Partnership chapter deals with profit and loss related problems. Profit and/or loss is shared among the partners based on the sum of money invested by individual partners and the time period of the investment.
For example: A partner who has invested the highest amount of money/resources will receive the highest share of the profit at the end of the year provided all the partners have invested for the same time period.
Ratio of Division of Gains:
1. Suppose A and B invest Rs. [latex]x[/latex] and Rs. [latex]y[/latex] respectively for a year in a business, then at the end of the year:
(A's share of profit) : (B's share of profit) = [latex]x[/latex] : [latex]y[/latex]
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. [latex]x[/latex] for 'p' months and B invests Rs. [latex]y[/latex] for 'q' months, then
(A's share of profit) : (B's share of profit) = [latex]x[/latex]p : [latex]x[/latex]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 [latex]t_{1} [/latex] period, second persons invested amount B for [latex]t_{2} [/latex] 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 = [latex]\frac{5600}{20} \times 6[/latex] = Rs 1680
B’s share = [latex]\frac{5600}{20} \times 9[/latex] = Rs 2520
C’s share = [latex]\frac{5600}{20} \times 5[/latex] = 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
= [latex]\frac{5}{5}[/latex] : [latex]\frac{3}{6}[/latex] : [latex]\frac{12}{8}[/latex]
= 1 : [latex]\frac{1}{2}[/latex] : [latex]\frac{3}{2}[/latex] : 2 : 1 : 3

1. Three persons started a business by investing Rs.6,00,000, Rs.8,00,000, Rs.14,00,000 respectively. Calculate the share of each person, out of annual profit of Rs.60,000?
Solution:

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 [latex]\frac{3}{14}[/latex] = 12857.1
Share of second person = 60,000 x [latex]\frac{4}{14}[/latex] = 17142.9
Share of third person = 60,000 x [latex]\frac{7}{14}[/latex] = 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 [latex]\frac{205}{699}[/latex]) = Rs. 20500;
Johnson's share = Rs.( 69900 x [latex]\frac{212}{699}[/latex]) = Rs. 21200;
Tom's share = Rs.( 69900 x [latex]\frac{282}{699}[/latex]) = 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. [latex]x[/latex]
Then, Q's capital = Rs.[latex]\frac{2}{3}x[/latex]
P's share = 2 x [latex]\frac{2}{3}x[/latex] = Rs. [latex]\frac{4}{3}x[/latex]
Therefore, Ratios of their capitals = [latex]\frac{4}{3}x[/latex] : [latex]\frac{2}{3}x[/latex] : [latex]x[/latex] = 4[latex]x[/latex] : 2[latex]x[/latex] : 3[latex]x[/latex]
Sum of the ratios = 4 + 2 + 3 = 9
Then, Q's share = Rs. (7000 x [latex]\frac{2}{9}[/latex]) = 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 [latex]\frac{150000}{50000}[/latex])
⇒ 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 [latex]x[/latex]months
Ratio of their capitals = 5 [latex]x[/latex] : 8 x 5 ⇒ 5 [latex]x[/latex] : 40
Therefore,
Ratio of their capitals = ratio of the profit made by them
⇒ 5 [latex]x[/latex] : 40 = 3 : 6
⇒ 5 [latex]x[/latex] x 6 = 40 x 3
⇒ 30[latex]x[/latex] = 120
⇒ [latex]x[/latex] = [latex]\frac{120}{30}[/latex]
⇒ [latex]x[/latex] = 4 months
Hence, Capital of A is continued for 4 months in the business.