CAT 2025 Lesson : Ratio & Partnership - Partnership

W, X, Y and Z invested Rs. $1,20,000$, Rs. $1,00,000$, Rs. $90,000$ and Rs. $70,000$ for $4, 6, 8$ and $10$ months respectively. If the profits at the end of the year was Rs. $75,000$, then the difference between the profits (in Rupees) that X and Y received was

Solution

Although not explicitly stated, profits are implied to be distributed in the ratio of the investments.

Substituting the investment and duration in the formula given above
$\text{P}_{\text{W}} : \text{P}_{\text{X}} : \text{P}_{\text{Y}} : \text{P}_{\text{Z}}$ $= 120000 \times 4 : 100000 \times 6 : 90000 \times 8 : 70000 \times 10$

$= 24 : 30 : 36 : 35$

X's share in the profit $= \dfrac{30}{24 + 30 + 36 + 35} = \dfrac{30}{125}$

Similarly, Y's share in profit $= \dfrac{36}{125}$

Difference between X's share and Y's share $= \left(\dfrac{36}{125} - \dfrac{30}{125} \right) \times 75000$

$= \dfrac{6}{125} \times 75000 =$ Rs. $3600$

Answer: Rs. $3600$

Larry and Sergey brought Rs. $25,000$ and Rs. $40,000$ as capital and started a company at the beginning of a year. $6$ months after the business commenced, the company was short of funds and Larry brought in additional capital of Rs. $30,000$. Two months later, when the company needed further capital, Sergey brought in Rs. $20,000$. It was decided that the profits would be distributed in the ratio of the average capital invested during this period. If Larry received Rs. $24,000$ of profit at the end of the year, then how much profit did Sergey receive?

Solution

As the capital amounts are in the multiples of $1000$ and we are concerned about ratios, we can remove 3 zeroes in each of the amounts.

Larry brought in capitals of Rs. $25$ and Rs. $30$ which stayed in the business for $12$ months and $6$ months respectively.

Sergey brought in capitals of Rs. $40$ and Rs. $20$ which stayed in the business for $12$ months and $4$ months respectively.

Ratio of Profits = Ratio of Capital Invested $= 25 \times 12 + 30 \times 6 : 40 \times 12 + 20 \times 4$
= 480 : 560

= 6 : 7

Let the total profit be $x$.

Larry's share in profit $= \dfrac{6}{(6 + 7)} \times x = 24000$ ⇒ $x = 52000$

Sergey's share in profit $= 52000 - 24000 = 28000$

Answer: Rs. $28,000$

If the ratio of amount invested by A, B and C is $1 : 2 : 3$ and the profits were split in the ratio of $3 : 4 : 5$, then what was the ratio of time period of the investments?

Solution

$\text{I}_{\text{A}}\text{D}_{\text{A}} : \text{I}_{\text{B}}\text{D}_{\text{B}} : \text{I}_{\text{C}}\text{D}_{\text{C}} = \text{P}_{\text{A}} : \text{P}_{\text{B}} : \text{P}_{\text{C}}$

⇒ $1 \times \text{D}_{\text{A}} : 2 \times \text{D}_{\text{B}} : 3 \times \text{D}_{\text{C}} = 3 : 4 : 5$

⇒ $\text{D}_{\text{A}} : \text{D}_{\text{B}} : \text{D}_{\text{C}} = \dfrac{3}{1} : \dfrac{4}{2} : \dfrac{5}{3} = 9 : 6 : 5$

Answer: $9 : 6 : 5$

A, B and C started a business at the beginning of a financial year with capitals of Rs. $20,000,$ Rs. $30,000$ and Rs. $50,000$ respectively. If B and C withdrew their investments and exited the business sometime during the year and the ratio in which profits were divided was $4 : 2 : 5$, then after how many months did B and C quit?

(1) $4$ and $6$            (2) $8$ and $6$            (3) $6$ and $4$            (4) $6$ and $8$           

Solution

Ratio of Capitals invested $= 20000 : 30000 : 50000 = 2 : 3 : 5$

Let $b$ and $c$ be the number of months for which the investment of B and C remained in the business. A's investment stayed in the business for the entire duration of $12$ months.

Ratio of time periods $= 12 : b : c$

Ratio of Capitals Invested $=$ Ratio of Profits

⇒ $2 \times 12 : 3 \times b : 5 \times c = 4 : 2 : 5$

Scaling up the first term in the RHS to $24$, we get

⇒ $24 : 3b : 5c = 24 : 12 : 30$

∴ $3b = 12$ and $5c = 30$

⇒ $b = 4$ and $c = 6$

Answer: $(1) 4$ and $6$

Viji started a business at the beginning of a year with a capital of Rs. $1,04,000$. Three months later, Raji joined the business and brought in a capital of Rs. $91,000$. The monthly salaries that Raji and Viji were to draw was Rs. $3,000$ and Rs. $2,000$ respectively. These salaries were, however, to be paid as a lump sum at the end of the year. If the company's total profit (before salary payments to partners) was Rs. $2,63,000$, then the amount due to Raji (Salary + Profits) is

Solution

Viji worked for $12$ months and earned a salary of $2000 \times 12 =$ Rs. $24,000.$

Raji worked for $9$ months and earned a salary of $3000 \times 9 =$ Rs. $27,000.$

Profit after deducting partners' salaries $= 2,63,000 - 24,000 - 27,000 =$ Rs. $2,12,000$

Profits are distributed to the partners only after paying their salaries.

Viji and Raji invested Rs. $1,04,000$ and Rs. $91,000$ for $12$ and $9$ months respectively.

Ratio of Profits $= 104 \times 12 : 91 \times 9 = 32 : 21$

Raji's share in profits $= \dfrac{21}{53} \times 212000 = 21 \times 4000 =$ Rs. $84,000$

Total amount due to Raji $=$ Profit + Salary $= 84,000 + 27,000 =$ Rs.$1,11,000$

Answer: Rs. $1,11,000$