Detailed Solution for: Ex. No. 1.2
1. Ajay, Atul and Anil started a business in a partnership by investing Rs. 12,000, Rs. 18,000 and Rs. 30,000 respectively. At the end of the year, they earned a profit of Rs. 15,200 in the business. Find the share of each in the profit.
Solution:
The ratio of their investments is the ratio of their profit shares.
Ratio of investment = $12000 : 18000 : 30000$
Dividing by $6000$, we get the simplified ratio: $2 : 3 : 5$
Total parts = $2 + 3 + 5 = 10$
Ajay's share = $\frac{2}{10} \times 15200 = 3040$
Atul's share = $\frac{3}{10} \times 15200 = 4560$
Anil's share = $\frac{5}{10} \times 15200 = 7600$
Answer: Ajay gets Rs. 3,040, Atul gets Rs. 4,560, and Anil gets Rs. 7,600.
2. Raghu, Madhu and Ramu started a business in a partnership by investing Rs. 60,000, Rs. 40,000 and Rs. 75,000 respectively. At the end of the year they found that they have incurred a loss of Rs. 24,500. Find how much loss each one had to bear.
Solution:
The loss is borne in the ratio of their investments.
Ratio = $60000 : 40000 : 75000$
Dividing by $5000$, we get: $12 : 8 : 15$
Total parts = $12 + 8 + 15 = 35$
Raghu's loss = $\frac{12}{35} \times 24500 = 12 \times 700 = 8400$
Madhu's loss = $\frac{8}{35} \times 24500 = 8 \times 700 = 5600$
Ramu's loss = $\frac{15}{35} \times 24500 = 15 \times 700 = 10500$
Answer: Raghu bears Rs. 8,400, Madhu bears Rs. 5,600, and Ramu bears Rs. 10,500.
3. A, B and C are in the partnership. A's capital was Rs. 65,000 and C's capital was Rs. 50,000. The total profit is Rs. 38,000; out of which B's profit was Rs. 15,000. What was B's capital?
Solution:
Total profit = Rs. $38000$
B's profit = Rs. $15000$
Profit of A and C combined = $38000 - 15000 = 23000$
The ratio of capitals is equal to the ratio of profits.
$\frac{\text{Capital of A and C}}{\text{Capital of B}} = \frac{\text{Profit of A and C}}{\text{Profit of B}}$
Capital of A and C combined = $65000 + 50000 = 115000$
$\frac{115000}{\text{Capital of B}} = \frac{23000}{15000}$
$\frac{115000}{\text{Capital of B}} = \frac{23}{15}$
$\text{Capital of B} = \frac{115000 \times 15}{23} = 5000 \times 15 = 75000$
Answer: B's capital was Rs. 75,000.
4. Paul and Qasim started a business with equal amount of capital. After 8 months Paul withdrew his amount and Raja entered in the business with same amount of capital. At the end of the year they found that they have incurred a loss of Rs. 24,500. Find how much loss each one had to bear.
Solution:
Let the capital invested by each person be $C$.
Paul invested for 8 months. Effective capital = $C \times 8 = 8C$
Qasim invested for 12 months. Effective capital = $C \times 12 = 12C$
Raja invested for the remaining 4 months (12 - 8). Effective capital = $C \times 4 = 4C$
Ratio of their shares = $8C : 12C : 4C = 2 : 3 : 1$
Total parts = $2 + 3 + 1 = 6$
Paul's loss = $\frac{2}{6} \times 24500 = \text{Rs. } 8166.67$
Qasim's loss = $\frac{3}{6} \times 24500 = \text{Rs. } 12250$
Raja's loss = $\frac{1}{6} \times 24500 = \text{Rs. } 4083.33$
Answer: The losses are Rs. 8,166.67 (Paul), Rs. 12,250 (Qasim), and Rs. 4,083.33 (Raja).
5. Amit and Rohit started a business by investing Rs. 20,000 each. After 3 months Amit withdrew Rs. 5000 and Rohit put in the same amount additionally. How should a profit of Rs. 12,800 be divided between them at the end of the year?
Solution:
Amit's equivalent capital for 1 year:
$(20000 \times 3) + (15000 \times 9) = 60000 + 135000 = 195000$Rohit's equivalent capital for 1 year:
$(20000 \times 3) + (25000 \times 9) = 60000 + 225000 = 285000$Ratio of profit = $195000 : 285000 = 195 : 285$
Dividing by $15$, we get: $13 : 19$
Total parts = $13 + 19 = 32$
Amit's share = $\frac{13}{32} \times 12800 = 13 \times 400 = 5200$
Rohit's share = $\frac{19}{32} \times 12800 = 19 \times 400 = 7600$
Answer: Amit gets Rs. 5,200 and Rohit gets Rs. 7,600.
6. John and Mathew started a business with their capitals in the ratio 8:5. After 8 months John added 25% of his earlier capital as further investments. At the same time Mathew withdrew 20% of his earlier capital. At the end of the year they earned Rs. 52,000 as profit. How should they divide it between them?
Solution:
Let their initial capitals be $8x$ and $5x$.
John's equivalent capital for the year:
$(8x \times 8) + (8x + 0.25 \times 8x) \times 4$
$= 64x + (10x \times 4) = 64x + 40x = 104x$Mathew's equivalent capital for the year:
$(5x \times 8) + (5x - 0.20 \times 5x) \times 4$
$= 40x + (4x \times 4) = 40x + 16x = 56x$Ratio of profit = $104x : 56x = 104 : 56$
Dividing by $8$, we get: $13 : 7$
Total parts = $13 + 7 = 20$
John's share = $\frac{13}{20} \times 52000 = 13 \times 2600 = 33800$
Mathew's share = $\frac{7}{20} \times 52000 = 7 \times 2600 = 18200$
Answer: John gets Rs. 33,800 and Mathew gets Rs. 18,200.
7. Ramesh, Vivek and Sunil started a business by investing the capitals in the ratio 4:5:6. After 3 months Vivek removed all his capital and after 6 months Sunil removed all his capital from the business. At the end of the year Ramesh received Rs. 6400 as profit. Find the profit earned by Vivek and Sunil.
Solution:
Let their initial capitals be $4x, 5x,$ and $6x$.
Ramesh invested for 12 months. Equivalent capital = $4x \times 12 = 48x$
Vivek invested for 3 months. Equivalent capital = $5x \times 3 = 15x$
Sunil invested for 6 months. Equivalent capital = $6x \times 6 = 36x$
Ratio of profit = $48x : 15x : 36x = 48 : 15 : 36$
Dividing by $3$, we get: $16 : 5 : 12$
Ramesh's share is given as Rs. $6400$. Since Ramesh has $16$ parts:
$16 \text{ parts} = 6400 \implies 1 \text{ part} = 400$Vivek's profit ($5$ parts) = $5 \times 400 = 2000$
Sunil's profit ($12$ parts) = $12 \times 400 = 4800$
Answer: Vivek earned Rs. 2,000 and Sunil earned Rs. 4,800.
8. Mr. Natarajan and Mr. Gopalan are partners in a company having capitals in the ratio 4:5 and profits received by them are in the ratio 5:4. If Gopalan invested capital in the company for 16 months, how long was Natrajan’s investment in the company?
Solution:
Let Natarajan's investment time be $T$ months.
Ratio of equivalent capitals = $(\text{Natarajan's Capital} \times T) : (\text{Gopalan's Capital} \times 16)$
Substitute the given capital ratio ($4:5$):
Ratio = $(4 \times T) : (5 \times 16) = 4T : 80$The ratio of equivalent capitals must equal the profit ratio ($5:4$):
$\frac{4T}{80} = \frac{5}{4}$$\frac{T}{20} = \frac{5}{4}$
$T = \frac{20 \times 5}{4} = 25$
Answer: Natarajan’s investment was in the company for 25 months.
9. Anita and Nameeta are partners in the business for some years. Their capitals are Rs. 3,00,000 and Rs. 2,00,000 respectively. Yogeeta wants to join the business with the capital of Rs. 4,00,000. They agree that the goodwill will be considered as two times the average of last three years profit. The profit of last three years are Rs. 60,000, Rs. 70,000 and Rs. 50,000 respectively. What are the amounts to be paid by Yogeeta and Anita and Nameeta as goodwill?
Solution:
Average profit for the last 3 years = $\frac{60000 + 70000 + 50000}{3} = \frac{180000}{3} = \text{Rs. } 60,000$
Total Goodwill of the firm = $2 \times \text{Average Profit} = 2 \times 60000 = \text{Rs. } 1,20,000$
Old ratio of Anita and Nameeta (based on capital) = $3,00,000 : 2,00,000 = 3 : 2$
New capital structure is $3,00,000 : 2,00,000 : 4,00,000 = 3 : 2 : 4$
Yogeeta's share in the firm = $\frac{4}{3 + 2 + 4} = \frac{4}{9}$
Yogeeta's share of the goodwill to be brought in = $\frac{4}{9} \times 1,20,000 = \text{Rs. } 53,333.33$
This goodwill is distributed to Anita and Nameeta in their sacrificing ratio, which is their old ratio $3 : 2$.
Anita's share of goodwill received = $\frac{3}{5} \times 53333.33 = \text{Rs. } 32,000$
Nameeta's share of goodwill received = $\frac{2}{5} \times 53333.33 = \text{Rs. } 21,333.33$
Answer: Yogeeta pays Rs. 53,333.33. Out of this, Anita receives Rs. 32,000 and Nameeta receives Rs. 21,333.33.
10. A, B and C are three partners with their capitals in the ratio 4:3:3. They decide to dissolve the partnership. The assets of the company are sold for Rs. 4,00,000 and liabilities (other than capital) of Rs. 60,000. They incur realisation expenses of Rs. 4,000. What is the amount that each partner gets as final settlement after dissolution?
Solution:
Total cash realized from sale of assets = Rs. $4,00,000$
Less: Payment of external liabilities = Rs. $60,000$
Less: Payment of realization expenses = Rs. $4,000$
Net cash available for distribution = $4,00,000 - 60,000 - 4,000 = \text{Rs. } 3,36,000$
Since no other information is provided, this remaining cash incorporates both their original capital and any realization profit/loss. It is distributed among the partners in their profit-sharing ratio, which is equal to their capital ratio $4 : 3 : 3$.
Total parts = $4 + 3 + 3 = 10$
A's final settlement = $\frac{4}{10} \times 336000 = \text{Rs. } 1,34,400$
B's final settlement = $\frac{3}{10} \times 336000 = \text{Rs. } 1,00,800$
C's final settlement = $\frac{3}{10} \times 336000 = \text{Rs. } 1,00,800$
Answer: A gets Rs. 1,34,400, B gets Rs. 1,00,800, and C gets Rs. 1,00,800.