \(x\)months

Ratio of their capitals = 5 \(x\) : 8 x 5 ⇒ 5 \(x\) : 40

Therefore,

Ratio of their capitals = ratio of the profit made by them

⇒ 5 \(x\) : 40 = 3 : 6

⇒ 5 \(x\) x 6 = 40 x 3

⇒ 30\(x\) = 120

⇒ \(x\) = \(\frac{120}{30}\)

⇒ \(x\) = 4 months

Hence, Capital of A is continued for 4 months in the business.