Question 1

In a partnership business, K and B start a business jointly. K invests Rs. 16,000 for 8 months, while B keeps his capital invested for 4 months. Out of the total profit, K receives 2/7 of the overall profit share. How much capital was contributed by…

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Introduction / Context: This is a classic aptitude question based on the concept of partnership in business. In such questions, the profit is distributed among partners in proportion to the product of the capital invested and the time for which it is invested, commonly known as capital time or money months. Here, we are given the share of profit of one partner and the investment details, and we have to determine the capital contributed by the other partner. Given Data / Assumptions: K and B are partners in a business. K invests Rs. 16,000 for 8 months. B keeps his capital invested for 4 months. K receives 2/7 of the total profit. Profit is distributed in the ratio of capital multiplied by time. Concept / Approach: The basic concept in partnership problems is that profit share is proportional to capital * time. If a partner invests more money or keeps it invested for more time, that partner should receive a higher share of the profit. Once we know the profit ratio between partners, we can equate it to the ratio of their capital time contributions and solve for the unknown capital. Step-by-Step Solution: Step 1: Compute K's capital time: 16,000 * 8 = 1,28,000 money months.Step 2: Let the capital of B be x. His capital time is x * 4 = 4x money months.Step 3: Since K receives 2/7 of the total profit, B must receive 5/7. Thus, the ratio of K : B profit shares is 2 : 5.Step 4: Profit share ratio equals capital time ratio, so 1,28,000 : 4x = 2 : 5.Step 5: Simplify 1,28,000 : 4x to 32,000 : x. Then 32,000 : x = 2 : 5 implies 32,000 / x = 2 / 5.Step 6: Cross multiply: 32,000 * 5 = 2x, hence 1,60,000 = 2x, so x = 80,000. Verification / Alternative check: Capital time for K is 1,28,000 and for B is 4 * 80,000 = 3,20,000. The ratio K : B is 1,28,000 : 3,20,000 = 2 : 5. Therefore, K receives 2 parts and B receives 5 parts of profit, so K's fraction of total profit is 2 / (2 + 5) = 2/7, which matches the given data. Why Other Options Are Wrong: Rs. 40,000, Rs. 64,000 and Rs. 96,000 do not give a capital time ratio of 2 : 5 when compared with K's investment. With these values, the resulting profit share of K would not be exactly 2/7 of the total profit. 'None of these' is also incorrect because one of the numerical options, Rs. 80,000, satisfies all the partnership conditions exactly. Common Pitfalls: Candidates often forget that profit distribution in partnership depends on both capital and time, not capital alone. Another frequent mistake is to interpret the 2/7 as the ratio between K and B, instead of K's fraction of the total profit. Mixing these interpretations leads to incorrect equations and wrong answers. Final Answer: The correct amount invested by B is Rs. 80,000.

Question 2

Vishal and Raghu jointly start a business. Vishal invests three times as much money as Raghu and keeps his money invested for twice the time as Raghu. If the gain is proportional to both money invested and time, Raghu receives Rs. 4,000 as his share…

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Introduction / Context: This question tests understanding of profit sharing in partnerships where both the amount of capital and the duration of investment affect the share of profit. The scenario involves two partners, Vishal and Raghu, with a specific relation between their capital amounts and the times for which they are invested. We are given Raghu's actual profit and must determine the total profit of the business. Given Data / Assumptions: Vishal and Raghu start the business together. Vishal invests three times the capital of Raghu. Vishal invests his money for double the time compared to Raghu. Raghu receives Rs. 4,000 as his share of profit. Profit is proportional to capital multiplied by time. Concept / Approach: The fundamental rule is that profit share is proportional to (capital * time). When one partner invests more money and keeps it invested longer, that partner receives a correspondingly larger share of the profit. By forming a ratio of their effective investments, we can find the ratio of their profit shares. Once we know how Raghu's share relates to total profit, we can directly compute the total profit. Step-by-Step Solution: Step 1: Let Raghu's capital be R and his time be T.Step 2: Then Vishal's capital is 3R and his time is 2T.Step 3: Raghu's capital time is R * T.Step 4: Vishal's capital time is 3R * 2T = 6RT.Step 5: Therefore, their profit share ratio Raghu : Vishal is RT : 6RT = 1 : 6.Step 6: Total profit is divided into 1 + 6 = 7 equal parts, with Raghu receiving 1 part and Vishal receiving 6 parts.Step 7: Given Raghu's share is Rs. 4,000, which equals 1 part, so total profit = 7 * 4,000 = Rs. 28,000. Verification / Alternative check: If total profit is Rs. 28,000, then Vishal's share should be 6 parts, that is 6 * 4,000 = Rs. 24,000. The ratio of 4,000 : 24,000 simplifies to 1 : 6, which matches the capital time ratio derived from the data. This confirms that the total profit of Rs. 28,000 is consistent with the partnership conditions. Why Other Options Are Wrong: If the total profit were Rs. 18,000, Raghu's share at 1/7 of the total would be about Rs. 2,571, not Rs. 4,000. For Rs. 14,000, his share would be only Rs. 2,000. For Rs. 8,000, his share would be approximately Rs. 1,143. None of these values match Raghu's given share of Rs. 4,000, so they are incorrect. 'None of these' is not needed because one of the given amounts is exactly correct. Common Pitfalls: Many learners incorrectly treat the profit ratio as simply 3 : 1 instead of combining the effect of both capital and time. Another mistake is to divide 4,000 by 3 or 2 directly without first determining the correct ratio of 1 : 6 and the total of 7 parts. Careful attention to both capital and time is essential in partnership questions. Final Answer: The total profit earned by the business is Rs. 28,000.

Question 3

K, L and M together start a business. K invests Rs. 6,000 for 5 months, L invests Rs. 3,600 for 6 months, and M invests Rs. 7,500 for 3 months. If the total profit at the end of the period is Rs. 7,410, what is the share of L in this profit?

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Introduction / Context: This question is a standard partnership problem in which three partners invest different amounts of money for different durations. To find the share of one partner in the total profit, we must calculate each partner's effective investment using the capital time concept and then distribute the profit in the same ratio. Given Data / Assumptions: Partner K invests Rs. 6,000 for 5 months. Partner L invests Rs. 3,600 for 6 months. Partner M invests Rs. 7,500 for 3 months. Total profit at the end of the period is Rs. 7,410. Profit is shared in proportion to capital multiplied by time. Concept / Approach: The core principle is that if partners invest money for different time periods, their shares of profit are proportional to the product of capital and duration. We first compute each partner's capital time, then form their ratio and divide the total profit according to this ratio. Finally, we extract L's share. Step-by-Step Solution: Step 1: Compute K's capital time: 6,000 * 5 = 30,000 money months.Step 2: Compute L's capital time: 3,600 * 6 = 21,600 money months.Step 3: Compute M's capital time: 7,500 * 3 = 22,500 money months.Step 4: The ratio of K : L : M is 30,000 : 21,600 : 22,500.Step 5: The total of ratio units is 30,000 + 21,600 + 22,500 = 74,100.Step 6: Each money month unit corresponds to profit 7,410 / 74,100 = 0.1 rupee.Step 7: L's share of profit is 21,600 * 0.1 = Rs. 2,160. Verification / Alternative check: Check K's share: 30,000 * 0.1 = Rs. 3,000. M's share: 22,500 * 0.1 = Rs. 2,250. The total profit is 3,000 + 2,160 + 2,250 = Rs. 7,410, which matches the given total. Therefore, the calculation is consistent and confirms that L's share is correct. Why Other Options Are Wrong: Rs. 1,640 is too low and does not correspond to the ratio 30,000 : 21,600 : 22,500. Rs. 2,500 and Rs. 3,000 are larger than L's fair share based on her capital time; Rs. 3,000 actually corresponds to K's share. 'None of these' is incorrect because Rs. 2,160 is a valid option that exactly matches the partnership ratio. Common Pitfalls: Students sometimes mistakenly use only capital values and ignore time, or sum all investments and divide directly. Another error is to divide the total profit in the wrong proportion, such as using 6,000 : 3,600 : 7,500 instead of the correct money month values. Always multiply capital by time before forming the ratio for partnership questions. Final Answer: The correct share of L in the total profit is Rs. 2,160.

Question 4

Three partners A, B and C share the profit of a business such that three times the share of A is equal to two times the share of B and also equal to twelve times the share of C. Based on this information, what is the ratio of the profits of A, B and…

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Introduction / Context: This is a conceptual partnership and ratio problem. Instead of dealing with actual rupee amounts, we are given proportional relationships between the profit shares of three partners A, B and C. Our task is to convert the given proportional statements into a clear numerical ratio of their profit shares. Given Data / Assumptions: Three partners A, B and C share the profit of a business. Three times the share of A is equal to two times the share of B. The same quantity is also equal to twelve times the share of C. We need the ratio of A : B : C. Concept / Approach: When multiple expressions are equal, we set them equal to a common variable and then solve for each individual quantity. Here, 3A, 2B and 12C are all equal to the same value. By expressing A, B and C in terms of that common value, we can derive a simple ratio between them. This is a standard technique in ratio and proportion questions in aptitude tests. Step-by-Step Solution: Step 1: Let 3A = 2B = 12C = k, where k is a common proportional constant.Step 2: From 3A = k, we get A = k / 3.Step 3: From 2B = k, we get B = k / 2.Step 4: From 12C = k, we get C = k / 12.Step 5: Now the ratio A : B : C is (k / 3) : (k / 2) : (k / 12).Step 6: Cancel k from all terms to get 1/3 : 1/2 : 1/12.Step 7: Convert to whole numbers using common denominator 12: A = 4/12, B = 6/12, C = 1/12.Step 8: Therefore, A : B : C = 4 : 6 : 1. Verification / Alternative check: To verify, assume an actual value for k, for example k = 12. Then A = 12 / 3 = 4, B = 12 / 2 = 6, C = 12 / 12 = 1. Check the condition: 3A = 3 * 4 = 12, 2B = 2 * 6 = 12, 12C = 12 * 1 = 12. All are equal, confirming that the ratio 4 : 6 : 1 is consistent with the original condition. Why Other Options Are Wrong: 3 : 2 : 12 directly uses the multipliers but does not represent the actual shares of A, B and C. 12 : 2 : 3 and 1 : 6 : 4 do not satisfy the condition 3A = 2B = 12C when tested. 'None of these' is incorrect because option 4 : 6 : 1 exactly matches the proportional relationship given in the problem. Common Pitfalls: A common error is to misread the condition as A : B : C = 3 : 2 : 12, which is wrong. Another mistake is to forget to express A, B and C separately in terms of a common variable before forming the ratio. Always remember that ratios should reflect the original quantities (A, B and C), not the multipliers applied to them in the condition. Final Answer: The correct ratio of the profits of A, B and C is 4 : 6 : 1.

Question 5

A man who recently died left a sum of Rs. 3,90,000 to be divided among his wife, five sons and four daughters. He directed that each son should receive three times as much as each daughter receives, and that each daughter should receive twice as muc…

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Introduction / Context: This problem is a ratio based distribution question under the aptitude topic of partnership and shares. Instead of business partners, the shares belong to family members, but the method is the same. The total sum is fixed and the relations between individual shares are given. Our job is to convert these relationships into algebraic form and find the exact share of the wife. Given Data / Assumptions: Total amount left by the man is Rs. 3,90,000. Beneficiaries are: wife, five sons and four daughters. Each son receives three times as much as each daughter. Each daughter receives twice as much as the mother (wife). All conditions must hold simultaneously. Concept / Approach: The key idea is to represent all individual shares in terms of a single variable. Since the wife's share is the smallest, it is convenient to express daughters' and sons' shares in multiples of the wife's share. Once we express every person's share in terms of the wife's share, we can sum all shares, equate the result to the total amount and solve for the wife's share. Step-by-Step Solution: Step 1: Let the wife's share be w rupees.Step 2: Each daughter receives twice the mother, so each daughter's share is 2w.Step 3: Each son receives three times a daughter, so each son's share is 3 * 2w = 6w.Step 4: There is one wife, five sons and four daughters.Step 5: Total share in terms of w is wife + sons + daughters = w + 5 * 6w + 4 * 2w.Step 6: Simplify: w + 30w + 8w = 39w.Step 7: This total 39w equals Rs. 3,90,000, so 39w = 3,90,000.Step 8: Solve for w: w = 3,90,000 / 39 = Rs. 10,000. Verification / Alternative check: If the wife receives Rs. 10,000, then each daughter receives 2 * 10,000 = Rs. 20,000 and each son receives 6 * 10,000 = Rs. 60,000. Total for five sons is 5 * 60,000 = Rs. 3,00,000. Total for four daughters is 4 * 20,000 = Rs. 80,000. Adding the wife's share, 3,00,000 + 80,000 + 10,000 = Rs. 3,90,000, which matches the given total amount. Why Other Options Are Wrong: If the wife received Rs. 14,000, total amount would be 39 * 14,000 = Rs. 5,46,000, which is too high. For Rs. 12,000, the total would be Rs. 4,68,000. For Rs. 9,000, the total would be Rs. 3,51,000. None of these align with the given amount of Rs. 3,90,000, so they are incorrect. 'None of these' is also incorrect because Rs. 10,000 matches the conditions exactly. Common Pitfalls: Candidates sometimes misinterpret which quantity is double or triple, leading to wrong expressions. Another frequent error is incorrectly counting the number of sons or daughters. Always verify by recomputing the total from the final assumed shares to ensure it matches the given sum. Final Answer: The wife's correct share of the estate is Rs. 10,000.

Question 6

Ravi, Anand and Pranay run a business firm in partnership. You are asked to determine Anand's share in the profit earned by them. (a) Ravi, Anand and Pranay invest amounts in the ratio 2 : 4 : 7. (b) Pranay's share in the profit is Rs. 8,750. Based …

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Introduction / Context: This is a data sufficiency style aptitude question involving a partnership. Rather than directly asking for a numerical answer, the question asks which given statements are sufficient to calculate Anand's share of profit. Understanding when profit ratios and actual amounts together allow you to compute an unknown share is the core skill tested here. Given Data / Assumptions: We need to find Anand's share in the profit. Statement (a): Ravi, Anand and Pranay invest in the ratio 2 : 4 : 7. Statement (b): Pranay's share in the profit is Rs. 8,750. Profit is distributed according to the investment ratio, assuming equal time period for all partners. Concept / Approach: Profit shares in partnership are proportional to investments (and time, if times differ). In this question we assume time is the same for all partners, so profit shares directly follow the ratio 2 : 4 : 7. To compute a specific partner's share numerically, we must know at least one actual share value or the total profit. A single ratio without any actual amount is not enough, while a single amount without the ratio also does not uniquely determine the required share. Step-by-Step Solution: Step 1: From statement (a), the investment (and hence profit) ratio is Ravi : Anand : Pranay = 2 : 4 : 7.Step 2: Knowing only the ratio does not give any specific rupee amount for Anand's share, so statement (a) alone is not sufficient.Step 3: From statement (b), we know only that Pranay's share is Rs. 8,750.Step 4: Without the ratio, that single value cannot tell us the share of Anand, so statement (b) alone is also not sufficient.Step 5: Using (a) and (b) together, we know that Pranay's share corresponds to 7 parts out of (2 + 4 + 7) = 13 parts.Step 6: If 7 parts equal Rs. 8,750, then 1 part is 8,750 / 7 = Rs. 1,250.Step 7: Anand's share corresponds to 4 parts, so Anand's share is 4 * 1,250 = Rs. 5,000, which can now be uniquely determined. Verification / Alternative check: Once we know that one part is Rs. 1,250, Ravi's share is 2 * 1,250 = Rs. 2,500 and Pranay's is 7 * 1,250 = Rs. 8,750 as given. Total profit is 2,500 + 5,000 + 8,750 = Rs. 16,250. These values are all consistent with the ratio and the given share, confirming the sufficiency of using both statements together. Why Other Options Are Wrong: 'Only a is sufficient' is incorrect because a ratio alone cannot give a numerical rupee value. 'Only b is sufficient' is incorrect because one partner's amount alone does not reveal another partner's share without knowing how profits are split. 'Neither a nor b is sufficient' is wrong because together they clearly determine Anand's share. 'None of these' is not needed because one of the listed options correctly states that both (a) and (b) are sufficient. Common Pitfalls: Test takers sometimes assume that knowing the ratio is enough, overlooking that the question asks for a specific rupee value, not just a ratio. Others may mistakenly think that one known amount automatically reveals all other amounts. Always check whether you can uniquely compute the required value with the given information. Final Answer: The correct conclusion is that both statements (a) and (b) together are sufficient to determine Anand's share of the profit.

Question 7

Deepa starts a business in 2009 by investing Rs. 50,000. In 2010 she adds Rs. 20,000 more to her capital, and her friend Sushma joins at that time with an investment of Rs. 70,000. In 2011 Deepa adds another Rs. 20,000, making her capital larger aga…

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Introduction / Context: This question involves a changing capital partnership over multiple years. Different partners join at different times and capital amounts change during the three year period from 2009 to 2011. To find Sushma's share of the profit, we need to carefully compute capital time contributions for each partner across each year and then distribute the total profit in that ratio. Given Data / Assumptions: Deepa invests Rs. 50,000 in 2009. Deepa increases her capital by Rs. 20,000 in 2010 (so it becomes Rs. 70,000 in 2010). Sushma joins in 2010 with Rs. 70,000. In 2011, Deepa adds another Rs. 20,000, making her investment Rs. 90,000 that year. Avanthi joins in 2011 with Rs. 70,000. The business runs from the start of 2009 to the end of 2011 (3 years). Total profit at the end of 3 years is Rs. 3,00,000. Concept / Approach: When capital changes or new partners join at different times, the profit sharing is based on the product of capital and the time for which that capital remains invested. We treat each year as one equal time unit and compute capital time for each partner in each year. Summing these gives relative weights for profit distribution. Then we compute Sushma's share by applying her share of total weight to the total profit. Step-by-Step Solution: Step 1: For 2009, only Deepa invests Rs. 50,000 for 1 year, so Deepa's capital time = 50,000 * 1 = 50,000.Step 2: For 2010, Deepa invests Rs. 70,000 for 1 year and Sushma invests Rs. 70,000 for 1 year, so Deepa contributes 70,000 and Sushma contributes 70,000 for that year.Step 3: For 2011, Deepa invests Rs. 90,000 for 1 year, Sushma invests Rs. 70,000 for 1 year and Avanthi invests Rs. 70,000 for 1 year.Step 4: Total capital time for Deepa = 50,000 + 70,000 + 90,000 = 2,10,000.Step 5: Total capital time for Sushma = 70,000 (2010) + 70,000 (2011) = 1,40,000.Step 6: Total capital time for Avanthi = 70,000 (2011) = 70,000.Step 7: The ratio Deepa : Sushma : Avanthi is 2,10,000 : 1,40,000 : 70,000 = 3 : 2 : 1.Step 8: Total ratio parts are 3 + 2 + 1 = 6. Each part of profit = 3,00,000 / 6 = Rs. 50,000.Step 9: Sushma gets 2 parts, so Sushma's share = 2 * 50,000 = Rs. 1,00,000. Verification / Alternative check: Check the shares: Deepa gets 3 * 50,000 = Rs. 1,50,000, Sushma gets 1,00,000 and Avanthi gets 50,000. Summing these gives 1,50,000 + 1,00,000 + 50,000 = Rs. 3,00,000, which matches the total profit. This confirms that the ratio and calculations are correct. Why Other Options Are Wrong: Rs. 75,000 and Rs. 85,000 do not correspond to any integer number of parts in the 3 : 2 : 1 ratio when the total profit is Rs. 3,00,000. Rs. 98,000 is close to 1,00,000 but does not match the exact computation. 'None of these' is incorrect because Rs. 1,00,000 is a valid option that satisfies all conditions precisely. Common Pitfalls: Many students treat the problem as if the capitals remain constant over the whole 3 years, ignoring the changes. Another error is to ignore Avanthi's late entry or to assume each year's profit is split separately. The correct method is to use capital time for the entire duration and then form a single ratio that accounts for every change in capital and membership. Final Answer: Sushma's correct share of the profit is Rs. 1,00,000.

Question 8

Ravi and Ramu are partners in a business. Ravi contributes one sixth of the total capital for 16 months. Ramu receives two thirds of the total profit. Assuming profit is shared in proportion to capital multiplied by time, for how many months was Ram…

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Introduction / Context: This question tests a deeper understanding of partnership problems where one partner's share of profit is given as a fraction of the total profit. We know the capital share and time for Ravi, and the fraction of total profit received by Ramu. Our goal is to use the proportionality of profit to capital time and find for how long Ramu's capital remained invested. Given Data / Assumptions: Ravi contributes one sixth of the total capital. Ravi's capital remains invested for 16 months. Ramu receives two thirds of the total profit. Total capital is shared between Ravi and Ramu. Profit shares are proportional to capital multiplied by time. Concept / Approach: If the total capital is C, Ravi's capital is C/6 and Ramu's capital is 5C/6. Profit is shared in the ratio of their capital time contributions. Since Ramu receives two thirds of the total profit, the ratio of Ramu's share to Ravi's share is 2 : 1. By equating this profit ratio to the ratio of their capital time products, we can solve for the unknown time period for which Ramu's money was used. Step-by-Step Solution: Step 1: Let total capital be C. Then Ravi's capital is C/6 and Ramu's capital is 5C/6.Step 2: Ravi's capital time = (C/6) * 16 = (16C) / 6.Step 3: Let t be the number of months Ramu's money was used. Then Ramu's capital time = (5C/6) * t = (5Ct) / 6.Step 4: Profit ratio Ramu : Ravi = (5Ct / 6) : (16C / 6) = 5t : 16.Step 5: Ramu receives two thirds of the total profit, while Ravi receives one third, so Ramu : Ravi = 2 : 1.Step 6: Equate the ratios: 5t : 16 = 2 : 1.Step 7: From 5t / 16 = 2 / 1, cross multiply to get 5t = 32.Step 8: Solve for t: t = 32 / 5 = 6.4 months. Verification / Alternative check: If t = 6.4 months, Ramu's capital time is (5C/6) * 6.4 = (32C) / 6 and Ravi's is (16C) / 6. The ratio is (32C / 6) : (16C / 6) = 32 : 16 = 2 : 1. Thus Ramu receives twice the share of Ravi, that is two thirds of the total profit, which matches the given condition. This confirms that t = 6.4 months is correct. Why Other Options Are Wrong: Time periods of 4 months, 5 months or 8 months do not satisfy the ratio 5t : 16 = 2 : 1. For example, if t = 8, the ratio becomes 40 : 16 = 5 : 2, which corresponds to Ramu receiving five sevenths of the profit, not two thirds. Therefore those values are inconsistent with the given fraction of profit. 'None of these' is not correct because one of the options, 6.4 months, satisfies the partnership equations exactly. Common Pitfalls: Many students assume the time period must be an integer and try to force one of the neat integer options instead of solving the equation properly. Others confuse the fraction of total profit (two thirds) with a direct ratio of capital amounts. Always set up the profit ratio correctly and equate it with the capital time ratio to obtain the correct time period, even if the answer is a decimal. Final Answer: The number of months for which Ramu's money was used is 6.4 months.

Question 9

A sum of Rs. 50,000 is divided into two parts. One part is lent at 10 percent simple interest per annum and the other part is lent at 20 percent simple interest per annum. At the end of one year, the total interest received is Rs. 7,000. How much mo…

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Introduction / Context: This is a straightforward simple interest and ratio type question. The total principal is split into two parts, each earning a different rate of simple interest. We are given the total interest earned after one year and must determine how much of the original sum was lent at the lower interest rate of 10 percent. Given Data / Assumptions: Total principal = Rs. 50,000. First part is lent at 10 percent simple interest per annum. Second part is lent at 20 percent simple interest per annum. Total interest received after 1 year = Rs. 7,000. Time period for both loans is the same, namely 1 year. Concept / Approach: Simple interest for one year is calculated as Principal * Rate * Time, where time is measured in years and rate is in decimal form. Because time is one year, interest is directly proportional to the principal and the rate. Let one part of the principal be x and the other part be (50,000 - x). We express the total interest as the sum of the interest on both parts and equate that to Rs. 7,000, then solve for x. Step-by-Step Solution: Step 1: Let the amount lent at 10 percent be x rupees.Step 2: Then the amount lent at 20 percent is 50,000 - x rupees.Step 3: Simple interest on x at 10 percent for 1 year = 0.10 * x = 0.1x.Step 4: Simple interest on (50,000 - x) at 20 percent for 1 year = 0.20 * (50,000 - x) = 10,000 - 0.2x.Step 5: Total interest after 1 year = 0.1x + (10,000 - 0.2x) = 10,000 - 0.1x.Step 6: The total interest is given as Rs. 7,000, so set 10,000 - 0.1x = 7,000.Step 7: Rearranging, -0.1x = 7,000 - 10,000 = -3,000.Step 8: Hence x = 3,000 / 0.1 = Rs. 30,000. Verification / Alternative check: If Rs. 30,000 is lent at 10 percent, it earns 3,000 interest in one year. The remaining amount is 50,000 - 30,000 = Rs. 20,000, lent at 20 percent, which earns 4,000 interest in one year. Total interest is 3,000 + 4,000 = Rs. 7,000, which exactly matches the given data. This verification confirms the correctness of the answer. Why Other Options Are Wrong: If Rs. 20,000 were lent at 10 percent, interest would be 2,000, and the remaining 30,000 at 20 percent would produce 6,000, giving a total of 8,000, not 7,000. If Rs. 10,000 were lent at 10 percent, total interest would be far smaller. If Rs. 40,000 were lent at 10 percent, then 10 percent on 40,000 is 4,000 and 20 percent on 10,000 is 2,000, giving only 6,000. Thus these values are inconsistent with the given interest of 7,000. 'None of these' is incorrect because Rs. 30,000 fits perfectly. Common Pitfalls: Some students mistakenly divide 7,000 in proportion to the interest rates without considering the constraint that the total principal is 50,000. Others may erroneously assume equal division. Always set up an equation based on the exact definition of simple interest and the given total interest. Final Answer: The amount of money lent at 10 percent interest is Rs. 30,000.

Question 10

In a partnership, the capitals of three partners P, Q and R are such that 6 times P's capital equals 8 times Q's capital and equals 12 times R's capital. Out of a total profit of Rs. 4,650, how much profit does R receive?

Accepted Answer

Introduction / Context: This question examines understanding of proportional relationships in partnership capital. Instead of providing the capitals directly, it relates them through multiple equations: 6 times P's capital equals 8 times Q's capital and 12 times R's capital. From this, we must determine the ratio of P, Q and R's investments and then the share of R in the total profit of Rs. 4,650. Given Data / Assumptions: 6 times the capital of P equals 8 times the capital of Q. 6 times the capital of P also equals 12 times the capital of R. Total profit to be divided among P, Q and R is Rs. 4,650. Profit sharing is directly proportional to the capital invested. Concept / Approach: When several products of capitals with constants are equal, we set them equal to a common variable and then derive each capital in terms of that variable. Once we have expressions for P, Q and R, we can obtain the ratio of their capitals. The share of R in the total profit is then equal to R's ratio share divided by the sum of all ratio shares, multiplied by the total profit. Step-by-Step Solution: Step 1: Let 6P = 8Q = 12R = k, where k is a common constant.Step 2: From 6P = k, we get P = k / 6.Step 3: From 8Q = k, we get Q = k / 8.Step 4: From 12R = k, we get R = k / 12.Step 5: The ratio P : Q : R is (k / 6) : (k / 8) : (k / 12).Step 6: Cancel k to obtain 1/6 : 1/8 : 1/12.Step 7: Express them with common denominator 24: P = 4/24, Q = 3/24, R = 2/24, giving P : Q : R = 4 : 3 : 2.Step 8: Total ratio parts = 4 + 3 + 2 = 9.Step 9: Share of R = (2 / 9) of total profit = (2 / 9) * 4,650.Step 10: Calculate (2 / 9) * 4,650 = 2 * 516.666... = approximately Rs. 1,033.33. Verification / Alternative check: We can compute all shares. One share (1 part) is 4,650 / 9 ≈ Rs. 516.67. Then P's share is 4 parts ≈ 4 * 516.67 = 2,066.67. Q's share is 3 parts ≈ 1,550.00 and R's share is 2 parts ≈ 1,033.33. Adding these, 2,066.67 + 1,550.00 + 1,033.33 ≈ 4,650, confirming that the ratio and calculations are consistent to two decimal places. Why Other Options Are Wrong: The other numerical options (Rs. 1,024, Rs. 1,148.14 and Rs. 1,245.32) do not correspond to 2/9 of 4,650. When plugged back into the ratio, they would not maintain the 4 : 3 : 2 balance among the three partners. 'None of these' is not correct because approximately Rs. 1,033.33 exactly matches the computed share of R. Common Pitfalls: A frequent mistake is to misinterpret 6P = 8Q = 12R as giving a ratio of 6 : 8 : 12 directly, which is actually the opposite of what is required. Another common error is to forget to convert fractions like 1/6, 1/8 and 1/12 into a clean integer ratio before applying the total profit. Always derive the actual capital ratio before dividing the profit. Final Answer: The share of R from the total profit is approximately Rs. 1,033.33.

Question 11

Umar and Avinash start a partnership with investments of Rs. 10,000 and Rs. 15,000 respectively. Due to a financial emergency, Avinash withdraws his entire investment after 8 months, while Umar keeps his capital invested for the full 12 months. In w…

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Introduction / Context: This is a partnership problem where partners invest different amounts for different durations. We must calculate the effective capital time for each partner and then determine the ratio in which the profit should be shared. Such questions are very common in aptitude tests and banking exams. Given Data / Assumptions: Umar invests Rs. 10,000 at the beginning and keeps it for the full 12 months. Avinash invests Rs. 15,000 at the beginning but withdraws his amount after 8 months. Profit is shared in proportion to capital multiplied by time. The period under consideration is 12 months (one year). Concept / Approach: When partners change their investments during the course of a business, we do not simply compare their capital amounts. Instead, we compare the product of capital and time, often referred to as money months. Once we calculate the capital time contributions, the ratio of these products gives the ratio of their profit shares. Step-by-Step Solution: Step 1: Umar's investment is Rs. 10,000 for 12 months.Step 2: Umar's capital time = 10,000 * 12 = 1,20,000 money months.Step 3: Avinash's investment is Rs. 15,000 for 8 months.Step 4: Avinash's capital time = 15,000 * 8 = 1,20,000 money months.Step 5: The ratio of Umar : Avinash in terms of capital time is 1,20,000 : 1,20,000.Step 6: Simplify this ratio by dividing both sides by 1,20,000 to get 1 : 1.Step 7: Therefore, the profit of the first year should be shared equally between the two partners. Verification / Alternative check: We can interpret in another way: Avinash invests 1.5 times the capital of Umar (15,000 versus 10,000) but only for two thirds of the time (8 months versus 12 months). Multiplying these factors, 1.5 * (8/12) = 1.5 * 2/3 = 1. This shows that his overall contribution is exactly equal to that of Umar, confirming the 1 : 1 profit sharing ratio. Why Other Options Are Wrong: Ratios 3 : 2, 2 : 3 and 1 : 2 suggest unequal contributions that would arise only if one partner had a higher weighted capital time. However, both partners contribute equally when capital and time are combined. 'None of these' is not correct because one of the offered ratios, 1 : 1, matches the computed profit ratio exactly. Common Pitfalls: Students often look only at capital amounts or only at time and assume that the partner who invested more money receives more profit. In this problem, ignoring time or capital would lead to incorrect conclusions. Always use capital multiplied by time to compare profit shares in such partnership questions. Final Answer: The profit for the first year should be shared between Umar and Avinash in the ratio 1 : 1.

Question 12

A sum of Rs. 2,489 is divided among A, B and C. If Rs. 12, Rs. 12 and Rs. 5 are subtracted from the shares of A, B and C respectively, then their resulting shares are in the ratio 5 : 3 : 4. What is the new share of C (that is, C's share after the d…

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Introduction / Context: This question is a ratio and distribution problem with a twist. The total amount is divided among three people A, B and C, but the ratio given applies not to their original shares but to the reduced amounts after specific deductions. We must reconstruct the original shares and then find the new share of C after the deduction of Rs. 5. Given Data / Assumptions: Total amount divided among A, B and C is Rs. 2,489. If Rs. 12 is subtracted from A's share, Rs. 12 from B's share and Rs. 5 from C's share, the resulting amounts are in the ratio 5 : 3 : 4. We are asked to find C's new share, which is C's amount after the deduction of Rs. 5. Concept / Approach: Let the reduced shares of A, B and C be 5k, 3k and 4k respectively. These are the amounts after the subtractions. Then we can express the original shares in terms of k and sum them up to match the given total of Rs. 2,489. From there, we solve for k and compute C's reduced share directly. This is a common technique in ratio problems involving increases or decreases. Step-by-Step Solution: Step 1: Let A's reduced share be 5k, B's reduced share be 3k and C's reduced share be 4k.Step 2: A's original share = 5k + 12, because Rs. 12 was subtracted to get 5k.Step 3: B's original share = 3k + 12.Step 4: C's original share = 4k + 5.Step 5: The sum of original shares is (5k + 12) + (3k + 12) + (4k + 5) = 12k + 29.Step 6: This sum equals the given total amount, so 12k + 29 = 2,489.Step 7: Subtract 29 from both sides: 12k = 2,489 - 29 = 2,460.Step 8: Solve for k: k = 2,460 / 12 = 205.Step 9: C's reduced share is 4k = 4 * 205 = Rs. 820. Verification / Alternative check: Compute the original shares using k = 205. A's original share = 5 * 205 + 12 = 1,037. B's original share = 3 * 205 + 12 = 627. C's original share = 4 * 205 + 5 = 825. Sum is 1,037 + 627 + 825 = Rs. 2,489, which matches the given total. After subtracting 12, 12 and 5, the shares become 1,025, 615 and 820. The ratio 1,025 : 615 : 820 simplifies to 5 : 3 : 4, confirming the correctness of the calculation. Why Other Options Are Wrong: Values such as 750, 1,060 and 1,475 do not fit either the derived value of 4k or the total sum when you reconstruct the original shares. Using any of these would break the 5 : 3 : 4 ratio or the total of Rs. 2,489. 'None of these' is not correct because Rs. 820 is exactly the reduced share of C derived from the given conditions. Common Pitfalls: One common mistake is to directly apply the ratio 5 : 3 : 4 to the total amount 2,489 without accounting for the subtractions. Another is to mistakenly interpret 5k, 3k and 4k as the original shares instead of the reduced ones. Always carefully note whether the ratio applies before or after an adjustment to the amounts. Final Answer: The new share of C after the deduction is Rs. 820.

Question 13

A and B invest in a business in the ratio 2 : 5. Fifty percent of the total profit is donated to charity, and the remaining 50 percent is shared between A and B according to their investment ratio. If A's share of the profit is Rs. 3.6 lakhs, what i…

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Introduction / Context: This partnership question involves both a charitable deduction and a profit division based on an investment ratio. A portion of the total profit goes to charity, and the rest is divided between two partners in a specified ratio. We are given A's final share and must find the original total profit of the business in lakhs. Given Data / Assumptions: Investment ratio of A : B is 2 : 5. Fifty percent of the total profit goes to charity. The remaining fifty percent is divided between A and B according to the ratio 2 : 5. A's share of profit is Rs. 3.6 lakhs. We are asked for the total profit in lakhs. Concept / Approach: Let the total profit be P lakhs. Then 50 percent of P, that is P/2, goes to charity. The remaining P/2 is shared by A and B in the ratio 2 : 5. A's share is therefore a specific fraction of the total profit. By equating that fraction to the given 3.6 lakhs, we can solve for P. This method uses proportional reasoning and basic fraction operations. Step-by-Step Solution: Step 1: Let total profit be P lakhs.Step 2: Half of the profit, P/2, is donated to charity.Step 3: The remaining profit P/2 is shared between A and B in the ratio 2 : 5.Step 4: Total ratio parts for A and B are 2 + 5 = 7.Step 5: A's share from the remaining profit is (2/7) of P/2.Step 6: So A's share = (2/7) * (P/2) = (1/7) * P.Step 7: We are told A's share is 3.6 lakhs, so (1/7) * P = 3.6.Step 8: Multiply both sides by 7 to get P = 3.6 * 7 = 25.2 lakhs. Verification / Alternative check: If total profit is 25.2 lakhs, charity receives half, that is 12.6 lakhs. The remaining 12.6 lakhs is shared between A and B in the ratio 2 : 5. A's share is (2/7) * 12.6 = 3.6 lakhs, which matches the value given in the question. This confirms that the computed total profit is consistent with all conditions. Why Other Options Are Wrong: If total profit were 12.6 lakhs, A's share would be only (1/7) * 12.6 = 1.8 lakhs. If it were 37.8 lakhs, A's share would be 37.8 / 7 ≈ 5.4 lakhs. For 16.8 lakhs, A's share would be 2.4 lakhs. None of these values match the given share of 3.6 lakhs, so they are incorrect. 'None of these' is not correct because 25.2 lakhs is exactly the required total profit. Common Pitfalls: Some candidates forget that A receives his portion only from the remaining half of the profit after charity, not from the entire profit. Others mistakenly divide A's share by 2 or 5 directly without forming the correct fraction. Carefully identifying the portion of the total profit that is available for sharing is crucial in such questions. Final Answer: The total profit earned by the business is 25.2 lakhs.

Question 14

Ram and Raj start a business with investments of Rs. 2,500 and Rs. 2,250 respectively. After 2 months, Ram withdraws Rs. 1,250 from his capital. After 1 more month, Raj withdraws Rs. 750 from his capital, and at that time Rakesh joins them with Rs. …

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Introduction / Context: This is a multi stage partnership question where partners change their capital contributions at different times and a new partner joins later. To determine the correct profit share for each person, we must calculate the capital time (money months) for each segment of the 6 month period and then form a ratio based on those capital time totals. Given Data / Assumptions: Ram invests Rs. 2,500 initially. Raj invests Rs. 2,250 initially. After 2 months, Ram reduces his investment by Rs. 1,250, so his new investment becomes Rs. 1,250. After 3 months from the start (that is, 1 month later), Raj withdraws Rs. 750, leaving Rs. 1,500 invested. At the same time (end of 3rd month), Rakesh joins with Rs. 3,500. The business runs for a total of 6 months. Total profit at the end is Rs. 2,540. Concept / Approach: When capital changes during the business period, we cannot use a single static ratio of capitals. Instead, we divide the total time into segments where capital remains constant for each partner, compute capital time for every partner in each segment and then sum these to obtain their overall contribution. The profit is shared in the ratio of these overall capital time values. Step-by-Step Solution: Step 1: From month 0 to month 2 (2 months), Ram has Rs. 2,500 and Raj has Rs. 2,250. Rakesh has not yet joined.Step 2: Ram's capital time for this segment = 2,500 * 2 = 5,000. Raj's capital time = 2,250 * 2 = 4,500.Step 3: From month 2 to month 3 (1 month), Ram reduces to Rs. 1,250, while Raj remains at Rs. 2,250. Rakesh has not joined yet.Step 4: In this segment, Ram's capital time = 1,250 * 1 = 1,250; Raj's capital time = 2,250 * 1 = 2,250.Step 5: From month 3 to month 6 (3 months), Ram has Rs. 1,250, Raj has Rs. 1,500, and Rakesh has Rs. 3,500.Step 6: In this segment, Ram's capital time = 1,250 * 3 = 3,750; Raj's capital time = 1,500 * 3 = 4,500; Rakesh's capital time = 3,500 * 3 = 10,500.Step 7: Total capital time for Ram = 5,000 + 1,250 + 3,750 = 10,000.Step 8: Total capital time for Raj = 4,500 + 2,250 + 4,500 = 11,250.Step 9: Total capital time for Rakesh = 10,500.Step 10: Ratio Ram : Raj : Rakesh = 10,000 : 11,250 : 10,500.Step 11: Divide all by 250 to simplify: 40 : 45 : 42.Step 12: Total ratio parts = 40 + 45 + 42 = 127.Step 13: Each part of profit = 2,540 / 127 = Rs. 20.Step 14: Ram's share = 40 * 20 = Rs. 800; Raj's share = 45 * 20 = Rs. 900; Rakesh's share = 42 * 20 = Rs. 840. Verification / Alternative check: Add the three computed shares: 800 + 900 + 840 = Rs. 2,540, which matches the given total profit. This confirms that the capital time ratio and the final distribution of profit are consistent and correctly calculated. Why Other Options Are Wrong: The distributions in the other options do not respect the capital time ratio 40 : 45 : 42 when converted back to parts of 20 rupees. For example, any option that changes Rakesh's share to 940 or swaps Ram and Raj's amounts will no longer add to Rs. 2,540 in the correct ratio. 'None of these' is incorrect because option with Ram - Rs. 800, Raj - Rs. 900, Rakesh - Rs. 840 perfectly matches the ratio derived from the capital time analysis. Common Pitfalls: Candidates often forget to segment the timeline correctly and may assume fixed capitals throughout the six months. Another frequent mistake is to ignore the effect of late entry by Rakesh or to average capitals instead of using capital time. Always break the problem into time segments and compute money months for each partner in each segment. Final Answer: The correct distribution of profit is Ram - Rs. 800, Raj - Rs. 900, Rakesh - Rs. 840.

Question 15

In a joint business, P and Q share profit according to their capital time contributions. What is the difference between the profit shares of P and Q at the end of one year? (a) P invests Rs. 80,000 and withdraws Rs. 20,000 after 6 months. (b) Q join…

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Introduction / Context: This is a data sufficiency question involving capital time in a partnership. Instead of asking for the actual numerical difference between the profits of P and Q, the question focuses on whether the given statements provide enough information to determine that difference. Understanding what information is essential and what is missing is the key to solving such questions. Given Data / Assumptions: We want the difference between the profit shares of P and Q at the end of one year. Statement (a): P invests Rs. 80,000 and withdraws Rs. 20,000 after 6 months (so his last six months capital is Rs. 60,000). Statement (b): Q joins the business 4 months after the start. Statement (c): During the last 6 months, Q's capital is 80 percent of P's capital. Profit is proportional to capital time. Concept / Approach: To determine the exact difference in profit between P and Q in rupees, we must know both the ratio of their capital time contributions and the total profit amount. The given statements provide some information about their capital time contributions but not the total profit. We will analyze each statement and combination to see if they allow computation of the monetary difference between their shares. Step-by-Step Solution: Step 1: From statement (a), P's capital is Rs. 80,000 for the first 6 months and Rs. 60,000 for the last 6 months.Step 2: This allows us to compute P's capital time but provides no information about Q's capital or the total profit.Step 3: From statement (b), we only know that Q joins after 4 months, so Q is in the business for 8 months.Step 4: Statement (b) alone does not give Q's capital amount or any relation to P's capital.Step 5: From statement (c), we know that in the last 6 months, Q's capital is 80 percent of P's capital, meaning Q's capital is 0.8 times P's capital for that period.Step 6: Combining (a), (b) and (c), we can determine the ratio of the capital time contributions of P and Q, and thus the ratio of their profit shares.Step 7: However, we still do not know the total profit of the business, so while the ratio of their shares can be computed, the absolute difference in rupees cannot be determined.Step 8: Therefore, the data is insufficient to find a unique numerical value for the difference between their profit shares. Verification / Alternative check: Even if we assume a total profit of T rupees and compute P : Q ratio from the given information, the difference between their shares will be some fraction of T. Without a specific value for T, the difference in rupees remains unknown and could vary with any change in total profit. This confirms that the given statements, even taken together, do not uniquely determine the required difference. Why Other Options Are Wrong: 'Only a' and 'Only c' are incorrect because neither provides enough information about Q's investment duration and the total profit. 'Both a and b' is insufficient because we still lack the exact value of Q's capital or total profit. 'None of these' is not correct since one of the listed options correctly describes the situation: the data is not sufficient. Common Pitfalls: Many learners confuse knowing the ratio of profit shares with knowing the numerical difference in rupees. Another frequent mistake is to assume some arbitrary total profit to compute a difference, which violates the requirement of uniqueness in data sufficiency questions. Always check whether the total profit or an equivalent piece of information is available before concluding that the difference can be determined. Final Answer: The correct conclusion is that the data is not sufficient to determine the numeric difference in profit shares between P and Q.

Question 16

Two persons A and B invest in a business with capitals of Rs. 1,15,000 and Rs. 75,000 respectively. They agree that 40 percent of the total profit will be divided equally between them, and the remaining 60 percent will be divided in the ratio of the…

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Introduction / Context: This question combines partnership ratios with a special profit sharing agreement. The partners first share a fixed percentage of the profit equally, and then distribute the remaining part in proportion to their capital investments. We are given the difference in the amounts received by A and B and asked to find the total profit. This is a good test of algebraic modeling of profit distribution. Given Data / Assumptions: A invests Rs. 1,15,000. B invests Rs. 75,000. Forty percent of the total profit is divided equally between A and B. Sixty percent of the total profit is divided in the ratio of their investments. A receives Rs. 500 more than B in total. We need to find the total profit. Concept / Approach: Let the total profit be P rupees. Then 0.4P is shared equally, and 0.6P is shared in the ratio of their investments. We first determine the ratio of investments and then express A's and B's shares from the second portion. The total amount for each partner is the sum of their equal share and their variable share. Setting the difference equal to Rs. 500, we can solve for P. Step-by-Step Solution: Step 1: Let total profit be P rupees.Step 2: Equal share portion is 40 percent of P, that is 0.4P.Step 3: Each of A and B receives half of 0.4P, so each gets 0.2P from the equal share portion.Step 4: Remaining profit is 0.6P, which is distributed in the ratio of investments 1,15,000 : 75,000.Step 5: Simplify the ratio by dividing by 5,000 to get 23 : 15.Step 6: A's variable share from this portion = 0.6P * 23 / 38.Step 7: B's variable share from this portion = 0.6P * 15 / 38.Step 8: Total share of A = 0.2P + 0.6P * 23 / 38.Step 9: Total share of B = 0.2P + 0.6P * 15 / 38.Step 10: Difference between A and B's shares = 0.6P * (23 - 15) / 38 = 0.6P * 8 / 38.Step 11: Simplify: 0.6P * 8 / 38 = (0.6 * 8 / 38)P = (4.8 / 38)P = (2.4 / 19)P.Step 12: We are told this difference is Rs. 500, so (2.4 / 19)P = 500.Step 13: Solve for P: P = 500 * 19 / 2.4 ≈ 3,958.33 rupees. Verification / Alternative check: To verify, take total profit P ≈ 3,958.33. Equal share portion is 0.4P ≈ 1,583.33, so each partner gets about 791.67 from this part. Remaining 0.6P ≈ 2,375 is divided in the ratio 23 : 15, so A gets 2,375 * 23 / 38 ≈ 1,437.50 and B gets 937.50. So total shares are approximately A = 791.67 + 1,437.50 = 2,229.17 and B = 791.67 + 937.50 = 1,729.17. The difference is very close to Rs. 500, confirming the computation. Rounded to two decimal places, the profit is best represented by 3,958.34 among the given options. Why Other Options Are Wrong: The other numerical options (3,599.34, 699.34 and 999.34) either are far too small or do not produce a difference of Rs. 500 between A and B when the distribution rules are applied. 'None of these' is incorrect because 3,958.34 matches the computed total profit to two decimal places. Common Pitfalls: Students sometimes wrongly assume that the entire profit is shared in proportion to investments and ignore the equal share portion, or they forget to halve the 40 percent when giving each partner's equal share. Another common mistake is miscomputing the ratio of investments. Always carefully distinguish between the equal portion and the proportional portion of the profit. Final Answer: The total profit of the business is approximately Rs. 3,958.34.

Question 17

Three brothers divide Rs. 1,620 among themselves in such a way that the share of the second brother is equal to 5/13 of the combined shares of the first and third brothers. What is the share of the second brother in rupees?

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Introduction / Context: This is a ratio and sharing problem in which the share of one person is given as a specific fraction of the combined shares of the other two. The total amount is known, and we are asked to find the exact share of the second brother. This is a standard problem that reinforces algebraic modeling of relationships between quantities. Given Data / Assumptions: Total amount shared among the three brothers is Rs. 1,620. Let the shares of the first, second and third brothers be A, B and C respectively. The share of the second brother is 5/13 of the sum of the shares of the first and third brothers. We have to determine B, the share of the second brother. Concept / Approach: We can represent the relationship B = (5/13)(A + C). Then we know that the total sum A + B + C equals Rs. 1,620. By expressing A + C in terms of B, we can substitute into the total and solve for B directly. This approach is efficient and avoids unnecessary introduction of extra variables. Step-by-Step Solution: Step 1: Let the shares of the three brothers be A, B and C.Step 2: Given that B = (5/13)(A + C).Step 3: From this, we get A + C = (13/5)B.Step 4: Total amount shared is A + B + C = 1,620.Step 5: Substitute A + C = (13/5)B into the total: (A + C) + B = (13/5)B + B.Step 6: Simplify: (13/5)B + B = (13/5)B + (5/5)B = (18/5)B.Step 7: Thus (18/5)B = 1,620.Step 8: Multiply both sides by 5: 18B = 1,620 * 5 = 8,100.Step 9: Divide by 18: B = 8,100 / 18 = Rs. 450. Verification / Alternative check: We can perform a quick check by computing A + C = (13/5)B = (13/5) * 450 = 13 * 90 = 1,170. Then A + B + C = (A + C) + B = 1,170 + 450 = Rs. 1,620, which matches the given total. The relation B = (5/13)(A + C) also holds because (5/13) * 1,170 = 5 * 90 = 450, which is exactly B. Why Other Options Are Wrong: Rs. 1,170 is actually the combined share of the first and third brothers, not the share of the second. Rs. 540 and Rs. 500 do not satisfy the relation B = (5/13)(A + C) when recomputed with the total of Rs. 1,620. 'None of these' is not correct because Rs. 450 perfectly satisfies all given conditions. Common Pitfalls: Some candidates misread the relationship and assume B is 5/13 of the total amount rather than of the sum of the other two shares. Others try to assign arbitrary values to A and C without properly using the equation. Always express the relationship algebraically and use the given total to solve for the required share. Final Answer: The share of the second brother is Rs. 450.

Question 18

Rs. 6,300 is divided among A, B and C in the ratio 1/2 : 1 : 3/5. If these fractional terms represent their proportional shares, what is the share of B in rupees?

Accepted Answer

Introduction / Context: This problem deals with distributing a fixed sum according to a ratio that is given in fractional form. Instead of simple integers, the shares of A, B and C are in the ratio 1/2 : 1 : 3/5. We must convert these fractions into a simple integer ratio and then use it to find B's share from the total amount of Rs. 6,300. Given Data / Assumptions: Total amount to be divided is Rs. 6,300. Ratio of shares of A : B : C is 1/2 : 1 : 3/5. We need to find the share of B. Concept / Approach: When a ratio is given in fractions, the first step is to convert them to a common denominator and express the ratio as whole numbers. Then the usual method of ratio distribution is applied: we sum the ratio parts, find how much money each part represents and multiply by the number of parts corresponding to B. This process ensures that the fractions are correctly handled. Step-by-Step Solution: Step 1: The given ratio is A : B : C = 1/2 : 1 : 3/5.Step 2: Convert all fractions to a common denominator, which is 10.Step 3: 1/2 = 5/10, 1 = 10/10, and 3/5 = 6/10.Step 4: The ratio in terms of numerators is A : B : C = 5 : 10 : 6.Step 5: Sum of the ratio parts = 5 + 10 + 6 = 21.Step 6: Each ratio part corresponds to 6,300 / 21 = 300 rupees.Step 7: B has 10 parts, so B's share = 10 * 300 = Rs. 3,000. Verification / Alternative check: We can verify by computing all shares. A's share = 5 * 300 = Rs. 1,500. B's share = 3,000. C's share = 6 * 300 = Rs. 1,800. Adding them up gives 1,500 + 3,000 + 1,800 = Rs. 6,300, which matches the total. The ratio 1,500 : 3,000 : 1,800 simplifies to 1/2 : 1 : 3/5 when expressed back as fractions relative to a common value, confirming the correctness of the distribution. Why Other Options Are Wrong: If B's share were 3,300, the remaining 3,000 would not fit the 5 : 10 : 6 ratio. Values like 2,700 and 4,200 also lead to inconsistent ratios when the remaining amounts are checked. 'None of these' is incorrect because Rs. 3,000 matches the required ratio and total sum perfectly. Common Pitfalls: Some learners directly interpret 1/2 : 1 : 3/5 as 1 : 2 : 3 without adjusting for the denominators, which is wrong. Others forget to divide the total by the sum of ratio parts and instead multiply incorrectly. Always convert fraction ratios to whole number ratios using a common denominator, then apply the standard ratio distribution method. Final Answer: The share of B from the total amount is Rs. 3,000.

Question 19

A sum of Rs. 3,170 is divided among X, Y and Z. If Rs. 13, Rs. 12 and Rs. 18 are subtracted from the shares of X, Y and Z respectively, then their resulting shares are in the ratio 20 : 18 : 21. What is the initial share of Z in rupees (that is, bef…

Accepted Answer

Introduction / Context: This is another ratio and distribution problem involving adjusted shares. The total amount is fixed, but the ratio provided applies to the amounts after specific deductions from each person's share. We need to use the given ratio of the adjusted amounts to reconstruct the original shares and then find Z's initial share before the deduction of Rs. 18. Given Data / Assumptions: Total amount divided among X, Y and Z is Rs. 3,170. When Rs. 13, Rs. 12 and Rs. 18 are subtracted from X, Y and Z respectively, their resulting shares are in the ratio 20 : 18 : 21. We are asked to find Z's original share before the deduction. Concept / Approach: We let the adjusted shares of X, Y and Z be 20k, 18k and 21k respectively. Then we add back the deducted amounts to express the original shares in terms of k. The sum of these original shares must equal the total amount given, 3,170. Solving for k gives us the adjusted and original shares, including Z's initial share. Step-by-Step Solution: Step 1: Let X's reduced share be 20k, Y's reduced share be 18k and Z's reduced share be 21k.Step 2: Then X's original share is 20k + 13.Step 3: Y's original share is 18k + 12.Step 4: Z's original share is 21k + 18.Step 5: The sum of original shares equals 3,170, so (20k + 13) + (18k + 12) + (21k + 18) = 3,170.Step 6: Combine like terms: 20k + 18k + 21k = 59k, and 13 + 12 + 18 = 43, so 59k + 43 = 3,170.Step 7: Subtract 43 from both sides to get 59k = 3,170 - 43 = 3,127.Step 8: Solve for k: k = 3,127 / 59 = 53.Step 9: Z's original share = 21k + 18 = 21 * 53 + 18 = 1,113 + 18 = Rs. 1,131. Verification / Alternative check: With k = 53, we can compute the original shares. X gets 20 * 53 + 13 = 1,060 + 13 = 1,073. Y gets 18 * 53 + 12 = 954 + 12 = 966. Z gets 1,131 as calculated. Summing these, 1,073 + 966 + 1,131 = Rs. 3,170, which matches the given total. The reduced shares are 1,073 - 13 = 1,060, 966 - 12 = 954 and 1,131 - 18 = 1,113, which are in ratio 20 : 18 : 21 when divided by 53. This confirms that our solution is consistent. Why Other Options Are Wrong: Values such as 1,530, 910 and 1,350 do not satisfy the equation when substituted back into the distribution. They lead to incorrect totals or ratios that do not match 20 : 18 : 21 after adjusting for the deductions. 'None of these' is incorrect because 1,131 fits all given conditions precisely. Common Pitfalls: One frequent mistake is to apply the ratio 20 : 18 : 21 directly to 3,170 without considering the subtractions, which leads to wrong answers. Another error is to forget to include all three deduction amounts when forming the total equation. Always define the post deduction amounts in terms of a common variable and then add back the deducted values before equating to the total. Final Answer: The initial share of Z is Rs. 1,131.

Question 20

Mohit and Anshu invest some money in a business in the ratio 8 : 15 respectively. After 9 months Mohit withdraws his capital. If the profits are finally received in the ratio 4 : 5 (Mohit : Anshu), for how many months did Anshu invest in the busines…

Accepted Answer

Introduction / Context: This question deals with a partnership where one partner withdraws his capital after a fixed time while the other partner keeps his investment for a different period. We know the ratio of their investments and the final profit ratio. Our goal is to determine the time period for which Anshu kept his capital invested, using the concept of capital time in partnership problems. Given Data / Assumptions: Mohit and Anshu invest in the ratio 8 : 15. Mohit withdraws his capital after 9 months. Profits are received in the ratio 4 : 5 (Mohit : Anshu). Profit is proportional to capital multiplied by time. We are asked to find the number of months Anshu's capital remained invested. Concept / Approach: Let Mohit's and Anshu's capitals be 8 units and 15 units respectively. Mohit's investment is for 9 months, while Anshu's is for t months. Profit shares will then be in the ratio 8 * 9 : 15 * t. This ratio is given to be 4 : 5. By equating these two ratios, we can solve for t, the time period for which Anshu invested his money. Step-by-Step Solution: Step 1: Assume Mohit's capital is 8 units and Anshu's capital is 15 units.Step 2: Mohit keeps his capital for 9 months, so his capital time is 8 * 9 = 72 unit months.Step 3: Let Anshu's time period be t months, so his capital time is 15 * t unit months.Step 4: Profit sharing ratio Mohit : Anshu = 72 : 15t.Step 5: Given profit ratio is 4 : 5, so 72 : 15t = 4 : 5.Step 6: Write as 72 / (15t) = 4 / 5.Step 7: Cross multiply: 72 * 5 = 4 * 15t.Step 8: This gives 360 = 60t.Step 9: Solve for t: t = 360 / 60 = 6 months. Verification / Alternative check: If Anshu invests for 6 months, his capital time is 15 * 6 = 90 unit months. Mohit's capital time is 72 unit months as calculated. The profit ratio Mohit : Anshu is then 72 : 90, which simplifies to 4 : 5 when both numbers are divided by 18. This exactly matches the given profit ratio, confirming that t = 6 months is correct. Why Other Options Are Wrong: If Anshu invested for 4 or 5 months, his capital time would be 15 * 4 = 60 or 15 * 5 = 75. Ratios 72 : 60 and 72 : 75 simplify to 6 : 5 and 24 : 25 respectively, not 4 : 5. If he invested for 8 months, the ratio would be 72 : 120 = 3 : 5, which is again inconsistent with the given 4 : 5. 'None of these' is not correct because 6 is one of the options and satisfies all conditions. Common Pitfalls: Some candidates directly compare the investment ratio 8 : 15 with the profit ratio 4 : 5 without accounting for time. Others may incorrectly assume that the time period for Anshu must be more than 9 months because Mohit withdrew, without focusing on the given profit ratio. Always use the formula profit share ratio = capital1 * time1 : capital2 * time2 and solve the resulting equation accurately. Final Answer: Anshu kept his capital invested in the business for 6 months.

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