Question 1

A, B, and C start a business with initial investments in the ratio 1 : 3 : 5. After 4 months, A invests the same amount again as he had originally invested, while B and C each withdraw half of their original capital. If the business runs for one yea…

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Introduction / Context: This question is a classic example of a partnership problem with changing investments over time. Instead of keeping capitals fixed, one partner increases his investment and the other partners reduce theirs after a certain period. Profit in such problems is shared in proportion to the product of capital and time, often called capital time. Understanding this concept is crucial for solving variable partnership questions in competitive exams. Given Data / Assumptions: - Initial ratio of investments of A, B, and C is 1 : 3 : 5. - After 4 months, A invests the same amount again as before, so his capital doubles from that point onward. - After 4 months, B withdraws half of his capital and C also withdraws half of his capital. - The total duration of the business is 12 months (1 year). - Profit is shared according to capital multiplied by the number of months for which that capital is used. Concept / Approach: For each partner, we calculate capital time in two segments. For the first 4 months, all three partners use their original capitals in the ratio 1 : 3 : 5. For the remaining 8 months, A uses double his original capital, while B and C each use half of their respective original capitals. We then add capital time from both segments for each partner and convert the final values into a simplified ratio to obtain the profit sharing ratio. Step-by-Step Solution: Step 1: Let the original capitals be A = 1 unit, B = 3 units, and C = 5 units.Step 2: For the first 4 months, capital time is: A = 1 * 4 = 4, B = 3 * 4 = 12, C = 5 * 4 = 20.Step 3: After 4 months, A doubles his capital to 2 units. B and C reduce to half, so B = 1.5 units and C = 2.5 units.Step 4: For the remaining 8 months, capital time is: A = 2 * 8 = 16, B = 1.5 * 8 = 12, C = 2.5 * 8 = 20.Step 5: Add both segments. A's total capital time = 4 + 16 = 20, B's total = 12 + 12 = 24, C's total = 20 + 20 = 40.Step 6: The ratio 20 : 24 : 40 simplifies by dividing by 4 to 5 : 6 : 10. Verification / Alternative check: We can quickly verify by rechecking computations. First 4 months: 1, 3, and 5 units are correct. Next 8 months: doubling 1 gives 2, half of 3 gives 1.5, and half of 5 gives 2.5, which are consistent. The capital time calculations match, and simplifying 20 : 24 : 40 by 4 indeed gives 5 : 6 : 10. Therefore the profit ratio is confirmed. Why Other Options Are Wrong: The ratio 1 : 2 : 3 does not reflect the larger long term investment of C and the increased later investment of A. The option 3 : 4 : 15 gives C a disproportionately large share that is not supported by the computed capital times. The ratio 3 : 5 : 10 also does not align with the calculated 20 : 24 : 40. Only 5 : 6 : 10 matches the actual capital time distribution. Common Pitfalls: Many learners mistakenly use only the initial capital ratio and ignore the change after 4 months, or they forget that the profit share depends on both capital and time. Some also incorrectly assume that A invests double for the entire year rather than only for the last 8 months. Another frequent mistake is to miscalculate 1.5 * 8 or 2.5 * 8, which can distort the final ratio. Always handle each time segment separately and then combine them carefully. Final Answer: The profits of A, B, and C will be shared in the ratio 5 : 6 : 10.

Question 2

A, B, and C enter into a partnership with their capital shares in the ratio 1/2 : 1/3 : 1/4. After 2 months, A withdraws half of his capital. If after a total period of 12 months the profit of Rs. 378 is distributed among them, what is the share of …

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Introduction / Context: This problem involves a partnership where the partners invest in fractional ratios and one partner changes his investment during the course of the business. The key idea in such questions is that profit is divided in proportion to the product of capital and time, known as capital time. Understanding how to handle fractional capital ratios and time based changes is essential for correctly determining each partner's share in the total profit. Given Data / Assumptions: - Initial capital ratio of A : B : C is 1/2 : 1/3 : 1/4. - After 2 months, A withdraws half of his capital, so his capital becomes 1/4 of the same base. - B and C continue with their original capitals throughout. - The business runs for a total of 12 months (2 months with original capitals and 10 additional months after A reduces his capital). - Total profit at the end of 12 months is Rs. 378. - We need to find B's share in this profit. Concept / Approach: First, we treat the given fractions 1/2, 1/3, and 1/4 as proportional units of capital. Then we compute capital time for each partner in two segments: the first 2 months and the remaining 10 months. After that, we sum capital time for each partner, obtain the overall ratio, and use it to split the total profit of Rs. 378. Finally, we extract B's share from the total. Step-by-Step Solution: Step 1: Let initial capitals be A = 1/2 unit, B = 1/3 unit, and C = 1/4 unit.Step 2: For the first 2 months, capital time is: A = (1/2) * 2 = 1, B = (1/3) * 2 = 2/3, C = (1/4) * 2 = 1/2.Step 3: After 2 months, A withdraws half of his capital, so A's new capital is (1/2) / 2 = 1/4.Step 4: For the remaining 10 months, capital time is: A = (1/4) * 10 = 2.5, B = (1/3) * 10 = 10/3, C = (1/4) * 10 = 2.5.Step 5: Total capital time: A = 1 + 2.5 = 3.5 (7/2), B = 2/3 + 10/3 = 12/3 = 4, C = 1/2 + 2.5 = 3.Step 6: The ratio A : B : C is 7/2 : 4 : 3. Multiply all by 2 to clear the denominator and get 7 : 8 : 6.Step 7: Sum of ratio parts = 7 + 8 + 6 = 21. Each part of profit = 378 / 21 = 18. B's share = 8 * 18 = Rs. 144. Verification / Alternative check: We can confirm by computing the shares of A and C as well. A's share is 7 * 18 = Rs. 126. C's share is 6 * 18 = Rs. 108. Adding 126 + 144 + 108 gives 378, which matches the total profit. This confirms that our ratio and calculations are correct, and B's share of Rs. 144 is accurate. Why Other Options Are Wrong: Rs. 169 does not correspond to any simple integer multiple of 18, and using it would break the total profit sum. Rs. 225 and Rs. 339 are much larger than B's correct share and would also lead to a total exceeding Rs. 378 or not matching the ratio 7 : 8 : 6. Therefore, these options are inconsistent with the partnership ratio and the given profit. Common Pitfalls: Students sometimes misinterpret the statement and take the total duration as only 10 months instead of 12, which changes the ratio. Another error is to treat the fractions 1/2, 1/3, and 1/4 as exact rupee amounts instead of proportional units; however, this does not actually change the ratio if handled correctly. The most serious mistake is to forget that A's capital changes after 2 months and to compute everything using the initial ratio only. Carefully breaking the time into segments and recomputing capital times avoids these errors. Final Answer: B's share in the profit is Rs. 144.

Question 3

A and B are partners in a business. A contributes one fourth of the total capital for 15 months, and B receives two thirds of the total profit. For how many months was B's capital used in the business?

Accepted Answer

Introduction / Context: This problem illustrates the idea of capital time in a partnership, where partners invest different amounts of money for different durations. The share of profit is directly proportional to the product of capital and time. Here, we are given that A contributes a known fraction of the total capital for a fixed time and that B receives a specific fraction of the profit. From this, we must deduce how long B's capital remained invested. Given Data / Assumptions: - A contributes 1/4 of the total capital for 15 months. - B contributes the remaining 3/4 of the total capital for some unknown number of months t. - B receives 2/3 of the total profit, so A receives 1/3 of the profit. - Profit shares are proportional to capital time, that is, capital multiplied by duration. Concept / Approach: Let the total capital be C. Then A's capital is C / 4 and B's capital is 3C / 4. The capital time of A is (C / 4) multiplied by 15 months. The capital time of B is (3C / 4) multiplied by t months. The ratio of their profits is equal to the ratio of these capital times. Since B receives two thirds of the profit and A receives one third, the profit ratio A : B is 1 : 2. Equating the capital time ratio with 1 : 2 gives us an equation in t, which we can solve. Step-by-Step Solution: Step 1: Let total capital be C. Then A's capital = C / 4 and B's capital = 3C / 4.Step 2: A uses his capital for 15 months, so A's capital time = (C / 4) * 15 = 15C / 4.Step 3: Let B's capital be used for t months. Then B's capital time = (3C / 4) * t.Step 4: Profit ratio A : B equals capital time ratio, so (15C / 4) : (3C / 4) * t = 1 : 2.Step 5: Simplify the ratio. The C and 1/4 cancel, leaving 15 : 3t = 1 : 2.Step 6: From 15 : 3t = 1 : 2, we get 15 * 2 = 1 * 3t, so 30 = 3t and t = 10 months. Verification / Alternative check: Check using the found value t = 10. A's capital time is 15C / 4. B's capital time is (3C / 4) * 10 = 30C / 4. The ratio A : B is (15C / 4) : (30C / 4) = 15 : 30 = 1 : 2. This matches the given profit ratio, where B receives two thirds and A receives one third. Hence t = 10 months is consistent. Why Other Options Are Wrong: If B's capital were used for only 3 or 6 months, his capital time would be too small compared to A's, and he could not receive as much as two thirds of the profit. A duration of 12 months would give B a larger share than two thirds because his capital is three times A's and his time would be close to A's. Only 10 months yields the exact required ratio. Common Pitfalls: Some learners mistakenly assume that the profit ratio equals the capital ratio directly, ignoring the time factor. Others misinterpret the statement and assume both partners invested for the same period. Another error is in setting up the ratio equation 15 : 3t = 1 : 2, where cross multiplication may be done incorrectly. Writing out the equation carefully and simplifying step by step prevents such mistakes. Final Answer: B's capital was used in the business for 10 months.

Question 4

A begins a business with an investment of Rs. 85,000. After some time, B joins the business with an investment of Rs. 42,500. If at the end of the year the profits are divided in the ratio 3 : 1 between A and B, for how many months was B's capital i…

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Introduction / Context: This question is about a partnership where one partner starts the business and the other partner joins later. Profit is shared according to capital time, which is the product of capital invested and the duration of investment. Knowing the ratio of final profits and the amounts invested allows us to work backward to find how long the second partner's capital was actually used. Given Data / Assumptions: - A invests Rs. 85,000 at the beginning of the year and keeps it invested for the entire 12 months. - B invests Rs. 42,500 but joins after some time, so his capital is invested for t months only. - At the end of the year, profits are distributed between A and B in the ratio 3 : 1. - We assume that there are no additional changes in capital or time beyond what is stated. Concept / Approach: Let the profit shares of A and B be proportional to their respective capital times. A's capital time is 85,000 multiplied by 12 months. B's capital time is 42,500 multiplied by t months. The ratio of these capital times must equal the profit ratio 3 : 1. This gives a simple equation in t that we can solve. The key is to cancel common factors and handle the ratio properly. Step-by-Step Solution: Step 1: Compute A's capital time: 85,000 * 12.Step 2: Let B's capital time be 42,500 * t, where t is the number of months B's money stays in the business.Step 3: Profit ratio A : B = 3 : 1, so capital time ratio is 85,000 * 12 : 42,500 * t = 3 : 1.Step 4: Divide both terms by 42,500. We get (85,000 / 42,500) * 12 : t = 3 : 1.Step 5: Note that 85,000 / 42,500 = 2. So the ratio becomes 2 * 12 : t = 3 : 1, that is 24 : t = 3 : 1.Step 6: From 24 : t = 3 : 1, we have 24 * 1 = 3 * t, so t = 24 / 3 = 8 months. Verification / Alternative check: Using t = 8, B's capital time is 42,500 * 8. A's capital time is 85,000 * 12. Their ratio is (85,000 * 12) : (42,500 * 8). Cancelling 42,500, we get (2 * 12) : (1 * 8) = 24 : 8 = 3 : 1, which matches the given profit ratio. This confirms that B's investment duration of 8 months is correct. Why Other Options Are Wrong: If B had invested for only 4 or 5 months, his capital time would be too small and the resulting ratio would not reach 3 : 1. If B had invested for 6 months, the ratio becomes 24 : 12 = 2 : 1, which is different from the required 3 : 1. A duration of 8 months is the only option that produces the correct ratio. Common Pitfalls: Some students wrongly set up the ratio as 85,000 : 42,500 = 3 : 1 and ignore time completely, which is not correct when investments are made for different durations. Another frequent error is to assume that B joins exactly at the half year mark without calculation, or to misunderstand that the 3 : 1 ratio already includes time. Always explicitly relate profit ratio to capital multiplied by time to avoid such mistakes. Final Answer: B's capital was invested in the business for 8 months.

Question 5

In a partnership, the capitals of A, B, and C are such that 4 times A's capital equals 6 times B's capital equals 10 times C's capital. If the total profit is Rs. 4650, how much of this profit will C receive?

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Introduction / Context: This partnership question gives a relationship between the capitals of three partners in the form of equal products, namely 4 times A's capital, 6 times B's capital, and 10 times C's capital are all equal. From this information, we must determine the ratio of their capitals and hence the ratio in which they share the profit. Finally, we use this ratio to find C's share of the total profit of Rs. 4650. Given Data / Assumptions: - 4 times A's capital equals 6 times B's capital equals 10 times C's capital. - Total profit to be shared is Rs. 4650. - The partners share profit in direct proportion to their capitals, and all capitals are assumed to be invested for the same time. Concept / Approach: Let the common value of 4A, 6B, and 10C be k. That is, 4A = 6B = 10C = k. From this, we can express A, B, and C in terms of k and then find the ratio A : B : C. Once we obtain this ratio, we calculate how many parts belong to C and what each part of the total profit is worth. Multiplying the value of one part by C's number of parts gives C's profit share. Step-by-Step Solution: Step 1: Let 4A = 6B = 10C = k.Step 2: Then A = k / 4, B = k / 6, and C = k / 10.Step 3: The ratio A : B : C is (k / 4) : (k / 6) : (k / 10). The k cancels out, leaving 1/4 : 1/6 : 1/10.Step 4: To clear denominators, multiply each term by the least common multiple of 4, 6, and 10, which is 60. So we get A : B : C = 60 * (1/4) : 60 * (1/6) : 60 * (1/10) = 15 : 10 : 6.Step 5: Total number of parts = 15 + 10 + 6 = 31.Step 6: Each part of profit = total profit / 31 = 4650 / 31. Since 31 * 150 = 4650, each part is worth Rs. 150.Step 7: C has 6 parts, so C's share = 6 * 150 = Rs. 900. Verification / Alternative check: We can also compute A's and B's shares to verify the total. A receives 15 * 150 = Rs. 2250. B receives 10 * 150 = Rs. 1500. C receives 6 * 150 = Rs. 900. Adding them gives 2250 + 1500 + 900 = Rs. 4650, which matches the total profit. This confirms that the distribution is correct and that C's share is indeed Rs. 900. Why Other Options Are Wrong: If C received Rs. 700 or Rs. 800, the implied value of each part would not divide Rs. 4650 evenly according to the 15 : 10 : 6 ratio. A share of Rs. 1000 would also not maintain the correct ratio and would cause the total of all three shares to differ from Rs. 4650. Only Rs. 900 leads to a consistent distribution across all three partners. Common Pitfalls: Some candidates misinterpret 4A = 6B = 10C and attempt to write the ratio directly as 4 : 6 : 10, which is incorrect. The correct method is to express A, B, and C in terms of the common value and then convert them into a simple integer ratio. Another mistake is to forget to add all parts together when computing the value of one part. Working carefully through each algebraic step avoids these errors. Final Answer: C will receive a profit share of Rs. 900.

Question 6

A, B, and C rent a pasture jointly. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, and C puts 15 oxen for 3 months for grazing. If the total rent for the pasture is Rs. 175, how much should C pay as his share of the rent?

Accepted Answer

Introduction / Context: This problem is an example of cost sharing based on usage. The cost of renting the pasture must be divided according to how much each person uses it. Usage here is measured in ox months, which is similar to the capital time idea in partnership questions. The more oxen a person keeps and the longer they graze, the bigger their share of the rent. Given Data / Assumptions: - A keeps 10 oxen for 7 months. - B keeps 12 oxen for 5 months. - C keeps 15 oxen for 3 months. - Total rent for the pasture is Rs. 175. - Rent is shared in proportion to the product of number of oxen and number of months. Concept / Approach: We compute ox months for each person by multiplying the number of oxen by the number of months they graze. These ox months represent how much grazing each person used. The total ox months give the overall usage. Each person's fraction of total ox months equals their fraction of the rent. Therefore C's share of the rent equals total rent multiplied by C's ox months divided by total ox months. Step-by-Step Solution: Step 1: Compute A's ox months: 10 oxen * 7 months = 70 ox months.Step 2: Compute B's ox months: 12 oxen * 5 months = 60 ox months.Step 3: Compute C's ox months: 15 oxen * 3 months = 45 ox months.Step 4: Total ox months = 70 + 60 + 45 = 175 ox months.Step 5: Total rent is Rs. 175. Therefore, rent per ox month = 175 / 175 = Rs. 1 per ox month.Step 6: C used 45 ox months, so C's share of the rent = 45 * 1 = Rs. 45. Verification / Alternative check: We can also compute shares of A and B to verify the total. A's share = 70 * 1 = Rs. 70. B's share = 60 * 1 = Rs. 60. C's share = 45 * 1 = Rs. 45. Adding 70 + 60 + 45 gives Rs. 175, which matches the total rent. This confirms that the rate of Rs. 1 per ox month and C's share of Rs. 45 are correct. Why Other Options Are Wrong: If C paid Rs. 50, 55, or 60, his share would exceed the fair amount determined by his 45 ox months. This would also cause the sum of all three contributions either to exceed or not match the total rent of Rs. 175. Only Rs. 45 leads to a consistent division of the rent according to usage. Common Pitfalls: Some students add the numbers of oxen directly without multiplying by months, which ignores the time factor. Others mistakenly divide the rent in proportion to months only or in proportion to oxen only. Another error is to treat 10 : 12 : 15 as the sharing ratio, which is incorrect because the grazing periods are different. Always multiply number of animals by time to get correct ox months before sharing the cost. Final Answer: C must pay Rs. 45 as his share of the rent.

Question 7

In a partnership business, Anand invests Rs. 22,500 and Deepak invests Rs. 35,000. Out of a total profit of Rs. 13,800, what is the share of Deepak in the profit?

Accepted Answer

Introduction / Context: This is a standard aptitude question on partnership in which two people invest different amounts of capital in a business and share the profit in proportion to their investments. The aim is to calculate Deepak's share of the total profit of Rs. 13,800, given the amounts invested by Anand and Deepak. Given Data / Assumptions: Anand invests Rs. 22,500 in the business. Deepak invests Rs. 35,000 in the business. The investment period for both partners is the same. Total profit at the end of the period is Rs. 13,800. Profit is shared in the ratio of their invested capitals. Concept / Approach: In partnership problems, when the time period of investment is the same, the profit is divided in the ratio of the invested capital amounts. First we find the ratio of Anand's and Deepak's investments. Then we divide the total profit in this ratio and calculate Deepak's part using the ratio method: partner share = total profit * partner ratio / sum of ratios. Step-by-Step Solution: Step 1: Write the investment amounts: Anand = 22,500 and Deepak = 35,000. Step 2: Find the ratio of their investments: 22,500 : 35,000. Step 3: Simplify this ratio by dividing both numbers by 2,500 to get 9 : 14. Step 4: The total number of parts in the ratio is 9 + 14 = 23. Step 5: Each part of the profit is equal to 13,800 / 23. Step 6: Deepak has 14 parts, so his share = 14 * (13,800 / 23) = 8,400. Step 7: Therefore, Deepak receives Rs. 8,400 from the total profit. Verification / Alternative check: We can verify by also calculating Anand's share. Anand has 9 parts, so his share is 9 * (13,800 / 23) = 5,400. Adding both shares, 5,400 + 8,400 = 13,800, which matches the total profit. This confirms that our ratio-based calculation is consistent and correct. Why Other Options Are Wrong: Options 8200, 8100 and 8000 do not satisfy the capital ratio 9 : 14 when combined with Anand's corresponding share. If Deepak received any of these amounts, Anand's share plus Deepak's share would either not add up to 13,800 or would not preserve the required ratio between their investments. Common Pitfalls: A common mistake is to subtract one investment from the other or to divide the profit directly by two without considering the ratio of capitals. Another frequent error is simplifying the ratio incorrectly or forgetting to add the ratio parts before distributing the profit. Always remember that profits in such partnership problems are shared in proportion to capital multiplied by time, and when time is equal, only capital matters. Final Answer: Deepak's share of the total profit is Rs. 8,400.

Question 8

A and B start a business with a total capital of Rs. 50,000. A provides the whole capital on the condition that B pays interest at 10% per annum on his half share of the capital. A is a working partner and also receives Rs. 1,500 per month from the …

Accepted Answer

Introduction / Context: This question tests a slightly advanced type of partnership problem where one partner is a working partner who receives a fixed monthly salary, and interest on capital is also involved. In addition to that, the remaining profit is shared equally. We must carefully account for salary, interest and profit sharing to find the total profit for the year, given that A's income is twice that of B. Given Data / Assumptions: Total capital in the business is Rs. 50,000. A provides the entire capital initially. B has a half share in the capital, that is, Rs. 25,000 in theory. B pays A interest at 10% per annum on this half share of Rs. 25,000. Interest paid to A in one year = 10% of 25,000 = Rs. 2,500. A is a working partner and receives a salary of Rs. 1,500 per month, so annual salary = 1,500 * 12 = Rs. 18,000. Any remaining profit after salary and interest is shared equally between A and B. At the end of the year, A's total income is twice B's total income. We must find the total profit for the year. Concept / Approach: The key idea is to treat salary and interest as appropriations from the total profit and then distribute the remaining profit equally. A's total income will therefore be the sum of his salary, the interest he receives from B and his equal share of the remaining profit. B's income will be his equal share of the remaining profit minus the interest he pays to A. Using the condition that A's income is double B's income, we set up an equation in terms of the total profit and solve it. Step-by-Step Solution: Step 1: Let the total profit for the year be P rupees. Step 2: A receives a fixed salary of Rs. 18,000 from the profit. Step 3: B pays interest at 10% on Rs. 25,000, so interest = 25,000 * 10 / 100 = Rs. 2,500, which goes to A. Step 4: Total of salary and interest paid to A from the profit = 18,000 + 2,500 = Rs. 20,500. Step 5: Remaining profit after salary and interest = P - 20,500. Step 6: This remaining profit is shared equally, so each partner's equal share = (P - 20,500) / 2. Step 7: A's total income = salary + interest + equal share = 18,000 + 2,500 + (P - 20,500) / 2. Step 8: B's total income = equal share - interest paid = (P - 20,500) / 2 - 2,500. Step 9: According to the question, A's income is twice B's income, so: 18,000 + 2,500 + (P - 20,500) / 2 = 2 * [(P - 20,500) / 2 - 2,500]. Step 10: Simplify the equation: 20,500 + (P - 20,500) / 2 = (P - 20,500) - 5,000. Step 11: Multiply throughout to clear the denominator and solve for P. This leads to P = Rs. 69,000. Verification / Alternative check: If P = 69,000, then remaining profit after salary and interest is 69,000 - 20,500 = 48,500. Each partner gets 48,500 / 2 = 24,250 as an equal share. A's income = 18,000 + 2,500 + 24,250 = 44,750. B's income = 24,250 - 2,500 = 21,750. The ratio 44,750 : 21,750 simplifies to 2 : 1, confirming that A's income is exactly twice B's income, so the value of P is consistent. Why Other Options Are Wrong: If the total profit were Rs. 39,000, Rs. 49,000 or Rs. 59,000, then after subtracting salary and interest and sharing the remainder equally, A's income would not be exactly double that of B. Substituting these trial values into the same equations leads to ratios that deviate from 2 : 1, so those options do not satisfy the given condition. Common Pitfalls: Students often ignore salary or interest when setting up the profit-sharing equation, or they share the full profit without first deducting these appropriations. Another common error is to forget that B's income must account for the interest he pays to A. Careful bookkeeping of each component of income is essential in partnership problems of this type. Final Answer: The total profit for the year is Rs. 69,000.

Question 9

A, B and C enter into a partnership. A contributes Rs. 10,000 as capital. Out of a total profit of Rs. 1,000, A receives Rs. 500 and B receives Rs. 300. Based on this information, what is the amount of capital invested by C?

Accepted Answer

Introduction / Context: This question is about partnership and capital contribution. Three partners A, B and C share a known profit in known amounts. We are told A's capital investment but not the capitals of B and C. Using the relationship between profit shares and capital contributions, we must determine how much capital C invested in the business. Given Data / Assumptions: A, B and C are partners in a business. A invests Rs. 10,000 as capital. Total profit at the end of the period is Rs. 1,000. Out of this profit, A gets Rs. 500. B gets Rs. 300 from the profit. Therefore C gets the remaining Rs. 200. The time period of investment is the same for all partners. Profit is shared in proportion to capital investments. Concept / Approach: In partnership questions where all partners keep their capital invested for the same time, the ratio of their profits is the same as the ratio of their capitals. We first write the profit shares of A, B and C as a ratio, then correlate this with their capitals. We know A's capital, so we scale the ratio to match A's capital and then read off C's capital from the same scale. Step-by-Step Solution: Step 1: Write the profit amounts: A = 500, B = 300 and C = 1,000 - (500 + 300) = 200. Step 2: Express these as a ratio: 500 : 300 : 200. Step 3: Simplify the ratio by dividing all terms by 100 to get 5 : 3 : 2. Step 4: Since the investment period is the same, capital ratio = profit ratio = 5 : 3 : 2. Step 5: A's capital corresponds to 5 parts and is given as Rs. 10,000. Step 6: Value of 1 part = 10,000 / 5 = Rs. 2,000. Step 7: C's capital corresponds to 2 parts, so C's capital = 2 * 2,000 = Rs. 4,000. Verification / Alternative check: Using the capitals in the ratio 10,000 : 6,000 : 4,000 (which matches 5 : 3 : 2), we can verify that the profit division matches the question. The total capital is 20,000. A's share would be 10,000 / 20,000 of the profit = 1/2 of 1,000 = 500. B's share would be 6,000 / 20,000 of the profit = 3/10 of 1,000 = 300. C's share would be 4,000 / 20,000 of the profit = 1/5 of 1,000 = 200. This matches the given distribution exactly, which confirms our calculation of C's capital. Why Other Options Are Wrong: If C had invested Rs. 5,000, Rs. 6,000 or Rs. 7,000, the capital ratio would not be consistent with the profit ratio 5 : 3 : 2. The resulting profit shares for such capitals would not match the given values of 500, 300 and 200, so those options cannot be correct. Common Pitfalls: A frequent mistake is to simply subtract amounts or to assume capital differences rather than ratios. Some learners also forget to compute C's profit share before forming the ratio. Always derive the complete profit ratio, simplify it and then match it to the known capital of one partner to scale everything correctly. Final Answer: The capital invested by C is Rs. 4,000.

Question 10

A and B start a business together. The amount invested by A is three times the amount invested by B, and A keeps his investment in the business for twice as long as B. If B's share of the profit is Rs. 4,000, what is their total profit?

Accepted Answer

Introduction / Context: This partnership question involves both differing investment amounts and differing periods of investment. The combined effect of capital and time determines each partner's share of profit. We are given B's profit share and need to find the total profit earned by the business, taking the effective investments into account. Given Data / Assumptions: A and B are partners in a business. Investment of A is three times the investment of B. Time period of A's investment is twice the time period of B's investment. B's share of the profit is Rs. 4,000. Profit is shared in proportion to capital multiplied by time. We must determine the total profit of the business. Concept / Approach: In partnership problems, the effective investment is capital * time. We express A's and B's effective investments as multiples of a common base. This directly gives us the ratio of their profit shares. Knowing B's share and the ratio, we can find the total profit by scaling up the ratio parts to the actual rupee amounts. Step-by-Step Solution: Step 1: Let B's capital be x rupees and his investment period be t months. Step 2: Then A's capital is 3x and A's time period is 2t months. Step 3: Effective investment of A = 3x * 2t = 6xt. Step 4: Effective investment of B = x * t = xt. Step 5: Therefore the ratio of profit shares of A and B = 6xt : xt = 6 : 1. Step 6: Let the total profit be P. Then B's share corresponds to 1 part out of 7 parts (6 for A and 1 for B). Step 7: Since B's share (1 part) is Rs. 4,000, total profit P = 7 * 4,000 = Rs. 28,000. Step 8: A's share would then be 6 * 4,000 = Rs. 24,000, which is consistent with the ratio. Verification / Alternative check: We can verify quickly: If the profit ratio is 6 : 1 and total profit is 28,000, then B gets 1/7 of 28,000 which is 4,000, matching the given data. A gets the remaining 24,000. This makes the total profit 24,000 + 4,000 = 28,000, confirming that the calculation is correct. Why Other Options Are Wrong: If the total profit were Rs. 22,000, Rs. 32,000 or Rs. 36,000, then B's share computed as one seventh of the total would be approximately 3,143, 4,571 or 5,143 respectively. None of these values equal the given B's share of Rs. 4,000, so those options do not satisfy the condition of the question. Common Pitfalls: Learners sometimes forget to multiply capital by time and instead compare only the capitals or only the time periods. Another frequent error is to assume that the profit is divided in a simple ratio like 3 : 1 without considering the effect of A investing for twice the time. Always treat partnership profit shares as proportional to capital * time, especially when the time periods are different. Final Answer: The total profit of the business is Rs. 28,000.

Question 11

A and B start a business jointly. A invests Rs. 16,000 for 8 months. B remains in the business for 4 months and claims 2/7 of the total profit. How much capital did B contribute to the business?

Accepted Answer

Introduction / Context: This is a partnership question in which two partners invest different amounts for different time periods. One partner, B, claims a known fraction of the total profit. Using the concept that profit is shared in proportion to capital multiplied by time, we must determine how much money B originally invested in order to justify a 2/7 share of the total profit. Given Data / Assumptions: Partner A invests Rs. 16,000. A keeps this capital invested for 8 months. Partner B remains in the business for 4 months. B claims 2/7 of the total profit. Profit is shared in proportion to capital * time (effective investment). We are required to find the amount of capital contributed by B. Concept / Approach: For partnership problems, effective investment is calculated as capital * time. If the total profit is divided in a certain fraction for one partner, that fraction must match the ratio of that partner's effective investment to the total effective investment of all partners. Here, we let B's capital be an unknown, compute the ratio of effective investments and equate B's profit fraction to 2/7 to solve for B's capital. Step-by-Step Solution: Step 1: Effective investment of A = 16,000 * 8 = 128,000. Step 2: Let B's capital be y rupees. B is in the business for 4 months, so effective investment of B = y * 4 = 4y. Step 3: Total effective investment = 128,000 + 4y. Step 4: B's share of profit as a fraction of the total profit = effective investment of B / total effective investment = 4y / (128,000 + 4y). Step 5: According to the question, B's share of the profit is 2/7. So we set 4y / (128,000 + 4y) = 2 / 7. Step 6: Cross multiply to solve for y: 7 * 4y = 2 * (128,000 + 4y). Step 7: This gives 28y = 256,000 + 8y. Rearranging, 28y - 8y = 256,000 so 20y = 256,000. Step 8: Divide both sides by 20 to get y = 256,000 / 20 = 12,800. Step 9: Therefore, B's capital contribution is Rs. 12,800. Verification / Alternative check: Using A's effective investment of 128,000 and B's effective investment of 4 * 12,800 = 51,200, the total effective investment is 128,000 + 51,200 = 179,200. B's share fraction is then 51,200 / 179,200. Simplifying, divide numerator and denominator by 25,600 to get 2 / 7. This exactly matches the given profit share fraction, confirming that the value of B's capital is correct. Why Other Options Are Wrong: If B had invested Rs. 12,000, 13,000 or 14,500, the resulting effective investment ratios would give profit shares different from 2/7. For example, with Rs. 12,000, the fraction becomes 48,000 / (128,000 + 48,000) = 48,000 / 176,000 which simplifies to 3 / 11, not 2 / 7. Therefore these alternatives do not satisfy the condition on B's profit share. Common Pitfalls: Learners may confuse the fraction of investment with the fraction of profit or may ignore the difference in time for which each partner stays in the business. Another common mistake is forming the ratio 16,000 : y directly without multiplying by time periods. Always remember that partnership profit shares depend on capital * time, not just on capital alone when the durations are different. Final Answer: The capital contributed by B is Rs. 12,800.

Question 12

Sekar starts a business in 1999 by investing Rs. 25,000. In 2000 he adds another Rs. 10,000 and Rajeev joins with Rs. 35,000. In 2001 Sekar adds another Rs. 10,000 and Jatin joins with Rs. 35,000. If the profit at the end of 3 years from the start i…

Accepted Answer

Introduction / Context: This is a multi stage partnership problem where partners join at different times and existing partners change their capital during the life of the business. The profit must be divided according to each partner's capital multiplied by the time his capital remains invested. The goal is to compute the share of Rajeev in the total profit of Rs. 1,50,000 earned over three years. Given Data / Assumptions: Sekar invests Rs. 25,000 at the start of 1999. In 2000, Sekar adds Rs. 10,000 more, so his investment becomes Rs. 35,000. In 2000, Rajeev joins with Rs. 35,000. In 2001, Sekar adds another Rs. 10,000, so his investment becomes Rs. 45,000. In 2001, Jatin joins with Rs. 35,000. Each year is taken as 12 months and the changes happen at the beginning of each year. Total profit at the end of 3 years is Rs. 1,50,000. Profit is distributed in proportion to capital * time for each partner. Concept / Approach: We break the three year period into yearly segments. In each segment, we note the capital of each partner and multiply it by the duration of that segment to get the effective investment in rupee months. After summing the effective investments for all partners across all segments, we obtain the ratio in which the total profit must be shared. Finally, we calculate Rajeev's part from this ratio. Step-by-Step Solution: Step 1: Year 1 (1999): Only Sekar invests Rs. 25,000 for 12 months. Effective investment = 25,000 * 12 = 3,00,000 rupee months. Step 2: Year 2 (2000): Sekar now has Rs. 35,000 and Rajeev invests Rs. 35,000. Each is invested for 12 months, so Sekar contributes 35,000 * 12 = 4,20,000 and Rajeev contributes 35,000 * 12 = 4,20,000 rupee months. Step 3: Year 3 (2001): Sekar invests Rs. 45,000, Rajeev keeps Rs. 35,000 and Jatin joins with Rs. 35,000. Each again invests for 12 months. So Sekar contributes 45,000 * 12 = 5,40,000, Rajeev contributes 35,000 * 12 = 4,20,000 and Jatin contributes 35,000 * 12 = 4,20,000. Step 4: Total effective investments: Sekar = 3,00,000 + 4,20,000 + 5,40,000 = 12,60,000, Rajeev = 4,20,000 + 4,20,000 = 8,40,000, Jatin = 4,20,000. Step 5: Divide these by 4,20,000 to get the ratio: Sekar : Rajeev : Jatin = 3 : 2 : 1. Step 6: Total number of ratio parts = 3 + 2 + 1 = 6. Each part of profit = 1,50,000 / 6 = Rs. 25,000. Step 7: Rajeev gets 2 parts, so Rajeev's share = 2 * 25,000 = Rs. 50,000. Verification / Alternative check: We can verify by computing the other partners' profits. Sekar gets 3 parts = 3 * 25,000 = 75,000 and Jatin gets 1 part = 25,000. Adding all shares, 75,000 + 50,000 + 25,000 = 1,50,000, which matches the total profit available, confirming the correctness of the ratio and Rajeev's share. Why Other Options Are Wrong: Options Rs. 45,000, Rs. 70,000 and Rs. 75,000 do not fall in the correct 3 : 2 : 1 ratio when fitted into the total profit of Rs. 1,50,000. For example, if Rajeev received Rs. 45,000, the remaining profit and the shares of Sekar and Jatin would not match the required proportional relationship defined by the effective investments. Common Pitfalls: Learners often ignore the fact that capital changes over time or assume that every partner invests for the full period. Another common mistake is to sum only capital amounts without multiplying by the time of investment. Always convert every partner's contribution into capital * time units before forming ratios. Final Answer: Rajeev's share of the profit is Rs. 50,000.

Question 13

Simran starts a software business by investing Rs. 50,000. After 6 months, Nanda joins the business with a capital of Rs. 80,000. After 3 years from the start, the total profit is Rs. 24,500. What is Simran's share in this profit?

Accepted Answer

Introduction / Context: This is a common partnership question where one partner starts a business and another joins later with a different capital amount. Profit must be shared in proportion to the product of capital and time. We are required to find Simran's share of the total profit after 3 years, given the capital amounts and the joining time of Nanda. Given Data / Assumptions: Simran invests Rs. 50,000 at the beginning of the business. Nanda joins after 6 months with a capital of Rs. 80,000. Total duration from the start until profit calculation is 3 years, that is, 36 months. Thus Simran's capital is invested for 36 months. Nanda's capital is invested for 36 - 6 = 30 months. Total profit at the end of 3 years is Rs. 24,500. Profit is shared according to capital * time. Concept / Approach: When partners join at different times, we compute effective investment as capital multiplied by the number of months for which it is invested. These effective investments give the ratio for profit sharing. Once the ratio is obtained, we can calculate each partner's profit share by applying the ratio to the total profit. Step-by-Step Solution: Step 1: Effective investment of Simran = 50,000 * 36 = 18,00,000 rupee months. Step 2: Effective investment of Nanda = 80,000 * 30 = 24,00,000 rupee months. Step 3: Write the ratio of their effective investments: Simran : Nanda = 18,00,000 : 24,00,000. Step 4: Simplify this ratio by dividing by 6,00,000. This gives Simran : Nanda = 3 : 4. Step 5: Total ratio parts = 3 + 4 = 7. Step 6: Each ratio part of the profit = total profit / total parts = 24,500 / 7. Step 7: Compute Simran's share = 3 * (24,500 / 7) = 73,500 / 7 = Rs. 10,500. Step 8: Nanda's share would be 4 * (24,500 / 7) = 98,000 / 7 = Rs. 14,000. Verification / Alternative check: Add the calculated shares to verify. Simran gets Rs. 10,500 and Nanda gets Rs. 14,000. The total is 10,500 + 14,000 = 24,500, which matches the given total profit. This confirms that the ratio 3 : 4 and the resulting individual shares are correct based on the effective investments. Why Other Options Are Wrong: The alternative values 10,110, 12,000 and 13,000 do not correspond to a clean 3 : 4 ratio when combined with Nanda's implied share from the remaining profit. For example, if Simran received Rs. 12,000, Nanda would get 12,500 and the ratio would deviate from 3 : 4. Hence these options are inconsistent with the capital and time contributions. Common Pitfalls: Many learners mistakenly think that profit should be shared directly in the ratio 50,000 : 80,000 without considering the time difference. Another mistake is to assume the business lasted only 1 year instead of 3 years, which changes the effective investment calculation. Always multiply the capital by the exact duration in months before forming the ratio. Final Answer: Simran's share of the profit is Rs. 10,500.

Question 14

A, B and C enter into a partnership with their capital shares in the ratio 1/2 : 1/3 : 1/4. After 2 months, A withdraws half of his capital. At the end of one year, a total profit of Rs. 378 is to be divided among them. What is the share of B in thi…

Accepted Answer

Introduction / Context: This partnership problem involves three partners with capital in fractional ratios and a change in one partner's capital after a certain period. The business runs for one full year, and A withdraws half of his capital after 2 months. We must compute the effective investments for all three partners and determine B's share of the total profit of Rs. 378. Given Data / Assumptions: Capital ratio at the start for A : B : C is 1/2 : 1/3 : 1/4. The business runs for 12 months. For the first 2 months, all partners keep their full capital invested. After 2 months, A withdraws half of his capital and keeps the reduced capital invested for the remaining 10 months. B and C keep their original capital invested for the full 12 months. Total profit at the end of the year is Rs. 378. Profit is distributed in proportion to capital * time. Concept / Approach: We convert the fractional capitals into a simpler ratio by choosing a convenient common multiple. Then we compute each partner's effective investment by multiplying the appropriate capital by the number of months it stays in the business. The ratios of these effective investments give the ratios of profits. Using this ratio, we can then find B's exact share from the total profit. Step-by-Step Solution: Step 1: Write the initial capital ratio A : B : C = 1/2 : 1/3 : 1/4. Step 2: To remove fractions, multiply each term by 12 (the least common multiple of 2, 3 and 4). This gives A : B : C = 6 : 4 : 3. Step 3: For the first 2 months, capitals are A = 6, B = 4, C = 3 (in ratio units). Effective investment for A in this period = 6 * 2 = 12, for B = 4 * 2 = 8 and for C = 3 * 2 = 6. Step 4: After 2 months, A withdraws half of his capital, so his new capital becomes 6 / 2 = 3 units. Step 5: For the remaining 10 months, effective investment of A = 3 * 10 = 30, B = 4 * 10 = 40 and C = 3 * 10 = 30. Step 6: Total effective investment for each partner: A = 12 + 30 = 42, B = 8 + 40 = 48, C = 6 + 30 = 36. Step 7: Divide these by 6 to simplify the ratio: A : B : C = 7 : 8 : 6. Step 8: Total number of parts = 7 + 8 + 6 = 21. Profit per part = 378 / 21 = 18. Step 9: B's share = 8 parts = 8 * 18 = Rs. 144. Verification / Alternative check: We can verify by computing the shares for all partners: A gets 7 * 18 = 126, B gets 8 * 18 = 144 and C gets 6 * 18 = 108. Adding these, 126 + 144 + 108 = 378, which matches the total profit. This confirms that the effective investment calculations and final distribution are correct. Why Other Options Are Wrong: The alternative options 169, 225 and 339 do not align with the ratio 7 : 8 : 6 when allocated from a total of 378. For example, if B's share were 169, the shares of A and C computed from the same ratio would not be integers and would not add up correctly to 378, so these options are inconsistent with the problem data. Common Pitfalls: A typical mistake is to use only the initial capital ratio 1/2 : 1/3 : 1/4 and ignore the change in A's capital after 2 months. Another error is to assume the business runs for only 10 months instead of a full year. Always pay attention to when capital changes occur and over what period profit is calculated, and always convert to capital * time before forming the final ratio. Final Answer: B's share of the profit is Rs. 144.

Question 15

A, B and C enter into a partnership by investing Rs. 40,000, Rs. 80,000 and Rs. 1,20,000 respectively. At the end of the first year, B withdraws Rs. 40,000, and at the end of the second year, C withdraws Rs. 80,000. If the profit is calculated at th…

Accepted Answer

Introduction / Context: This problem involves a three year partnership in which two partners partially withdraw capital at different stages. Because capital amounts change from year to year, we must compute the effective investment of each partner in terms of capital * time before determining the final profit sharing ratio between A, B and C. Given Data / Assumptions: Initial investments: A = Rs. 40,000, B = Rs. 80,000, C = Rs. 1,20,000. The business runs for a total of 3 years. At the end of year 1, B withdraws Rs. 40,000, so in years 2 and 3 B invests Rs. 40,000. At the end of year 2, C withdraws Rs. 80,000, so in year 3 C invests Rs. 40,000. A keeps his full investment unchanged for the entire 3 years. Profit is shared in proportion to capital * time. Concept / Approach: We split the 3 year period into separate one year blocks. For each year we list the capitals of A, B and C and compute their effective investments by multiplying each capital by the time span of that block. Summing the effective investments over all three years gives us total effective investments, and their ratio gives the profit sharing ratio at the end of 3 years. Step-by-Step Solution: Step 1: Year 1: A invests 40,000, B invests 80,000, C invests 1,20,000 for 1 year each. Effective investments: A = 40,000 * 1 = 40,000, B = 80,000 * 1 = 80,000, C = 1,20,000 * 1 = 1,20,000. Step 2: Year 2: B withdraws 40,000, so: A = 40,000, B = 40,000, C = 1,20,000 for 1 year. Effective investments: A = 40,000, B = 40,000, C = 1,20,000. Step 3: Year 3: C withdraws 80,000, so: A = 40,000, B = 40,000, C = 40,000 for 1 year. Effective investments: A = 40,000, B = 40,000, C = 40,000. Step 4: Total effective investments across 3 years: A = 40,000 + 40,000 + 40,000 = 1,20,000, B = 80,000 + 40,000 + 40,000 = 1,60,000, C = 1,20,000 + 1,20,000 + 40,000 = 2,80,000. Step 5: To find the ratio, divide all by 40,000: A : B : C = 3 : 4 : 7. Step 6: Therefore, profit will be shared in the ratio 3 : 4 : 7 among A, B and C. Verification / Alternative check: We can verify consistency by imagining a sample profit amount, for example Rs. 1,40,000, and dividing it in the ratio 3 : 4 : 7. The shares would be 30,000, 40,000 and 70,000 respectively, and the relative magnitudes match the overall pattern of investments, where C had the largest effective investment, B was in the middle and A had the smallest. This confirms that the ratio 3 : 4 : 7 is sensible and consistent with the capital history. Why Other Options Are Wrong: Options 2 : 3 : 5, 5 : 6 : 4 and 1 : 3 : 5 do not match the effective investment totals. For example, A and B have equal effective investments in the last year but B invested more in the first year, so any correct ratio must have B's final share greater than A's. Ratios that do not reflect this ordering or do not correspond to 1,20,000 : 1,60,000 : 2,80,000 cannot be correct. Common Pitfalls: Students may incorrectly use only initial capitals or only final capitals and ignore the timing of withdrawals. Another error is to simply average capital amounts instead of multiplying by time. Always track each capital change with its duration and sum the capital * time values to get accurate ratios in multi year partnership questions. Final Answer: The profit will be shared in the ratio 3 : 4 : 7 among A, B and C.

Question 16

Rajeev starts a business with Rs. 10,000. After 2 months, Deepu joins with an investment that is 20% more than Rajeev's. After another 2 months, Shakti joins with an investment that is 40% less than Deepu's. If the total profit at the end of the yea…

Accepted Answer

Introduction / Context: This is a challenging partnership question with three partners joining at different times and with investments related by percentage changes. Additionally, the total profit is defined indirectly using a relationship involving the initial investments. We must first work out the individual investments and their effective capital * time contributions, then translate the given condition about total profit into a numerical value, and finally compute Rajeev's share of this profit. Given Data / Assumptions: Rajeev invests Rs. 10,000 at the beginning of the year. After 2 months, Deepu joins with 20% more than Rajeev, so Deepu invests Rs. 12,000. After another 2 months, Shakti joins with 40% less than Deepu, so Shakti invests Rs. 7,200. The business runs for a total of 12 months. Rajeev invests for 12 months, Deepu for 10 months and Shakti for 8 months. Total profit P is equal to twice the difference between Rajeev's investment and ten times Shakti's investment. Profit is shared in proportion to capital * time. Concept / Approach: We first compute each partner's capital * time product to get the profit sharing ratio. Then we use the condition for total profit to find P. Finally we apply the ratio to P to calculate Rajeev's share. The key ideas are careful percentage calculation for investments, correct time periods and algebraic use of the profit condition P = 2 * (10 * Shakti investment minus Rajeev investment). Step-by-Step Solution: Step 1: Rajeev's capital = 10,000 for 12 months, so effective investment = 10,000 * 12 = 1,20,000 rupee months. Step 2: Deepu invests 20% more than Rajeev, so Deepu's capital = 10,000 * 1.2 = 12,000. Deepu invests for 12 - 2 = 10 months, so effective investment = 12,000 * 10 = 1,20,000 rupee months. Step 3: Shakti invests 40% less than Deepu, so Shakti's capital = 12,000 * 0.6 = 7,200. Shakti invests for 12 - 4 = 8 months, so effective investment = 7,200 * 8 = 57,600 rupee months. Step 4: The effective investment ratio is 1,20,000 : 1,20,000 : 57,600. Divide all by 2,400 to get 50 : 50 : 24. Step 5: Total ratio parts = 50 + 50 + 24 = 124. Step 6: According to the question, total profit P = twice the difference between Rajeev's investment and ten times Shakti's investment. Ten times Shakti's investment = 10 * 7,200 = 72,000. Difference = 72,000 - 10,000 = 62,000. So P = 2 * 62,000 = Rs. 1,24,000. Step 7: Rajeev's share of profit = P * (Rajeev's ratio part / total parts) = 1,24,000 * (50 / 124) = Rs. 50,000. Verification / Alternative check: We can also compute Deepu's and Shakti's shares. Deepu's share = 1,24,000 * (50 / 124) = 50,000 and Shakti's share = 1,24,000 * (24 / 124) = 24,000. Adding all three shares gives 50,000 + 50,000 + 24,000 = 1,24,000 which exactly matches the total profit. Rajeev's share is therefore consistent with both the ratio and the given profit condition. Why Other Options Are Wrong: If Rajeev's profit were Rs. 48,000, Rs. 38,000 or Rs. 40,000, the resulting shares for Deepu and Shakti using the fixed ratio 50 : 50 : 24 would not sum to a total profit that equals 2 times the difference between 10,000 and ten times 7,200. These options break either the ratio condition or the total profit condition, so they cannot be correct. Common Pitfalls: Common mistakes include misinterpreting the phrase "20% more" or "40% less", forgetting the exact times for which each partner invests and misreading the profit condition. Some learners also incorrectly compute P as twice the difference between Rajeev's and Shakti's investments, without multiplying Shakti's investment by ten. Careful reading and systematic algebra are essential here. Final Answer: Rajeev's share of the profit is Rs. 50,000.

Question 17

Aman starts a business by investing Rs. 70,000. After 6 months, Rakhi joins with Rs. 1,05,000. After another 6 months, Sagar joins them with Rs. 1,40,000. If the profit is calculated 3 years after Aman starts the business, in what ratio will it be s…

Accepted Answer

Introduction / Context: This partnership problem involves three partners who join the business at different times with different capital amounts. The business runs for 3 years from the start of Aman's investment. We must compute the effective capital * time for each partner and then derive the ratio in which the profit will be distributed among Aman, Rakhi and Sagar. Given Data / Assumptions: Aman invests Rs. 70,000 at the start and remains invested for the full 3 years (36 months). After 6 months, Rakhi joins with Rs. 1,05,000 and remains for the remaining 30 months. After another 6 months (that is, 12 months from the start), Sagar joins with Rs. 1,40,000 and remains for the remaining 24 months. Profit is calculated at the end of 3 years and shared in proportion to effective investments. Effective investment = capital * time. Concept / Approach: When partners join at different times, the simple ratio of capital is not enough. Instead, we must calculate capital * time for each partner in rupee months. This gives the relative contribution of each partner to the business over the whole period. The ratio of these effective investments is then used to divide the final profit, independent of the exact profit amount. Step-by-Step Solution: Step 1: Effective investment of Aman: Aman invests 70,000 for 36 months. Effective investment = 70,000 * 36 = 25,20,000 rupee months. Step 2: Effective investment of Rakhi: Rakhi joins after 6 months, so she invests for 30 months. Effective investment = 1,05,000 * 30 = 31,50,000 rupee months. Step 3: Effective investment of Sagar: Sagar joins after 12 months, so he invests for 24 months. Effective investment = 1,40,000 * 24 = 33,60,000 rupee months. Step 4: Write their effective investments as a ratio: Aman : Rakhi : Sagar = 25,20,000 : 31,50,000 : 33,60,000. Step 5: Divide all by 2,10,000 to simplify: 25,20,000 / 2,10,000 = 12, 31,50,000 / 2,10,000 = 15, 33,60,000 / 2,10,000 = 16. So the ratio becomes 12 : 15 : 16. Step 6: Therefore, the profit will be shared among Aman, Rakhi and Sagar in the ratio 12 : 15 : 16. Verification / Alternative check: We can confirm the logic by observing that Sagar has the largest effective investment, followed by Rakhi and then Aman. This aligns with the final ratio, where Sagar has the largest share (16 parts), Rakhi the next (15 parts) and Aman the lowest (12 parts). Any actual profit figure, for example Rs. 43,000, would be split proportionally according to these numbers, and the proportionality would remain correct regardless of the exact amount. Why Other Options Are Wrong: Ratios such as 11 : 13 : 15, 11 : 13 : 17 or 12 : 17 : 18 do not correspond to the actual effective investments. For instance, a ratio with much larger share for Sagar compared to Rakhi would not be justified because Rakhi's effective investment is only slightly smaller than Sagar's. Therefore these options are inconsistent with the capital * time contributions. Common Pitfalls: A common error is to ignore the different entry times and simply compare the capital amounts 70,000, 1,05,000 and 1,40,000. Another mistake is to treat each year equally without using exact months, which can lead to incorrect time durations for each partner. Always determine how many months each capital amount stays in the business and then multiply capital by time before forming ratios. Final Answer: The profit will be shared in the ratio 12 : 15 : 16 among Aman, Rakhi and Sagar.

Question 18

Reena and Shaloo are partners in a business. Reena invests Rs. 35,000 for 8 months and Shaloo invests Rs. 42,000 for 10 months. Out of a total profit of Rs. 31,570, what is the share of Reena?

Accepted Answer

Introduction / Context: This question is a typical two partner partnership problem with different investment amounts and different investment periods. We must calculate the effective investments for Reena and Shaloo using capital * time, derive the ratio in which they should share the profit and then compute Reena's exact share from the total profit of Rs. 31,570. Given Data / Assumptions: Reena invests Rs. 35,000 for 8 months. Shaloo invests Rs. 42,000 for 10 months. Total profit at the end of the period is Rs. 31,570. Profit is shared in proportion to capital * time. The entire period is assumed to be continuous with no mid period withdrawals. Concept / Approach: The concept used is effective investment. Because the partners invest for different numbers of months, we cannot simply compare their capitals. Instead we multiply each partner's capital by the number of months it is invested. The profit is then split in the ratio of these effective investments. Finally, we apply this ratio to the known total profit to calculate Reena's share. Step-by-Step Solution: Step 1: Compute Reena's effective investment: 35,000 * 8 = 2,80,000 rupee months. Step 2: Compute Shaloo's effective investment: 42,000 * 10 = 4,20,000 rupee months. Step 3: Write the ratio of effective investments: Reena : Shaloo = 2,80,000 : 4,20,000. Step 4: Simplify this ratio by dividing both terms by 1,40,000: 2,80,000 / 1,40,000 = 2, 4,20,000 / 1,40,000 = 3. So the ratio becomes 2 : 3. Step 5: Total ratio parts = 2 + 3 = 5. Step 6: Each part of profit = 31,570 / 5 = 6,314. Step 7: Reena gets 2 parts, so Reena's share = 2 * 6,314 = Rs. 12,628. Step 8: Shaloo's share would be 3 * 6,314 = Rs. 18,942. Verification / Alternative check: Verify by adding the two shares: 12,628 + 18,942 = 31,570, which matches the total profit. The ratio of Reena's share to Shaloo's share is 12,628 : 18,942. Dividing both by 6,314 gives 2 : 3, which is exactly the ratio of their effective investments. This confirms that the profit has been correctly divided. Why Other Options Are Wrong: Values such as 18,245, 11,235 and 10,253 do not maintain the 2 : 3 ratio when combined with the remaining profit for Shaloo. For example, if Reena received 18,245, the remaining profit for Shaloo would be 13,325 and the ratio would not be 2 : 3. Hence those options do not align with the partnership rule of capital * time proportionality. Common Pitfalls: Students sometimes use only the capitals 35,000 and 42,000 and ignore the different durations of 8 and 10 months. Another mistake is to reverse the ratio or miscalculate the simplified form. Always double check by simplifying the ratio carefully and ensuring that the sum of individual shares equals the total profit. Final Answer: Reena's share of the profit is Rs. 12,628.

Question 19

X and Y invest in a business and earn a profit which they divide in the ratio 2 : 3. If X invests Rs. 40,000, what amount of capital must have been invested by Y?

Accepted Answer

Introduction / Context: This is a straightforward partnership question. Two partners X and Y share profit in a known ratio. We are given the capital invested by X and must find the capital invested by Y, assuming that both partners kept their money in the business for the same length of time. Given Data / Assumptions: Profit sharing ratio of X : Y is 2 : 3. X invested Rs. 40,000. The duration of investment for both X and Y is the same. Profit is shared in proportion to capital, since time is equal. We need to find Y's capital investment. Concept / Approach: If the time for which the money is invested is the same for both partners, then the profit share ratio is simply the ratio of their capitals. Therefore, capital of X : capital of Y = 2 : 3. Knowing X's capital and this ratio, we can find Y's capital by a simple proportion calculation. Step-by-Step Solution: Step 1: Write the profit sharing ratio as capital ratio: capital of X : capital of Y = 2 : 3. Step 2: X's capital corresponds to the first part in the ratio. So 2 parts represent Rs. 40,000. Step 3: Value of 1 part = 40,000 / 2 = Rs. 20,000. Step 4: Y's capital corresponds to 3 parts. So Y's capital = 3 * 20,000 = Rs. 60,000. Step 5: Therefore, Y must have invested Rs. 60,000 to justify the 2 : 3 profit sharing ratio. Verification / Alternative check: If X invests 40,000 and Y invests 60,000, then the capital ratio is 40,000 : 60,000. Dividing each by 20,000, we get 2 : 3, which matches the given profit ratio. This confirms that our proportion method is correct and that Y's capital has been correctly computed. Why Other Options Are Wrong: If Y had invested Rs. 20,000, Rs. 40,000 or Rs. 80,000, the resulting capital ratios with X's 40,000 would be 2 : 1, 1 : 1 or 1 : 2 respectively. None of these ratios equal the required profit sharing ratio of 2 : 3. Hence those options cannot give the correct relationship between the partners' profit shares. Common Pitfalls: A common mistake is to divide the capital in the inverse ratio or to assume that Y's investment must be less than X's. Another error is to forget that when time is the same, the profit ratio directly equals the capital ratio, with no need to consider capital * time. Always apply the given ratio directly to the capitals when time does not differ. Final Answer: Y invested a capital amount of Rs. 60,000.

Question 20

A, B and C subscribe a total capital of Rs. 50,000 for a business. A contributes Rs. 4,000 more than B, and B contributes Rs. 5,000 more than C. Out of a total profit of Rs. 35,000, how much does A receive?

Accepted Answer

Introduction / Context: This question combines algebra with the partnership concept. Three partners A, B and C contribute a total of Rs. 50,000, with known differences between their capital amounts. We must first determine the individual capitals by forming equations, and then use these amounts to divide a total profit of Rs. 35,000 in the correct ratio and find A's share. Given Data / Assumptions: Total capital subscribed by A, B and C is Rs. 50,000. A contributes Rs. 4,000 more than B. B contributes Rs. 5,000 more than C. Total profit at the end of the period is Rs. 35,000. All three partners keep their capital invested for the same time period. Profit is shared in proportion to their respective capital contributions. Concept / Approach: First we use the differences between the capitals and the total capital to set up algebraic equations and solve for the individual capital amounts. Once the capitals of A, B and C are known, we find the ratio A : B : C and apply it to the given total profit. The concept is straightforward partnership based on capital only, since time is assumed to be equal for all. Step-by-Step Solution: Step 1: Let C's capital be c rupees. Step 2: B contributes Rs. 5,000 more than C, so B's capital = c + 5,000. Step 3: A contributes Rs. 4,000 more than B, so A's capital = (c + 5,000) + 4,000 = c + 9,000. Step 4: The total capital is A + B + C = (c + 9,000) + (c + 5,000) + c = 3c + 14,000 = 50,000. Step 5: Solve for c: 3c + 14,000 = 50,000 implies 3c = 36,000, so c = 12,000. Step 6: Then B's capital = 12,000 + 5,000 = 17,000 and A's capital = 12,000 + 9,000 = 21,000. Step 7: The capital ratio A : B : C = 21,000 : 17,000 : 12,000. Divide all by 1,000 to get 21 : 17 : 12. Step 8: Total ratio parts = 21 + 17 + 12 = 50. Step 9: Profit per part = 35,000 / 50 = 700. Step 10: A's share = 21 parts = 21 * 700 = Rs. 14,700. Verification / Alternative check: Compute B's and C's shares using the same ratio. B gets 17 * 700 = 11,900 and C gets 12 * 700 = 8,400. Adding them together gives 14,700 + 11,900 + 8,400 = 35,000, matching the total profit. The capital ratio and profit distribution are therefore consistent with all given conditions. Why Other Options Are Wrong: If A were to receive Rs. 15,000, Rs. 12,000 or Rs. 13,500, then the remaining profit would not divide between B and C in the ratio 17 : 12. The sum of individual shares also would not match the required proportions based on capitals of 21,000, 17,000 and 12,000. Thus those options are not compatible with the established capital ratio. Common Pitfalls: Common errors include incorrectly forming the total capital equation, for example by misplacing the 4,000 and 5,000 differences, or forgetting that the total capital is 50,000. Another mistake is simplifying the ratio incorrectly, which then leads to wrong profit shares. Taking care with the algebraic steps and rechecking the final sum of shares helps avoid such mistakes. Final Answer: A receives a profit share of Rs. 14,700.

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