Question 1

Among five friends A, B, C, D and E, who is the tallest? Statement I: D is taller than A and C. Statement II: B is shorter than E but taller than D.

Accepted Answer

Introduction / Context: This data sufficiency problem concerns comparative heights among five friends. The question asks you to identify the tallest person among A, B, C, D, and E. Two statements describe relative height relationships. The challenge is to see whether either statement alone, or both together, provide enough information to determine who is tallest in the group. Given Data / Assumptions: - The friends are A, B, C, D and E. - Question: Who is the tallest among them? - Statement I: D is taller than A and C. - Statement II: B is shorter than E but taller than D. - No one else beyond A, B, C, D and E is being considered when we say "tallest". Concept / Approach: To solve ordering problems like this, we build partial orders of individuals based on given comparisons. We start with one statement, derive as much ordering as possible, then do the same for the second statement. If neither statement alone yields a complete picture, we combine the information from both to see whether we can now identify a unique tallest person. The key is to keep track of all inequalities and see whether any person is known to be taller than all others. Step-by-Step Solution: Step 1: Use statement I alone. It tells us D is taller than A and C, so D > A and D > C. We still know nothing about B and E relative to D, A, or C. Therefore, we cannot decide who is tallest among all five friends. Statement I alone is not sufficient. Step 2: Use statement II alone. It states that B is shorter than E but taller than D, so E > B > D. However, we still know nothing about where A and C stand in comparison to E or B. So, statement II alone is also not sufficient. Step 3: Combine both statements. From statement I we have D > A and D > C. From statement II we have E > B > D. Step 4: Joining these, we get a chain: E > B > D > A and D > C. That means E is taller than B, D, A, and C. There is no information about anyone taller than E. Step 5: Therefore E is taller than every other friend in the group. So E is the tallest. Verification / Alternative check: We can quickly test whether any other person could possibly be tallest while still satisfying the statements. A or C cannot be taller, because statement I explicitly says that D is taller than both A and C, and statement II says B is taller than D, and E is taller than B. That already places A and C strictly below D, B and E. B cannot be tallest because E is stated to be taller than B. D cannot be tallest because B is taller than D. Thus E is the only candidate left, and the conclusion is unique. Why Other Options Are Wrong: - Option a is wrong because statement I alone leaves B and E completely unranked relative to D, A, and C, so we cannot find the tallest. - Option b is wrong because statement II alone does not locate A and C in the order, again preventing a unique answer. - Option c is incorrect because neither statement by itself is sufficient; both are required together. - Option e is wrong because we have shown that both statements combined clearly identify E as the tallest friend. Common Pitfalls: Some students prematurely conclude that E is tallest using statement II alone, forgetting that A or C might still be taller in the absence of more information. Others do not fully integrate the inequalities when combining the statements and therefore underestimate what can be concluded. Always write down or mentally picture the complete order implied by each statement and then merge them systematically. Final Answer: Only when both statements are used together can we conclude that E is the tallest among the five friends. Thus, the correct data sufficiency choice is option D.

Question 2

In which year was Rahul born? Statement I: At present Rahul is 25 years younger than his mother. Statement II: Rahul's brother, who was born in 1964, is 35 years younger than his mother.

Accepted Answer

Introduction / Context: This data sufficiency question concerns ages and years of birth. The goal is to determine the exact year in which Rahul was born. Two statements give information about age differences between Rahul, his brother, and their mother, including a known year of birth for the brother. You must decide whether either statement alone, or both together, are enough to determine Rahul's birth year. Given Data / Assumptions: - Question: In which year was Rahul born? - Statement I: Rahul is at present 25 years younger than his mother. - Statement II: Rahul's brother was born in 1964 and is 35 years younger than his mother. - Age differences (gaps) between family members remain constant over time. - Current year is not explicitly given; we only use relative age differences and the brother's birth year. Concept / Approach: We can interpret the information in terms of year of birth. If someone is k years younger than another person, then the younger person's birth year equals the older person's birth year plus k. Statement II gives us the mother's birth year directly because we know the brother's birth year and how much younger he is than his mother. Then we can use this to work out Rahul's birth year if we also know Rahul's age gap with his mother from statement I. Step-by-Step Solution: Step 1: Analyze statement I alone. It says Rahul is 25 years younger than his mother. This gives only the age difference and does not reveal any specific calendar year. Without knowing either the current year or the year of birth of the mother, we cannot determine Rahul's birth year. So statement I alone is not sufficient. Step 2: Analyze statement II alone. It says Rahul's brother was born in 1964 and is 35 years younger than his mother. Therefore, mother's birth year = 1964 - 35 = 1929. Step 3: Statement II alone gives us the mother's birth year but does not say how much younger Rahul is than his mother. Without that gap, we cannot compute Rahul's birth year. So statement II alone is also not sufficient. Step 4: Combine both statements. From statement II, mother's birth year is 1929. From statement I, Rahul is 25 years younger than his mother, so Rahul's birth year = 1929 + 25 = 1954. Step 5: With both age gaps and one known birth year, we can now determine a unique birth year for Rahul. Verification / Alternative check: We can verify the consistency of the numbers. If the mother was born in 1929, then the brother born in 1964 is indeed 35 years younger (1964 - 1929 = 35). Rahul, being 25 years younger than the mother, must be born in 1929 + 25 = 1954, and the gap between the brother and Rahul is 10 years, which is plausible. There is no contradiction, and the data give a unique year of birth. Why Other Options Are Wrong: - Option a is wrong because statement I alone does not yield any year, only a difference in ages. - Option b is wrong because statement II alone only tells us the mother's birth year but nothing specific about Rahul's gap. - Option c is incorrect because neither statement alone is sufficient; they must be used together. - Option e is wrong because, when combined, the two statements clearly allow us to calculate Rahul's birth year, so the data are sufficient. Common Pitfalls: A typical error is to misread the phrase "years younger" and try to subtract instead of adding when finding a younger person's birth year. Another pitfall is to assume that knowing one child's year of birth automatically gives the other's year, even when the age difference between the two children is not specified. In data sufficiency, you must track exactly which differences are known and only combine them when logically justified. Final Answer: Both statements I and II together are necessary to find that Rahul was born in 1954, so the correct data sufficiency choice is option D.

Question 3

In what proportion should Raj, Karan and Altaf distribute the profit among themselves? Statement I: Raj receives two fifths of the total profit. Statement II: Karan and Altaf together have contributed 75% of the total investment.

Accepted Answer

Introduction / Context: This is a partnership style data sufficiency question. Raj, Karan and Altaf share some profit, and we are asked to find the proportion in which this profit is distributed among them. Two statements give partial information about Raj's share and about the combined investment shares of Karan and Altaf. The goal is to decide whether this information is sufficient to determine a unique profit sharing ratio Raj : Karan : Altaf. Given Data / Assumptions: - Question: In what proportion would Raj, Karan and Altaf distribute the profit? - Statement I: Raj gets two fifths of the total profit. - Statement II: Karan and Altaf together have made 75% of the total investment. - Usual partnership assumption: if not stated otherwise, profit shares are typically proportional to investments, but here no direct explicit statement links the two in the given text. Concept / Approach: To determine the exact profit sharing ratio among three people, we need either direct information about each person's share of profit or a clear connection between profit and some other quantity (such as investment), along with the proportions of that quantity for each person. Data sufficiency analysis checks whether the statements provide enough information to obtain a unique triplet ratio Raj : Karan : Altaf. If there remain multiple consistent distributions, then the data are not sufficient. Step-by-Step Solution: Step 1: Use statement I alone. It tells us that Raj receives two fifths of the total profit, so Raj's profit share is 2/5, and the combined share of Karan and Altaf is 3/5. However, this does not tell us how that 3/5 is split between Karan and Altaf. Many different distributions such as Raj : Karan : Altaf = 2 : 1 : 2 or 2 : 2 : 1 would satisfy the condition that Raj has 2/5 of total profit. Thus, statement I alone is not sufficient. Step 2: Use statement II alone. It says that Karan and Altaf together contributed 75% of the total investment, so Raj contributed 25%. This allows us to express the investment proportions as 25% for Raj and 75% for Karan plus Altaf combined, but it does not state how that 75% is divided between Karan and Altaf, nor does it explicitly state that profit is distributed in proportion to investment. So statement II alone is not sufficient to determine the exact profit distribution. Step 3: Combine statements I and II. With both statements, we know Raj's investment share (25%) and Raj's profit share (2/5 = 40%). This suggests that Raj's profit share is not necessarily directly proportional to his investment share, and we still do not know how Karan and Altaf split either the investment or the profit between themselves. Step 4: Even if we assumed that profit and investment are related, we still lack any information that uniquely fixes the ratio between Karan and Altaf. Many combinations of Karan and Altaf shares can satisfy both the total investment constraint and the total profit constraint while changing the internal split between these two partners. Verification / Alternative check: Try constructing a concrete example. Suppose total profit is 1 unit. From statement I, Raj gets 2/5, so Karan plus Altaf together get 3/5. Now suppose total investment is also 1 unit. Statement II says Raj invests 1/4 and Karan plus Altaf together invest 3/4. You can choose infinitely many ways to split the 3/5 profit between Karan and Altaf while still keeping their combined investment at 3/4. This shows that the internal ratio Karan : Altaf is not determined uniquely, so the overall profit sharing ratio among all three remains undetermined. Why Other Options Are Wrong: - Option a is wrong because statement I alone does not fix how Karan and Altaf split the profit. - Option b is wrong since statement II alone only gives combined investment share, not the profit distribution, and not the internal ratio between Karan and Altaf. - Option c is incorrect because neither statement alone is sufficient; both leave multiple possibilities. - Option d is wrong because even combining both statements still leaves us with more than one possible profit distribution ratio among the three partners. Common Pitfalls: A frequent mistake is to assume that knowing Raj's profit share and the combined investment share of Karan and Altaf automatically fixes all three profit shares. Another pitfall is to assume without careful thought that profit must always be proportional to investment, and that this somehow determines internal splits without additional data. In data sufficiency questions, you should avoid introducing extra assumptions and focus strictly on what can be logically inferred from the given statements. Final Answer: Even when both statements are considered together, the exact profit sharing ratio among Raj, Karan and Altaf cannot be determined uniquely. So the correct data sufficiency choice is option E.

Question 4

What will be the total weight of 10 poles if each pole has the same weight? Statement I: One fourth of the weight of a pole is 5 kilograms. Statement II: The total weight of three poles is 20 kilograms more than the total weight of two poles.

Accepted Answer

Introduction / Context: This is a straightforward arithmetic data sufficiency question involving equal weights. You are asked to find the total weight of 10 identical poles. Two statements provide different ways of expressing the weight of a single pole. The primary goal is to determine whether one or both of the statements are sufficient to compute the total weight, not to perform unnecessary extra calculations. Given Data / Assumptions: - All 10 poles have the same weight. - Question: What is the total weight of 10 such poles? - Statement I: One fourth of the weight of a pole is 5 kilograms. - Statement II: The total weight of three poles is 20 kilograms more than the total weight of two poles. - We assume basic algebra holds and weights add linearly. Concept / Approach: If we denote the weight of one pole by w kilograms, then the total weight of 10 poles is 10w. Each statement gives a linear equation involving w. If any single statement allows us to solve uniquely for w, then that statement alone is sufficient. If both statements individually allow us to solve for w, then either statement alone is sufficient and we choose the appropriate data sufficiency option accordingly. Step-by-Step Solution: Step 1: Analyze statement I. It states that one fourth of the weight of a pole is 5 kilograms. In algebraic form, this is (1/4) * w = 5. Step 2: Solving for w gives w = 5 * 4 = 20 kilograms. Then the total weight of 10 poles is 10 * 20 = 200 kilograms. Hence, statement I alone is sufficient to answer the question. Step 3: Now analyze statement II. It states that the total weight of three poles is 20 kilograms more than the total weight of two poles. In algebraic form, 3w = 2w + 20. Step 4: Subtracting 2w from both sides yields w = 20 kilograms. Again, the total weight of 10 poles is 10 * 20 = 200 kilograms. Therefore, statement II alone is also sufficient to answer the question. Step 5: Since each statement alone leads to a unique value of w and hence the total weight of 10 poles, either statement alone is sufficient, and using both together is not necessary. Verification / Alternative check: We can confirm that both statements give the same weight for one pole. From both I and II we obtain w = 20 kilograms. This consistency reassures us that there is no hidden contradiction and that both statements describe the same situation in different ways. Furthermore, once w is known, the total weight of 10 poles is uniquely determined as 200 kilograms. Why Other Options Are Wrong: - Option a is wrong because it claims only statement I is sufficient, ignoring that statement II alone is also adequate. - Option b is wrong for the symmetric reason: it credits only statement II while statement I is equally sufficient. - Option d is incorrect because we do not need to combine the statements; either alone suffices. - Option e is wrong because the data are not insufficient or contradictory; each statement individually gives a clear answer. Common Pitfalls: Some students assume that if two statements are given, both must be used together, which is not the case in data sufficiency problems. Others may fail to translate the verbal descriptions into algebra correctly. For instance, misunderstanding "20 kilograms more than the total weight of two poles" may lead to incorrect equations. Carefully forming and solving the linear equations for each statement separately avoids these errors. Final Answer: Either statement I alone or statement II alone is sufficient to determine the total weight of 10 poles, which is 200 kilograms. Therefore, the correct data sufficiency choice is option C.

Question 5

In a code language, "lee pee tin" means "Always keep smiling". What is the code for the word "Rose"? Statement I: "tin lut lee" means "Always keep left". Statement II: "dee pee" means "Rose smiling".

Accepted Answer

Introduction / Context: This data sufficiency question uses a simple code language in which each word is represented by a code word. You are given the code for the phrase "Always keep smiling" and two additional coded sentences. The target is to identify which statement allows you to determine the code for the single word "Rose". The focus is on sufficiency, so you only need to know whether you can find the code, not necessarily to compute anything else about the language. Given Data / Assumptions: - Base code: "lee pee tin" means "Always keep smiling". - Question: What is the code for "Rose"? - Statement I: "tin lut lee" means "Always keep left". - Statement II: "dee pee" means "Rose smiling". - Each word corresponds to exactly one code word, and each code word corresponds to exactly one meaning within this language. Concept / Approach: The standard approach in code language questions is to compare sentences that share words. When two sentences share some words, the intersection of their codes must correspond to those shared words. Once the codes for certain words are known, the remaining code words in a sentence can be matched to the remaining meanings in that sentence. In data sufficiency, we must see if statement I alone, statement II alone, or a combination is required to determine the code for "Rose". Step-by-Step Solution: Step 1: Use the base code and statement I. From the base, "lee pee tin" means "Always keep smiling". From statement I, "tin lut lee" means "Always keep left". Step 2: The common English words between these two sentences are "Always" and "keep". The common code words are "lee" and "tin". So "lee" and "tin" represent "Always" and "keep" in some order. Step 3: That leaves the word "smiling" in the base sentence corresponding to the code "pee". It also leaves the word "left" in statement I corresponding to the code "lut". Up to this point, we have decoded "smiling" and "left", but we have not yet seen the word "Rose" in any sentence. Therefore, statement I alone does not tell us the code for "Rose". Step 4: Now use the base code and statement II. From the base, "lee pee tin" means "Always keep smiling". From statement II, "dee pee" means "Rose smiling". Step 5: The common English word here is "smiling", and the common code word is "pee". Hence "pee" corresponds to "smiling", which is consistent with what we inferred earlier. Step 6: In statement II, "dee pee" codes "Rose smiling". Since "smiling" corresponds to "pee", the remaining code word "dee" must correspond to the remaining English word "Rose". Thus, using statement II with the base information, we can determine that "dee" is the code for "Rose". Statement II alone is therefore sufficient. Verification / Alternative check: The mapping we obtain is internally consistent: "pee" matches "smiling" in both the base sentence and statement II. Then "dee" naturally maps to "Rose". Statement I, while useful for finding codes for "left" and confirming the patterns, is not needed to determine the code for "Rose". There is no alternative interpretation that would assign a different code to "Rose" under these constraints. Why Other Options Are Wrong: - Option a is wrong because statement I does not include the word "Rose" at all, so it cannot be used alone to find its code. - Option c is incorrect because only statement II alone is sufficient; statement I alone is not. - Option d is wrong because statements I and II together are not required; II alone already solves the problem. - Option e is clearly wrong because we have shown that the code for "Rose" can be determined exactly from statement II plus the base information. Common Pitfalls: Learners sometimes assume that if two statements are given, they must be combined, overlooking that one might already be sufficient. It is also easy to ignore the base sentence and attempt to decode using only the new statements, but in data sufficiency the base information is always present and must be used. Another common mistake is to confuse which code word is common and therefore represents the shared English word. Final Answer: Statement II alone is sufficient to identify that "dee" is the code for "Rose", while statement I alone is not sufficient. Hence the correct data sufficiency choice is option B.

Question 6

Manoj, Prabhakar, Mash and Kamal are four friends. Who among them is the heaviest? Statement I: Prabhakar is heavier than Manoj and Kamal but lighter than Akash. Statement II: Manoj is lighter than both Prabhakar and Mash but heavier than Kamal.

Accepted Answer

Introduction / Context: This data sufficiency problem asks you to identify the heaviest person among four friends: Manoj, Prabhakar, Mash and Kamal. Two statements give partial comparisons of their weights and mention one additional person, Akash, who is outside the group of interest. The task is to decide whether either statement alone, or the two statements together, allow us to determine exactly which of the four friends is the heaviest. Given Data / Assumptions: - Friends within the question: Manoj (M), Prabhakar (P), Mash (H) and Kamal (K). - Question: Who is the heaviest among M, P, H and K? - Statement I: Prabhakar is heavier than Manoj and Kamal but lighter than Akash (A). So A > P > M and P > K. - Statement II: Manoj is lighter than both Prabhakar and Mash but heavier than Kamal. So P > M and H > M and M > K. - Akash is not one of the four friends we are being asked about; he only provides a comparison point for Prabhakar. Concept / Approach: To solve ordering questions, we convert the verbal comparisons into chains of inequalities. For data sufficiency, we check what can be deduced from each statement alone and then from both together. The key question is whether we can identify a single friend among M, P, H and K who is definitely heavier than all three others. If more than one candidate could be heaviest while still satisfying the statements, the data are not sufficient. Step-by-Step Solution: Step 1: Analyze statement I alone. From P being heavier than Manoj and Kamal, we know P > M and P > K. However, we have no information about Mash (H) in this statement. Mash could be heavier than P, equal to P, or lighter than P. Hence we cannot determine who is heaviest among the four friends from statement I alone. Step 2: Analyze statement II alone. Manoj is lighter than both Prabhakar and Mash but heavier than Kamal. So we have P > M, H > M, and M > K. Step 3: From statement II we know that Kamal is not the heaviest and that Manoj is not the heaviest because both P and H are heavier than Manoj. Between Prabhakar and Mash, however, we do not know who is heavier. Either P or H could be the heaviest friend. So statement II alone is also not sufficient. Step 4: Combine both statements. From statement I we have P > M and P > K. From statement II we have H > M and M > K and P > M. Step 5: Merging these inequalities gives an order for three of the friends: P and H are both heavier than M, and M is heavier than K. We still have no direct comparison between P and H, so both orders P > H and H > P satisfy all the given conditions. Step 6: Since we cannot decide whether Prabhakar or Mash is heavier, we cannot determine a single heaviest friend among M, P, H and K even when both statements are considered together. Verification / Alternative check: To confirm this, consider two possible scenarios consistent with both statements. In the first, suppose H is heavier than P. Then the order could be H > P > M > K. In the second, suppose P is heavier than H. Then the order could be P > H > M > K. Both scenarios satisfy all the given comparisons, but they disagree about who is the heaviest. Because the data admit more than one possible answer, the information is not sufficient. Why Other Options Are Wrong: - Option a is wrong because statement I alone does not mention Mash at all, so we cannot identify the heaviest among the four friends. - Option b is wrong because statement II alone only tells us that both P and H are heavier than Manoj and Kamal, but not which of P and H is heavier. - Option c is incorrect because neither statement alone is sufficient; each leaves ambiguity. - Option d is wrong because even using both statements together does not remove the ambiguity between Prabhakar and Mash; both can still be the heaviest in different consistent arrangements. Common Pitfalls: A common mistake is to forget that Akash is not part of the set of four friends whose relative weights are being compared. Another pitfall is to assume, without evidence, that Prabhakar is heavier than Mash simply because he is mentioned first or because statement I talks about Prabhakar in relation to someone outside the group. In data sufficiency, you must avoid adding any extra assumptions beyond what is explicitly stated. Final Answer: Even when both statements are used together, we cannot uniquely identify the heaviest friend among Manoj, Prabhakar, Mash and Kamal. Therefore, the correct data sufficiency choice is option E.

Question 7

In a class of 60 students, each boy contributes rupees equal to the number of girls and each girl contributes rupees equal to the number of boys. If the total contribution collected is Rs. 1600, how many girls are there in the class?

Accepted Answer

Introduction / Context: This is a quantitative reasoning question involving two unknowns: the number of boys and the number of girls in a class. Each student contributes an amount of money equal to the number of students of the opposite gender. You are given the total number of students in the class and the total contribution collected. The task is to determine how many girls are in the class by forming and solving algebraic equations. Given Data / Assumptions: - Total number of students in the class is 60. - Let the number of boys be B and the number of girls be G. - Each boy contributes rupees equal to the number of girls, that is G rupees per boy. - Each girl contributes rupees equal to the number of boys, that is B rupees per girl. - Total contribution collected is Rs. 1600. - The class is composed only of boys and girls, so B + G = 60. Concept / Approach: We translate the verbal description into two equations. One comes from the fact that the total number of students is 60. The other comes from equating the total rupee contribution to 1600. The total amount collected from boys is B * G, and from girls is G * B, for a total of 2 * B * G. This leads to a system of equations in B and G that can be solved using basic algebra and quadratic equations. Finally, we check which of the possible solutions matches the options given and makes sense for the number of girls. Step-by-Step Solution: Step 1: From the total number of students, we have B + G = 60. Step 2: Total contribution from boys is B * G (each boy pays G rupees), and total contribution from girls is G * B (each girl pays B rupees). Step 3: Therefore total contribution = B * G + G * B = 2 * B * G. Step 4: We are told that this total is Rs. 1600, so 2 * B * G = 1600, which gives B * G = 800. Step 5: Now we have a system of two equations: B + G = 60 and B * G = 800. Step 6: Express G in terms of B as G = 60 - B, and substitute into B * G = 800 to get B * (60 - B) = 800. Step 7: Expand this to obtain 60B - B^2 = 800, or B^2 - 60B + 800 = 0. Step 8: Solve the quadratic equation B^2 - 60B + 800 = 0. The discriminant is 60^2 - 4 * 1 * 800 = 3600 - 3200 = 400. Step 9: The square root of 400 is 20, so B = (60 ± 20) / 2, giving B = 40 or B = 20. Step 10: For B = 40, G = 60 - 40 = 20. For B = 20, G = 60 - 20 = 40. Verification / Alternative check: Check both possible pairs. If there are 40 boys and 20 girls, then each boy contributes 20 rupees and each girl contributes 40 rupees. Total contribution is 40 * 20 + 20 * 40 = 800 + 800 = 1600. If there are 20 boys and 40 girls, each boy contributes 40 rupees and each girl contributes 20 rupees, giving the same total 20 * 40 + 40 * 20 = 1600. Thus both (B, G) = (40, 20) and (20, 40) satisfy the conditions, but the question specifically asks for the number of girls and provides answer options among which only 40 appears. Hence the number of girls is 40, which corresponds to the valid solution B = 20, G = 40. Why Other Options Are Wrong: - Option 30 does not satisfy B + G = 60 when combined with any integer B that also gives B * G = 800. - Option 25 likewise fails to satisfy both the sum and the product conditions simultaneously. - Option 15 cannot pair with any B such that both the sum is 60 and the total contribution equals 1600. Common Pitfalls: A common mistake is to forget that boys and girls play symmetric roles in the equations, which leads to two possible solutions. Some students pick the first solution they find without checking the other, or they mistakenly believe only one solution can exist. Others may set up the product equation incorrectly, missing the factor of 2. Careful algebra and a quick verification of all solutions against the answer options help avoid these errors. Final Answer: The configuration that fits both the total number of students and the total contribution and matches one of the given options is the one with 40 girls in the class.

Question 8

In a certain year, the first day of a month is a Tuesday and the last day of that same month is a Monday. Which month can it be?

Accepted Answer

Introduction / Context: This is a calendar reasoning question. You are told the day of the week for the first and last days of a month, and must identify which month has that pattern. Such questions test your understanding of how days of the week cycle through a month depending on how many days the month contains. Given Data / Assumptions: - The first day of an unnamed month is a Tuesday. - The last day of that same month is a Monday. - We are to choose among the months January, February, March and August. - Unless otherwise stated, we usually assume a non leap year when reasoning about February. Concept / Approach: Days of the week repeat every 7 days. If a month has N days, and the first day is a given weekday, then the last day is the weekday that is N − 1 days after the first day (because the count starts at day 1). In modulo 7 arithmetic, we can express the day difference between the first and last days as (N − 1) mod 7. The question tells us that the first day is Tuesday and the last day is Monday, so the net shift in weekdays is from Tuesday to Monday, which is effectively a shift of minus one day, or equivalently plus six days modulo 7. Step-by-Step Solution: Step 1: Let N be the number of days in the month. Step 2: The first day of the month is Tuesday. After N − 1 days, we reach the last day. The weekday shift is (N − 1) mod 7. Step 3: The last day is Monday, which is one day before Tuesday. That means the overall shift from first to last day is −1 day, which is the same as +6 days modulo 7. Step 4: Thus we require (N − 1) ≡ 6 (mod 7), or equivalently N ≡ 0 (mod 7). This means that the month must have a number of days that is a multiple of 7. Step 5: Among the months listed, January has 31 days, March has 31 days, and August has 31 days. None of these is a multiple of 7. February in a non leap year has 28 days, and 28 is a multiple of 7 (28 = 4 × 7). Step 6: Therefore, only February with 28 days can satisfy the condition that N is a multiple of 7 and hence produce a shift from Tuesday to Monday between the first and last days. Verification / Alternative check: We can verify by direct counting using February as an example. If February 1 is a Tuesday and February has 28 days, then February 8 is also a Tuesday, February 15 is a Tuesday, and February 22 is a Tuesday. Then February 28 falls on a Monday. This matches the pattern described in the question. For a 31 day month, starting on Tuesday, the last day would be four days later (because 30 mod 7 is 2, so the net shift is +2), leading to a Thursday rather than a Monday, which contradicts the problem statement. Why Other Options Are Wrong: - January (31 days) cannot produce a last day that is Monday when the first day is Tuesday, because the shift from day 1 to day 31 is 30 days, which is equivalent to +2 weekdays, not −1. - March, also with 31 days, fails for the same reason as January. - August, again with 31 days, similarly does not result in a Monday on the last day if the first day is Tuesday. Common Pitfalls: Some students attempt to test each option by intuitively cycling through weekdays without using modular arithmetic, which can be error prone. Others forget that the shift is determined by N − 1 days rather than N days. Applying the simple condition N ≡ 0 (mod 7) keeps the reasoning clean and avoids off by one mistakes. Final Answer: Only February, with 28 days in a non leap year, can have its first day on a Tuesday and its last day on a Monday. Therefore the correct answer is February.

Question 9

What day of the week is the fourteenth of a given month? Statement I: The last day of the month is a Wednesday. Statement II: The third Saturday of the month falls on the seventeenth.

Accepted Answer

Introduction / Context: This is a day of week data sufficiency question. You are asked to determine the weekday of the fourteenth day of a month. Two statements describe the weekday of the last day of the month and the date of the third Saturday. Your goal is to decide whether either statement alone, or both together, provide enough information to determine the weekday of the fourteenth. Given Data / Assumptions: - Question: What day of the week is the fourteenth of a given month? - Statement I: The last day of the month is a Wednesday. - Statement II: The third Saturday of the month was on the seventeenth. - No specific month or year is given; only relative day information is provided. Concept / Approach: The weekday of any specific date can be determined if we know the weekday of another nearby date together with the difference in days. Statement II relates the seventeenth to a Saturday, which is only three days away from the fourteenth, making it especially powerful. By contrast, knowing only the weekday of the last day of a month does not directly tie to the fourteenth without knowing the total number of days in that particular month. Step-by-Step Solution: Step 1: Consider statement I alone. It tells us that the last day of the month is a Wednesday, but we do not even know whether the month has 28, 29, 30, or 31 days. Step 2: Since both the exact length of the month and its starting weekday are unknown, many different configurations of weekdays are consistent with the last day being Wednesday. Therefore, we cannot uniquely determine the weekday of the fourteenth from statement I alone. Statement I is not sufficient. Step 3: Now consider statement II alone. It says that the third Saturday of the month is on the seventeenth. That means that the date 17 falls on a Saturday. Step 4: Working backwards, the sixteenth is a Friday, the fifteenth is a Thursday, and the fourteenth is a Wednesday, because each previous date is one day earlier in the week. Step 5: Therefore, using statement II alone, we can determine that the fourteenth is a Wednesday. Verification / Alternative check: We can cross check the logic: if the seventeenth is the third Saturday, earlier Saturdays would be on the third and tenth. The weekly spacing is consistent. Starting from 17 (Saturday), we move backward three days to reach 14 (Wednesday). This calculation does not depend on the total number of days in the month, so the result remains valid regardless of the month length. Thus statement II alone gives a unique weekday for the fourteenth. Why Other Options Are Wrong: - Option a is wrong because statement I alone does not let us deduce the weekday of the fourteenth due to the uncertainty about the month length and the starting day. - Option c is incorrect because it suggests that both statements individually are sufficient, while in fact only statement II is. - Option d is wrong because we do not need to combine the statements; statement II alone already gives the answer. - Option e is clearly wrong since at least one statement (II) is sufficient to solve the problem. Common Pitfalls: Some students mistakenly try to infer something about the fourteenth from the last day of the month without knowing the length of the month, which leads to multiple possibilities. Another pitfall is overcomplicating the analysis of statement II by trying to construct a full calendar rather than simply counting backwards three days from the seventeenth. Recognizing that dates separated by three days fall on weekdays separated by three positions is the key here. Final Answer: Statement II alone is sufficient to determine that the fourteenth is a Wednesday, while statement I alone is not sufficient. Hence, the correct data sufficiency choice is option B.

Question 10

Is Arun taller than Sachin? Statement I: Dinesh is of the same height as both Arun and Sachin. Statement II: Sachin is not shorter than Dinesh.

Accepted Answer

Introduction / Context: This data sufficiency question asks a yes or no question: Is Arun taller than Sachin? Two statements give information about heights involving a third person, Dinesh. The goal is to see whether either statement alone, or both together, allow us to answer definitively whether Arun is taller than Sachin, shorter than Sachin, or equal in height, and thus to answer the yes or no question. Given Data / Assumptions: - Question: Is Arun taller than Sachin? - Statement I: Dinesh is of the same height as Arun and Sachin. - Statement II: Sachin is not shorter than Dinesh. - All height comparisons are exact; there are no rounding issues. Concept / Approach: In data sufficiency, for a yes or no question, information is sufficient if it allows us to answer definitively yes or definitely no. If Arun and Sachin are found to be of equal height, the answer to "Is Arun taller than Sachin?" is a definite "no", which counts as sufficient. We must analyze each statement separately, then consider them together if needed, while resisting the temptation to combine them prematurely. Step-by-Step Solution: Step 1: Interpret statement I. It says that Dinesh is of the same height as Arun and Sachin. This implies Arun, Sachin and Dinesh are all of equal height. Step 2: If Arun and Sachin are of equal height, then Arun is not taller than Sachin. The answer to the question "Is Arun taller than Sachin?" is a definite "no". Step 3: Because statement I alone leads to a clear and unique answer, it is sufficient to answer the question. Step 4: Now analyze statement II alone. It says Sachin is not shorter than Dinesh, which means Sachin is either taller than or equal in height to Dinesh. Step 5: Statement II alone does not provide any comparison between Arun and Dinesh, and therefore no direct comparison between Arun and Sachin. Arun could be taller than Dinesh, equal to Dinesh, or shorter than Dinesh. Accordingly, Sachin could still be taller than, equal to, or shorter than Arun. Therefore, statement II alone is not sufficient to answer the question. Step 6: Since statement I alone is sufficient and statement II alone is not, there is no need to consider both together for sufficiency. Verification / Alternative check: To double check, imagine the height values. From statement I, let Dinesh, Arun, and Sachin each have height h. Then Arun is clearly not taller than Sachin; they are equal. Any value of h works, and the equality remains. Statement II alone allows multiple possibilities, such as Dinesh having height h, Sachin having height h, but Arun having any height greater or smaller than h, so we cannot resolve the question from II alone. Why Other Options Are Wrong: - Option b is wrong because statement II alone does not identify whether Arun is taller than Sachin or not. - Option c is incorrect because only statement I alone is sufficient; statement II alone is not. - Option d is wrong because both statements together are not needed; I alone suffices. - Option e is wrong because the data are not insufficient; the first statement clearly answers the question. Common Pitfalls: Students sometimes forget that equality provides a definitive "no" answer to a "taller than" question. Another pitfall is misreading "not shorter" as "strictly taller", while it actually means "greater than or equal to". Also, some may try to combine both statements automatically without first checking sufficiency of each statement on its own, which is contrary to the standard data sufficiency procedure. Final Answer: Statement I alone is sufficient to answer the question with a definite "no", while statement II alone is not sufficient. Therefore, the correct data sufficiency choice is option A.

Question 11

When is Manohar's birthday this year? Statement I: It falls between January 13 and January 15, and January 13 is a Wednesday. Statement II: It is not on a Friday.

Accepted Answer

Introduction / Context: This data sufficiency question asks you to determine the exact date of Manohar's birthday in January of a particular year. Two statements limit the possible dates and give information about the weekday of certain days. The objective is to decide whether one statement alone, or a combination of both, gives enough information to identify the unique date of his birthday. Given Data / Assumptions: - Question: On which date in January does Manohar's birthday fall this year? - Statement I: It is between January 13 and January 15, and January 13 is a Wednesday. - Statement II: It is not on a Friday. - The phrase "between January 13 and January 15" is interpreted in the usual logical puzzle sense as meaning the date is 14 January, which lies strictly between 13 and 15. Concept / Approach: The key is to interpret the language of the statements precisely. In many aptitude and data sufficiency problems, when a birthday is said to be "between" two specific dates, it is commonly taken to mean the date that lies strictly in the middle. If that interpretation gives a unique date from statement I alone, the second statement may be redundant. On the other hand, statement II only tells us that the birthday is not on a particular weekday, which is very weak information by itself. Step-by-Step Solution: Step 1: Analyze statement I. The birthday is said to be between January 13 and January 15. In the usual integer date sense, the only day that lies strictly between the 13th and 15th is the 14th. Step 2: Therefore, statement I alone tells us that Manohar's birthday is on 14 January. The added information that January 13 is a Wednesday may be intended to hint at the weekday sequence, but it is not necessary for determining the date. Step 3: Hence, statement I alone is sufficient to answer the question. Step 4: Now consider statement II alone. It says only that Manohar's birthday is not on a Friday. Without any information about the possible dates or other weekdays, the birthday could be any date in January that is not a Friday. This leaves many possible dates, so statement II alone is clearly not sufficient. Step 5: Since statement I alone suffices and statement II alone does not, we do not need to combine them to answer the question. Verification / Alternative check: Even if we try to use the weekday information given in statement I, with January 13 being a Wednesday, this leads to 14 January being a Thursday and 15 January being a Friday. Combining this with statement II would only rule out Friday (15 January), leaving the 13th or 14th as candidates if we misinterpreted "between" as inclusive. However, the standard wording in such puzzles treats "between 13 and 15" as the single day 14. Under that interpretation, statement II is redundant and statement I alone already identifies 14 January as the birthday. Why Other Options Are Wrong: - Option b is wrong because statement II alone does not narrow the birthday down to a specific date at all. - Option c is incorrect since only statement I alone is sufficient; statement II alone is not. - Option d is wrong because we do not need both statements; I alone suffices. - Option e is clearly wrong because the date can be determined from statement I alone. Common Pitfalls: Learners sometimes misinterpret "between 13 and 15" to mean on one of 13, 14 or 15, which would make statement I ambiguous. It is important in exam style questions to recognize the conventional usage where "between" two distinct dates refers to the date strictly in the middle. Another mistake is to overvalue statement II and try to combine both statements without first checking the sufficiency of I alone. Final Answer: Statement I alone is sufficient to determine that Manohar's birthday is on 14 January, whereas statement II alone is not sufficient. Therefore, the correct data sufficiency choice is option A.

Question 12

How many visitors saw the exhibition yesterday? Statement I: Each entry pass holder can take up to three persons along. Statement II: In all, 243 passes were sold yesterday.

Accepted Answer

Introduction / Context: This data sufficiency problem is about counting visitors to an exhibition. You are asked to find how many visitors attended, given information about how many passes were sold and how many people each pass holder is allowed to bring. The challenge is to see whether the information is enough to determine a unique number of visitors, or whether multiple different totals remain possible. Given Data / Assumptions: - Question: How many visitors saw the exhibition yesterday? - Statement I: Each entry pass holder can take up to three persons with him or her. - Statement II: In all, 243 passes were sold yesterday. - Every visitor enters through a pass holder, and there are no constraints on whether each pass holder must bring three persons or could bring fewer (including zero additional persons). Concept / Approach: We treat the total number of visitors as the sum of all pass holders plus all additional persons they bring. Statement I gives us the maximum number of people per pass but not the actual number for each pass holder. Statement II tells us how many passes were sold. For data sufficiency, if we cannot determine a unique total number of visitors because of flexible conditions, then the data are not sufficient. Step-by-Step Solution: Step 1: Analyze statement I alone. We know that each pass holder can take up to three persons along. So each pass could admit the pass holder plus anywhere from zero to three additional people, which means between 1 and 4 visitors per pass. But we do not know how many passes were sold, so statement I alone does not allow us to compute the total number of visitors. Step 2: Analyze statement II alone. We know that 243 passes were sold yesterday. However, we have no information about how many people each pass allows or how many people actually came with each pass holder. So statement II alone also does not provide enough information to determine the total number of visitors. Step 3: Combine statements I and II. With 243 passes sold and each pass holder allowed to bring up to three additional persons, the minimum number of visitors occurs when no pass holder brings anyone else. In that case, the number of visitors would be 243. Step 4: The maximum number of visitors occurs when every pass holder brings the full three additional persons. Then each pass admits 1 + 3 = 4 visitors, and the total number of visitors would be 243 * 4 = 972. Step 5: In between these extremes, any combination of pass holders bringing 0, 1, 2, or 3 additional persons can occur, so many different visitor totals between 243 and 972 are possible. Therefore, even with both statements together, we cannot determine a unique number of visitors. Verification / Alternative check: We can test specific cases. If half the pass holders bring one extra person and the other half bring none, the total visitor count will be some integer strictly between 243 and 972. There is nothing in the statements that forbids this scenario. If some bring two and some bring three, another total is possible. Because the problem allows up to three persons, and does not state that everyone always brings three, all these variations are consistent with the data, which proves that the total number of visitors is not fixed by the given information. Why Other Options Are Wrong: - Option a is wrong because statement I alone does not specify how many passes were sold, leaving the total completely open ended. - Option b is wrong because statement II alone does not constrain how many visitors per pass there are. - Option c is incorrect because neither statement alone is sufficient. - Option d is wrong because even the combination of statements I and II allows a wide range of possible visitor counts; they do not pin down a single value. Common Pitfalls: Many students loosely assume that each pass holder actually brings three persons because that is the maximum allowed, but the wording "up to three" clearly includes scenarios where fewer are brought. Another error is to confuse the number of passes with the number of visitors. In data sufficiency questions, you must read constraints very literally and never assume that the maximum or minimum is always reached unless explicitly stated. Final Answer: Even when both statements are considered together, the exact number of visitors cannot be determined. Therefore, the correct data sufficiency choice is option E.

Question 13

Vipin and Javed have salaries in the proportion 4 : 3 respectively. What is Vipin's salary? Statement I: Javed's salary is 75% of Vipin's salary. Statement II: Javed's salary is Rs. 4500.

Accepted Answer

Introduction / Context: This data sufficiency question relates to salaries given in a ratio. You are told that the salaries of Vipin and Javed are in the proportion 4 : 3. Two additional statements describe the relationship between their salaries numerically. Your job is to determine whether either statement alone, or both together, are enough to find the actual salary of Vipin in rupees. Given Data / Assumptions: - From the question: Vipin's salary : Javed's salary = 4 : 3. - Question: What is Vipin's salary? - Statement I: Javed's salary is 75% of Vipin's salary. - Statement II: Javed's salary is Rs. 4500. - The ratio 4 : 3 and the 75% relation are consistent with each other. Concept / Approach: When given a ratio and a specific amount for one of the terms in the ratio, we can find the other term by simple proportion. The initial ratio 4 : 3 effectively tells us that if Vipin's salary is 4x, then Javed's salary is 3x for some positive value x. Statement I restates the same proportional relationship in percent form, while statement II provides an actual rupee amount for Javed. Data sufficiency asks which combination of these pieces of information is necessary to compute Vipin's salary as a concrete number. Step-by-Step Solution: Step 1: Use the ratio from the question. Let Vipin's salary be 4x and Javed's salary be 3x. Step 2: Analyze statement I alone. It says Javed's salary is 75% of Vipin's salary. In algebraic form, this is 3x = 0.75 * 4x. Step 3: Simplify 0.75 * 4x to get 3x, so statement I does not introduce any new numerical amount; it simply confirms the same ratio 4 : 3 that we already have from the question. Step 4: Therefore, statement I alone does not allow us to compute x or Vipin's salary in rupees. It is not sufficient. Step 5: Now analyze statement II alone. It tells us Javed's salary is Rs. 4500. From the ratio, Javed's salary = 3x, so 3x = 4500. Step 6: Solve for x: x = 4500 / 3 = 1500. Step 7: Thus, Vipin's salary is 4x = 4 * 1500 = Rs. 6000. Statement II, together with the ratio given in the question, is enough to determine Vipin's salary. Step 8: Statement I does not add any fresh information beyond the original ratio, so statement II alone is sufficient. Verification / Alternative check: We can check that statement I is consistent with the derived salaries. If Vipin earns Rs. 6000 and Javed earns Rs. 4500, then Javed's salary is indeed 75% of Vipin's salary, since 4500 / 6000 = 0.75. The original ratio condition 4 : 3 is also satisfied because 6000 : 4500 reduces to 4 : 3. The calculations are consistent, confirming that statement II gives a unique answer for Vipin's salary when combined with the ratio from the question stem. Why Other Options Are Wrong: - Option a is wrong because statement I alone merely restates the same ratio in percentage form and gives no absolute values, so we cannot find Vipin's exact salary. - Option c is incorrect because only statement II alone is sufficient, while statement I alone is not. - Option d is wrong because the two statements are not both needed; statement II plus the original ratio is enough. - Option e is wrong because the data are not insufficient; they clearly allow us to solve for Vipin's salary as Rs. 6000. Common Pitfalls: Some students mistakenly think that any statement which seems to give a new equation is automatically sufficient. However, statement I here is just another way of expressing the same ratio, not new numerical data. Another error is to forget that the main question already provides a ratio, and to treat each statement as if it were the only information available. In data sufficiency, the original question text is always part of the information set for evaluating each statement. Final Answer: Statement II alone is sufficient to determine Vipin's salary as Rs. 6000, whereas statement I alone is not sufficient. Therefore, the correct data sufficiency choice is option B.

Question 14

How is D related to A? Statement I: B is the brother of A. Statement II: B is the son of D.

Accepted Answer

Introduction / Context: This data sufficiency question tests understanding of family relationships and the limits of information provided by two short statements. The task is to determine how D is related to A, that is, whether D is a parent, grandparent, uncle, or has some other fixed relation, based only on the information in the two statements. The key idea is to check whether any single statement or the combination of both statements forces one unique relation between D and A, without allowing alternative possibilities that are still consistent with the given data. Given Data / Assumptions: Statement I: B is the brother of A. Statement II: B is the son of D. Family relationships are assumed to be in the usual sense, with brother meaning a male sibling. No extra information about parents, step relations, or half siblings is given. We must identify the relation of D to A in a unique and definite way. Concept / Approach: The standard approach in data sufficiency is: First, analyse statement I alone and see whether it is enough. Next, analyse statement II alone. Then, combine both statements to check whether together they yield a unique answer. If even the combination does not lead to a single fixed relation, the data are considered insufficient. Here, the core concept is that if multiple different family structures fit the statements, then the relation of D to A cannot be uniquely determined. Step-by-Step Solution: Step 1: From statement I, B is the brother of A. So A and B share at least one common parent, but we are told nothing about D. Step 2: From statement I alone, D is never mentioned, so statement I alone is clearly not sufficient to decide the relation of D to A. Step 3: From statement II, B is the son of D. So D is a parent of B, but there is no mention of A. Step 4: From statement II alone, we know B is a child of D, but we do not know whether A is also a child of D, a child of only one parent common with B, or only a step or half sibling. Step 5: Now combine both statements. We know B is the brother of A, and B is the son of D. This means that B is a male child of D and that A is a sibling of B. Step 6: However, A could be a full child of D, a stepchild, or a child of a different parent who is married to D. All these cases still satisfy both statements. Step 7: Because there are multiple possibilities, D may be the father or mother of A, or possibly a step parent. No unique relation can be fixed. Verification / Alternative check: Alternative scenario 1: Suppose A is also a child of D. Then D is a parent of A. Alternative scenario 2: Suppose A is the child of the other parent only, and B is the only child of D with that partner. Then A is still a sibling of B, but D is a step parent to A, not a biological parent. Both scenarios satisfy statement I and statement II, yet the relation of D to A is not the same. This confirms that the relation is not uniquely determined even when both statements are used together. Why Other Options Are Wrong: Option a is wrong because statement I does not mention D at all. Option b is wrong because statement II only tells us about B and D, not about A. Option c is wrong because neither statement alone can ever be enough, so it is impossible that either alone is sufficient. Option d is wrong because even with both statements, multiple family configurations are possible, so the relation is not uniquely fixed. Common Pitfalls: Many students quickly assume that if B is the brother of A and B is the son of D, then A must also be the child of D, so D is a parent of A. This assumes that all siblings share both parents, which is not guaranteed by the statements. The question does not exclude step siblings or half siblings. In data sufficiency problems, one must avoid adding hidden assumptions. Unless the question clearly states that all siblings are born to the same parents, you must allow more than one logically possible family structure. As soon as more than one family structure fits the data, the information is not sufficient to give a unique answer about the required relation. Final Answer: Even using both statements I and II together, the relation of D to A cannot be uniquely determined, so the data are not sufficient. Correct option: Even using both statements I and II together, the data are not sufficient to answer the question.

Question 15

Four students Rakesh, Raju, Ravi and Ranga scored different marks in a test. Rakesh scored more than Raju. Ravi scored less than Rakesh. Raju scored more than Ravi. Ranga scored more than Raju but less than Rakesh. Who scored the highest marks among…

Accepted Answer

Introduction / Context: This question is a simple ranking and ordering problem based on comparative statements about marks scored by four students. Such questions are common in verbal reasoning and logical reasoning tests. The objective is to translate the given comparisons into a clear order and then identify who is at the top of the order. Careful attention to each inequality is essential in order to avoid mistakes when more than two people are compared. Given Data / Assumptions: There are four students: Rakesh, Raju, Ravi and Ranga. Rakesh scored more than Raju. Ravi scored less than Rakesh. Raju scored more than Ravi. Ranga scored more than Raju but less than Rakesh. All four have different scores. Concept / Approach: The approach is to interpret every sentence as an inequality involving the marks. If we let the marks be M(Rakesh), M(Raju), M(Ravi) and M(Ranga), each statement gives a relation such as M(Rakesh) greater than M(Raju). By combining these inequalities we can get a complete ordering from highest to lowest. Once that ordering is clear, the person with the maximum marks can be directly identified. No numerical values are required; only relative comparison is used. Step-by-Step Solution: Step 1: From the first statement, Rakesh scored more than Raju, so M(Rakesh) greater than M(Raju). Step 2: From the third statement, Raju scored more than Ravi, so M(Raju) greater than M(Ravi). Step 3: Combining these two, we get M(Rakesh) greater than M(Raju) greater than M(Ravi). Step 4: The second statement, Ravi scored less than Rakesh, is consistent with this chain since M(Ravi) is already less than M(Rakesh). Step 5: The last statement tells us that Ranga scored more than Raju but less than Rakesh, so M(Rakesh) greater than M(Ranga) greater than M(Raju). Step 6: Now combine Ranga into the previous chain. We already have M(Rakesh) greater than M(Raju) greater than M(Ravi) and M(Rakesh) greater than M(Ranga) greater than M(Raju). Step 7: The consistent full order from highest to lowest is M(Rakesh) greater than M(Ranga) greater than M(Raju) greater than M(Ravi). Step 8: Therefore, Rakesh is the highest scorer among the four students. Verification / Alternative check: We can quickly verify by imagining sample numerical marks that satisfy all the inequalities. For example, assign marks: Rakesh 90, Ranga 85, Raju 80 and Ravi 70. Check each condition: Rakesh scored more than Raju (90 greater than 80), Ravi scored less than Rakesh (70 less than 90), Raju scored more than Ravi (80 greater than 70), and Ranga scored more than Raju but less than Rakesh (85 between 80 and 90). All conditions are satisfied, and Rakesh clearly has the maximum marks. Why Other Options Are Wrong: Option a (Ranga) is wrong because Ranga scored less than Rakesh, not more. Option b (Raju) is wrong because Raju scored less than both Ranga and Rakesh. Option d (Ravi) is wrong because Ravi scored less than Raju and definitely less than Rakesh. Common Pitfalls: A common mistake is to read the statements quickly and mix up the order, especially when many names appear together. Some students may forget to place Ranga correctly between Rakesh and Raju. Others may ignore one of the conditions and conclude that Ranga or Raju scored the highest. To avoid such errors, it is helpful to write down the inequalities and then merge them systematically. Another pitfall is assuming equal marks, but the question clearly implies different scores, which ensures a strict ordering. Final Answer: From the combined inequalities, Rakesh is at the top of the order. Correct option: Rakesh.

Question 16

Rohit, Kajal, Tanmay and Suman are four friends. Who is the oldest among them? Statement I: The total age of Kajal and Tanmay together is more than the age of Suman. Statement II: The total age of Rohit and Kajal together is less than the age of Sum…

Accepted Answer

Introduction / Context: This is a typical data sufficiency question involving comparison of ages. Four friends Rohit, Kajal, Tanmay and Suman are given, and we are asked to identify who is the oldest among them. Instead of direct ages, we are given two total age comparisons. The goal is not to compute exact ages, but to judge whether the information given is enough to deduce uniquely which person is the oldest. Given Data / Assumptions: There are four friends: Rohit, Kajal, Tanmay and Suman. Statement I: Age of Kajal plus age of Tanmay is greater than age of Suman. Statement II: Age of Rohit plus age of Kajal is less than age of Suman. All ages are positive real numbers. We need to decide whether the oldest person can be uniquely identified. Concept / Approach: Data sufficiency problems require us to test each statement separately and then in combination: Check statement I alone for sufficiency. Check statement II alone. Check both together. If more than one age configuration satisfies both statements and leads to different oldest persons, then even the combined data are not sufficient. We are interested in uniqueness, not in just one possible scenario. Step-by-Step Solution: Step 1: Let the ages be R, K, T and S for Rohit, Kajal, Tanmay and Suman respectively. Step 2: From statement I, K + T greater than S. Step 3: This only tells us that the combined age of Kajal and Tanmay is more than Suman. It does not tell us whether Suman is older or younger than either Kajal or Tanmay individually, and it says nothing about Rohit. So statement I alone is not sufficient. Step 4: From statement II, R + K less than S. Step 5: Since R and K are positive, this implies that S is older than each of R and K individually, because for any positive numbers a and b, if a plus b is less than c, then each of a and b is less than c. Thus R less than S and K less than S. Step 6: Statement II alone gives that Suman is older than both Rohit and Kajal, but it gives no information about Tanmay relative to Suman, so statement II alone is not sufficient. Step 7: Now combine both statements. We have K + T greater than S and R + K less than S. However these are aggregate comparisons. They do not prevent Tanmay from being older than Suman, nor do they force Suman to be older than Tanmay. Step 8: Construct one scenario: let S be 30 years, K be 10 years, T be 25 years and R be 15 years. Then K + T is 35 greater than 30, and R + K is 25 less than 30. Here Tanmay, with 25, is younger than Suman, so Suman is the oldest. Step 9: Construct another scenario: let S be 28 years, K be 5 years, T be 26 years and R be 20 years. Then K + T is 31 greater than 28, and R + K is 25 less than 28. Now Tanmay, with 26, is the oldest. Step 10: Since both scenarios satisfy both statements and produce different oldest persons, the combined data are not sufficient. Verification / Alternative check: The reasoning can be verified algebraically. From K + T greater than S, there is no restriction on T relative to S as long as K is small enough. Similarly, from R + K less than S, S can be chosen just slightly larger than R + K, again allowing T to take a wide range of values. Because T can be either less than S or greater than S while still satisfying both inequalities, the identity of the oldest person remains ambiguous. Why Other Options Are Wrong: Option a is wrong because statement I alone does not mention Rohit at all and does not compare individual ages. Option b is wrong because statement II alone compares Suman with Rohit and Kajal, but says nothing about Tanmay relative to Suman. Option c is wrong because neither statement alone is sufficient; each leaves the possibility that different persons might be the oldest. Option d is wrong because even the combined data allow more than one person to be the oldest. Common Pitfalls: Many candidates mistakenly assume that if Suman is older than Rohit and Kajal and the sum of Kajal and Tanmay is greater than Suman, then Suman must be the oldest. This ignores the possibility that Tanmay may individually be older than Suman while still satisfying the sum condition. In data sufficiency questions it is crucial to test alternative numeric examples rather than trusting an initial impression. Final Answer: Even when statements I and II are used together, they do not uniquely determine who is the oldest. Correct option: Even using both statements I and II together, the data are not sufficient to answer the question.

Question 17

On which date in April is Gautam's birthday? Statement I: Gautam was born exactly 28 years after his mother was born. Statement II: His mother will be 55 years 4 months and 5 days old on August 18 of this year.

Accepted Answer

Introduction / Context: This data sufficiency question asks us to determine the exact date in April on which Gautam was born. Instead of giving a direct date, the question supplies information about Gautam's mother and about the age difference between Gautam and his mother. We must see whether this information, individually or together, allows us to pin down a unique date in April for Gautam's birthday. Given Data / Assumptions: We need the date in April of Gautam's birthday. Statement I: Gautam was born exactly 28 years after his mother was born. Statement II: Gautam's mother will be 55 years 4 months and 5 days old on August 18 of this year. Age expressions such as 55 years 4 months and 5 days are interpreted in the usual calendar sense. There is no leap year complication in the month day differences for the period considered, other than the normal calendar. Concept / Approach: The strategy is: Use statement II to work backwards from August 18 to find the exact birth date of the mother. Then apply statement I to shift forward by 28 years to get Gautam's birth date, which will have the same day and month but a later year. Finally, check whether either statement alone can give the date in April, and whether both together are required. Because the question asks only for the day in April, the year is not needed for the final answer. Step-by-Step Solution: Step 1: From statement II, the mother is 55 years 4 months and 5 days old on August 18 of this year. Step 2: Subtract 5 days from August 18. Counting backwards, 5 days before August 18 is August 13. Step 3: Subtract 4 months from August 13. Four months before August is April. So four months before August 13 is April 13. Step 4: Therefore, the mother's date of birth is April 13, in a year that is 55 years earlier than this year. Step 5: Statement I says that Gautam was born exactly 28 years after his mother. This means that the day and month of his birth are the same as those of his mother, but the year is 28 years later. Step 6: So Gautam is also born on April 13, 28 years after his mother's birth year. Step 7: Since we only need the date in April, we conclude that Gautam's birthday falls on the thirteenth of April. Step 8: Statement I on its own does not give any specific date, since the mother's birthday is not known from that statement alone. Step 9: Statement II on its own gives only the mother's date of birth, not the age difference between mother and son, so it does not directly provide Gautam's birthday. Verification / Alternative check: We can verify the calculation of the mother's date of birth by moving forward instead of backward. Start from April 13. Adding 4 months takes us to August 13. Adding 5 days to August 13 gives August 18. Thus 4 months and 5 days after April 13 is August 18. If the mother is 55 years old on that date, then she was born exactly 55 years earlier on April 13. This confirms that our backward calculation is correct. Next, the age gap of 28 years simply shifts the year while keeping the same month and day. Therefore Gautam's birthday must also fall on April 13, which matches our conclusion. Why Other Options Are Wrong: Option a is wrong because statement I alone does not tell us when the mother was born, so we cannot find the date in April. Option b is wrong because statement II alone gives only the mother's birth date, not the relationship between her age and Gautam's age. Option c is wrong because neither statement alone is sufficient; both are needed to link the mother's date of birth to Gautam's date of birth. Option e is wrong because, as shown, the combination of both statements leads to a unique date, April 13. Common Pitfalls: Students sometimes make errors while working with months and days, especially when counting backwards. A common mistake is to confuse 4 months and 5 days with 4 months and 15 days or to shift the month incorrectly. Another pitfall is to assume that the difference in years alone is enough, without carefully calculating the day and month of the mother's birth. In data sufficiency, one must also resist the temptation to treat a single statement as sufficient just because it looks rich in information. Always make sure that the required target quantity, here the date in April of Gautam's birthday, can be uniquely derived. Final Answer: Using both statements together, we find that Gautam's birthday is on the thirteenth of April. Correct option: The data in both statements I and II together are sufficient to answer the question.

Question 18

What is the price range of ordinary wall clocks of company X? Statement I: The price range of ordinary wrist watches of company X is from Rs 400 to Rs 600. Statement II: The price range of ordinary wall clocks of company X is 50 percent of the price…

Accepted Answer

Introduction / Context: This data sufficiency problem deals with price ranges and percentage relationships. We are asked to determine the price range of ordinary wall clocks produced by company X. Rather than giving the range directly, the question provides information about the price range of wrist watches and a percentage relation between watches and clocks. Our task is to decide whether each statement alone, or both together, allow us to compute the range for wall clocks. Given Data / Assumptions: We want the lowest and highest prices of ordinary wall clocks of company X. Statement I: Ordinary wrist watches of company X range from Rs 400 to Rs 600. Statement II: The price range of ordinary wall clocks of company X is 50 percent of the price range of their ordinary wrist watches. 50 percent is interpreted as half of each bound in the given range. Prices are non negative and the same company is considered in both statements. Concept / Approach: The key concept is simple ratio and percentage application. The range for wrist watches is already known from statement I. Statement II establishes that each price in the wall clock range is 50 percent of the corresponding wrist watch price. To compute the numerical clock range, we need both the percent factor and the original numeric range. We then check whether either statement by itself is enough to determine the range, and whether the combination yields a unique answer. Step-by-Step Solution: Step 1: From statement I, the wrist watch price range is from Rs 400 to Rs 600. Step 2: Statement I alone does not talk about wall clocks, so it cannot give the wall clock price range directly. Step 3: From statement II, we know that the price range of wall clocks is 50 percent of the price range of wrist watches. However, statement II alone does not give us the actual wrist watch price range in rupees. Step 4: Therefore, statement II alone only tells us that wall clock prices are half of whatever numbers the wrist watch range has, but does not provide concrete numerical values. Step 5: Now combine both statements. We know from statement I that watches range from Rs 400 to Rs 600. From statement II, wall clocks are at 50 percent of this range. Step 6: Calculate the lower end of the wall clock range: 50 percent of 400 is 0.5 * 400 = 200. So the minimum price for wall clocks is Rs 200. Step 7: Calculate the upper end of the range: 50 percent of 600 is 0.5 * 600 = 300. So the maximum price for wall clocks is Rs 300. Step 8: With both statements together, we can uniquely determine that the wall clock price range is from Rs 200 to Rs 300. Verification / Alternative check: As a quick check, one can think in terms of the ratio of prices. If every ordinary wall clock costs half as much as a corresponding ordinary wrist watch, then the entire interval of prices is scaled down by a factor of 0.5. Scaling the lower bound 400 by 0.5 gives 200, and scaling the upper bound 600 by 0.5 gives 300. This confirms our earlier result. We also verify logical sufficiency: without the numerical values in statement I, statement II remains purely relative and cannot produce concrete rupee amounts. Without the relation in statement II, statement I gives us only wrist watch prices, not wall clock prices. Why Other Options Are Wrong: Option a is wrong because statement I gives no information about wall clocks. Option b is wrong because statement II has a percentage relation but no numeric base to apply it to. Option c is wrong because neither I nor II alone is sufficient; both are required. Option e is wrong because, as shown, the combined data give a clear and unique range. Common Pitfalls: A frequent mistake is to think that statement II alone is sufficient simply because it mentions wall clocks and a percentage. However, a percentage without a base value does not yield specific numbers. Another pitfall is misinterpreting 50 percent of the range, thinking it refers to some difference rather than scaling the actual prices. Remember that if every clock is sold at half the price of a similar watch, both the minimum and maximum prices are also halved. For data sufficiency, you must not only be able to compute the answer but also be certain that no missing information remains. Final Answer: Using both statements together, the wall clock price range is from Rs 200 to Rs 300. Correct option: The data in both statements I and II together are sufficient to answer the question.

Question 19

How many sons does D have? Statement I: The father of A has three children. Statement II: B is the brother of A and B is a son of D.

Accepted Answer

Introduction / Context: This data sufficiency question focuses on family relationships and counts of children. We are asked to determine how many sons a person D has. Two statements are provided involving A, B, and the father of A. The challenge is to understand what can and cannot be concluded from these pieces of information, and to recognise when the information is not enough to determine a unique number of sons. Given Data / Assumptions: We need the number of sons of D. Statement I: The father of A has three children. Statement II: B is the brother of A and B is a son of D. Brother means a male sibling in the usual sense. No explicit statement connects the father of A in statement I to D in statement II. All children mentioned are living, and there may be additional children not mentioned. Concept / Approach: The method in data sufficiency is to evaluate: Statement I alone. Statement II alone. Both statements together. If there exists more than one possible family structure consistent with the statements but with different numbers of sons for D, then the data are not sufficient. We must avoid silently adding assumptions such as equating the father of A with D unless it is clearly stated. Step-by-Step Solution: Step 1: From statement I, the father of A has three children. Let that father be F. Then F has exactly three children, one of whom is A. The other two children are not identified by name or gender. Step 2: Statement I alone does not even mention D, so it clearly cannot tell us how many sons D has. Therefore statement I alone is not sufficient. Step 3: From statement II, B is the brother of A and is a son of D. Thus B is a male child of D, and B is a sibling of A. Step 4: From statement II alone, D has at least one son, namely B. However, D may have other sons and daughters not mentioned. Hence we cannot determine the exact number of sons of D from statement II alone. Step 5: Now combine both statements. One possible interpretation is that F, the father of A in statement I, might be the same person as D. But this is not explicitly stated. Even if we assume that D is the father of A, we only know that D has three children in total. Step 6: Among these three children, we know from statement II that B is a son and sibling of A. However, the genders of A and the third child are not specified. Step 7: Different configurations are possible. For example, D could have one son B and two daughters including A, giving one son. Alternatively, D could have two sons, B and the third child, with A as a daughter, giving two sons. It is also possible that A is also a son, giving three sons. Step 8: Since multiple consistent family patterns produce different numbers of sons for D, even the combined information does not uniquely determine the required number. Verification / Alternative check: Construct scenario one: Let D be the father of A. Suppose A is female, B is male and the third child is female. Then D has one son. Construct scenario two: Again let D be the father of A. Suppose A is female, B is male and the third child is male. Then D has two sons. Construct scenario three: Suppose both A and B are male and the third child is female. Then D has two sons again. In every case, the given statements I and II remain valid, but the number of sons changes or at least is not fixed to one unique value. Why Other Options Are Wrong: Option a is wrong because statement I does not mention D at all. Option b is wrong because statement II only guarantees at least one son for D, not the exact count. Option c is wrong because neither statement alone gives a unique answer. Option d is wrong because even after combining the statements, several different distributions of sons and daughters are possible. Common Pitfalls: A common mistake is to assume that the father of A mentioned in statement I must be D. Even if we make that assumption, students often conclude that D has three children and then wrongly infer that the number of sons is fixed, sometimes assuming an equal split between sons and daughters without any basis in the statements. Data sufficiency questions require strict logical discipline. Only what is explicitly given or is a direct consequence of what is given should be used. Any additional assumptions about genders or identities of parents can lead to incorrect conclusions about sufficiency. Final Answer: The exact number of sons of D cannot be determined from the given statements. Correct option: Even using both statements I and II together, the data are not sufficient to answer the question.

Question 20

In city X, buses are always punctual. How long, at the most, will Mr Roy have to wait for a bus? Statement I: Mr Roy arrives at the bus stand at 9:00 a.m. Statement II: There is a bus at 10:00 a.m. and possibly another bus even earlier.

Accepted Answer

Introduction / Context: This question examines understanding of the phrase at the most and the idea of punctuality in public transport. Mr Roy arrives at a bus stand in city X, where buses are always punctual. We have information about his arrival time and about at least one scheduled bus. The goal is to determine the maximum possible waiting time for Mr Roy, given that a bus is guaranteed at a particular time and may possibly arrive earlier. Given Data / Assumptions: Buses in city X are always punctual, so they arrive exactly at their scheduled times. Statement I: Mr Roy arrives at the bus stand at 9:00 a.m. Statement II: There is a bus at 10:00 a.m., and there may possibly be another bus even earlier than 10:00 a.m. We are asked for the maximum time Mr Roy might have to wait. The question is about the worst case waiting time, not the minimum. Concept / Approach: The key concept is to distinguish between minimum and maximum waiting time. The phrase at the most refers to the longest duration Mr Roy might need to wait, given the information. Since a bus is definitely scheduled at 10:00 a.m. and buses are punctual, Mr Roy is guaranteed to catch that bus at the latest. If there is an earlier bus, his actual waiting time will be less, but the maximum possible waiting time is determined by the latest guaranteed bus that he might need to take. Step-by-Step Solution: Step 1: From statement I, we know that Mr Roy is present at the bus stand at 9:00 a.m. Step 2: Statement I alone does not tell us when the next bus is due, so we cannot determine his waiting time from this statement alone. Step 3: From statement II, we know that there is a bus at 10:00 a.m., and there might also be another bus before 10:00 a.m., but the time of any earlier bus is not specified. Step 4: Statement II alone does not mention when Mr Roy arrives, so it cannot give his waiting time. Thus statement II alone is not sufficient. Step 5: Now combine both statements. Mr Roy arrives at 9:00 a.m., and buses are punctual. We are told there is definitely a bus at 10:00 a.m. Step 6: Because of punctuality, the 10:00 a.m. bus will arrive exactly at 10:00 a.m. So in the worst case, if there is no earlier bus that he can catch, he will wait from 9:00 a.m. to 10:00 a.m. Step 7: The waiting time from 9:00 a.m. to 10:00 a.m. is 1 hour. Step 8: If there is an earlier bus, Mr Roy will catch it and wait less than 1 hour. But the question asks for the maximum possible waiting time, which is based on the latest guaranteed bus time compatible with his arrival. Step 9: Therefore, the maximum time Mr Roy might need to wait is 1 hour. Verification / Alternative check: Assume there is an earlier bus at 9:20 a.m. In that case, his waiting time is 20 minutes. Assume another scenario where there is no earlier bus, and the only bus after 9:00 a.m. is at 10:00 a.m. Then his waiting time is 60 minutes. Since the question asks at the most, we consider the scenario in which he has to wait the longest while remaining consistent with the given information. The latest guaranteed bus is at 10:00 a.m., so 1 hour is the maximum waiting time. Why Other Options Are Wrong: Option a is wrong because, without any bus schedule, statement I alone cannot give a waiting time. Option b is wrong because, without knowing when Mr Roy arrives, statement II alone cannot produce a waiting duration. Option c is wrong because neither statement on its own allows computation of waiting time; we need both arrival time and bus time. Option e is wrong because, with both statements combined, we clearly get a maximum waiting time of 1 hour. Common Pitfalls: Some students confuse the phrase at the most with at least. At least would refer to the minimum guaranteed wait, while at the most is about the worst case. Others ignore the possibility of an earlier bus and think that punctuality implies only the 10:00 a.m. bus. However, the phrase possibly another bus even earlier clearly allows earlier departures. The safe approach is to base the maximum waiting time on the latest guaranteed bus after the arrival time, ignoring optional earlier buses for the maximum calculation. Final Answer: Using both statements together, the maximum time that Mr Roy may have to wait for a bus is 1 hour. Correct option: The data in both statements I and II together are sufficient to answer the question.

Data Sufficiency Questions