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.

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.

- 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

- 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

- 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

- 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.

- 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.

- 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

- 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

- 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.