Question 1

At what time did Sonali leave her home for the office?  Consider the following statements:  I. Sonali received a phone call at 9:15 a.m. while she was at home. II. The car of Sonali reached the office at 10:15 a.m., which was 45 minutes after she le…

Accepted Answer

Introduction / Context: This data sufficiency question deals with travel time between home and office. The time at which Sonali left her home needs to be determined. Two statements give information about a phone call she received at home and the time her car reached the office. The problem tests understanding of time interval calculations and the ability to decide which statements are sufficient on their own. Given Data / Assumptions: Statement I: Sonali received a phone call at 9:15 a.m. while she was at home. Statement II: The car of Sonali reached the office at 10:15 a.m., and this was 45 minutes after she left her residence. Travel time from her home to the office for this trip is constant at 45 minutes as implied in statement II. Times are given in the same day and time zone, with no breaks in the journey. Concept / Approach: The second statement directly relates the arrival time at the office to the departure time from home using a fixed journey duration. If that duration and arrival are known, the departure time can be calculated by simple subtraction. The first statement, however, gives only a point in time when Sonali was still at home but does not link that moment to the departure time, so we must examine each statement separately and then together. Step-by-Step Solution: Step 1: Use statement II alone. The car reached the office at 10:15 a.m., forty five minutes after Sonali left home. Let the departure time from home be T. Then T plus 45 minutes equals 10:15 a.m. Step 2: Solve for T. Subtract 45 minutes from 10:15 a.m. Ten fifteen minus forty five minutes is 9:30 a.m. Therefore, Sonali left her residence at 9:30 a.m. Step 3: This calculation is unique and uses only information from statement II. Hence statement II alone is sufficient to answer the question. Step 4: Now examine statement I alone. Knowing that Sonali received a phone call at 9:15 a.m. while she was at home tells us that she had not yet left by that time. She could have left at 9:16 a.m., 9:30 a.m., or any other time after 9:15 a.m. No further constraint is provided, so her exact departure time cannot be determined from this statement alone. Step 5: Even if we combine both statements, statement II already provides the complete answer. Statement I does not change the derived departure time; it only confirms that she was indeed at home fifteen minutes before leaving, which is consistent with leaving at 9:30 a.m. Verification / Alternative check: To verify sufficiency of statement II, consider whether any other departure time could lead to arrival at 10:15 a.m. with a fixed travel time of 45 minutes. Any such journey must start exactly 45 minutes earlier, at 9:30 a.m. There is no alternative. Conversely, with statement I, there is no information about the travel duration or arrival time, so multiple departure times are possible. This confirms that II alone is sufficient and I alone is not. Why Other Options Are Wrong: Option A is wrong because it gives sufficiency to statement I, which does not fix a unique departure time. Option C claims that either statement alone is sufficient, which is incorrect since I alone is not enough. Option D suggests that both statements together are necessary, but this is not true because II alone already yields the exact answer. Option E, claiming that even both together are not sufficient, is clearly false since we obtain a precise time of 9:30 a.m. when using statement II. Common Pitfalls: A common mistake is to overvalue statement I simply because it mentions a time when Sonali is at home. Learners may wrongly assume that the call at 9:15 a.m. is very near departure, but the question does not say that the call occurred immediately before she left. Another error is to confuse the phrase after forty five minutes with some vague time window, when it actually denotes a fixed travel duration. Reading the timing relations carefully avoids such errors. Final Answer: The data in statement II alone are sufficient to answer the question, while statement I alone is not sufficient, so the correct option is B.

Question 2

What is the total monthly salary of Vasu?  Consider the following statements:  I. The basic salary of Vasu is Rs 100 more than the basic salary of Rajan, who also works in the same company. II. Other allowances drawn by Rajan, in addition to his bas…

Accepted Answer

Introduction / Context: This data sufficiency question concerns salaries and allowances of two employees, Vasu and Rajan. The goal is to decide which combination of three statements allows us to compute Vasu total monthly salary, including basic salary and allowances. The question tests how well the learner can translate verbal descriptions of salary structure into equations and see which data points are essential. Given Data / Assumptions: Vasu and Rajan work in the same company. Statement I: The basic salary of Vasu is Rs 100 more than the basic salary of Rajan. Statement II: Rajan draws allowances of Rs 2000 per month, and these allowances are Rs 50 less than the allowances drawn by Vasu. Statement III: The basic salary of Rajan is Rs 1550 per month. Total salary is taken as basic salary plus allowances for each employee. Concept / Approach: Let RB and VB denote the basic salaries of Rajan and Vasu, and RA and VA denote their monthly allowances. Vasu total salary is VB + VA. We must examine which statements give enough numeric information to evaluate VB and VA. If any combination leaves one of these variables undetermined, that combination is not sufficient. We are not concerned with computing the final number for the exam, but doing so acts as a useful check. Step-by-Step Solution: Step 1: Translate statement I into an equation. If RB is the basic salary of Rajan, then VB = RB + 100. Step 2: Translate statement II. Rajan has allowances RA = 2000, and these allowances are Rs 50 less than Vasu allowances. So VA = RA + 50 = 2000 + 50 = 2050. Step 3: Translate statement III. It directly gives RB = 1550. Step 4: Check statement II alone. Knowing only that Vasu allowances are Rs 2050 does not provide any information about VB. Therefore, statement II alone is not sufficient to compute the total salary of Vasu. Step 5: Check statements II and III together. From III, RB = 1550. From II, VA = 2050. However, Vasu basic salary VB still depends on statement I, since we only know the relation between VB and RB from I. Without I, VB remains unknown, so the combination of II and III is not sufficient. Step 6: Check statements I and II together. With I, we know VB = RB + 100, and with II, VA = 2050. Without having a numeric value for RB, VB remains unknown, and therefore VB + VA cannot be computed. So I and II together are also not sufficient. Step 7: Combine all three statements. Using III, RB = 1550. Substituting into I gives VB = 1550 + 100 = 1650. From II we already know VA = 2050. Therefore Vasu total monthly salary is VB + VA = 1650 + 2050 = 3700 rupees. With all three statements, the total is uniquely determined. Verification / Alternative check: To verify, we can reason from the perspective of unknowns and equations. The key unknowns affecting Vasu salary are VB and VA. Statement I relates VB to RB, so VB depends on knowing RB. Statement II gives VA numerically but still requires VB. Statement III gives RB numerically. Only when all three statements are used do we have numeric values for both VB and VA. Any subset of statements leaves VB or VA unspecified, which means Vasu total salary is not fixed. This confirms that all three statements are required. Why Other Options Are Wrong: Option A is incorrect because statement II alone does not say anything about Vasu basic salary. Option B, which gives sufficiency to II and III, fails since VB still cannot be calculated without the link provided by statement I. Option C, involving I and II, is also not enough because RB remains unknown without statement III. Only option D correctly recognises that statements I, II, and III together are required to answer the question. Common Pitfalls: One common mistake is to misread statement II and assume that the allowances figure includes Vasu basic salary or that Vasu salary itself is Rs 50 more than some number. Another typical error is to think that relative information such as Rs 100 more is enough without a base figure. In salary questions, relative differences must be anchored by at least one concrete value before totals can be found. Final Answer: The total monthly salary of Vasu can be determined only when statements I, II, and III are all used together, so the correct option is D.

Question 3

How many gift boxes were sold on Monday?  Consider the following statements:  I. The number of gift boxes sold on Monday was 10 percent more than the number of boxes sold on the previous day, that is, Sunday. II. Every third visitor to the shop purc…

Accepted Answer

Introduction / Context: This data sufficiency question combines percentage increase with visitor based sales information. The objective is to find out how many gift boxes were sold on Monday. Two statements relate Monday sales to Sunday sales and relate Sunday sales to the number of visitors. The core skill is to see whether we can convert these into a unique numeric value and which statements are required for that. Given Data / Assumptions: Statement I: Monday sales in gift boxes were 10 percent more than Sunday sales. Statement II: On Sunday, every third visitor purchased exactly one box and there were 1500 visitors. All visitors are counted once, and every third visitor means one out of every three visitors bought one box, with no other sales on Sunday. Concept / Approach: Let S be the number of boxes sold on Sunday and M be the number of boxes sold on Monday. Statement I relates M to S via a percentage increase. Statement II allows us to calculate S from visitor data. To decide sufficiency, we check whether S and M can be uniquely determined from each statement alone or only from their combination. Step-by-Step Solution: Step 1: From statement II, there are 1500 visitors on Sunday and every third visitor buys one box. That means one box for every three visitors, so S = 1500 divided by 3. Step 2: Compute S. Since 1500 divided by 3 is 500, Sunday sales equal 500 gift boxes. Step 3: Using statement II alone, we know S but we do not know M. No information is given about Monday sales in this statement, so II alone is not sufficient. Step 4: From statement I, Monday sales are 10 percent more than Sunday sales. In algebraic form, M = S + 10 percent of S = 1.10 * S. Step 5: Using statement I alone, we have only the relation between M and S. Without a numerical value for S, M remains undetermined. Therefore statement I alone is also not sufficient. Step 6: Combine statements I and II. From statement II we have S = 500. Substituting into M = 1.10 * S gives M = 1.10 * 500 = 550. Now Monday sales are uniquely fixed at 550 gift boxes. Step 7: Since neither statement alone yields M but the combination does, both statements together are necessary to answer the question. Verification / Alternative check: To verify, imagine varying S while still satisfying statement I. For any positive S, statement I allows a corresponding M equal to 1.10 * S. Without statement II, there are infinitely many possible pairs (S, M). With statement II alone, S is fixed at 500, but M can still vary because the link to Monday is missing. Only when both pieces of information are used does the pair (S, M) become fixed as (500, 550). Why Other Options Are Wrong: Option A is wrong because statement I alone does not give any numeric base for Sunday sales. Option B is wrong because statement II alone does not mention Monday. Option C, which suggests that either I or II alone is sufficient, is clearly incorrect. Option E claims that even both statements together are not sufficient, which contradicts the clear computation of 550 boxes for Monday. Only option D correctly recognises that both statements together are necessary and sufficient. Common Pitfalls: Learners may confuse the phrase every third visitor with one third of all visitors, but here the interpretation gives the same result as long as the total is divisible by three. Another common oversight is to treat a percentage relation as if it alone fixes both quantities, ignoring the need for an absolute reference value. Careful separation of relative statements from absolute numerical statements helps to avoid these mistakes. Final Answer: The data in both statements I and II together are necessary to answer the question, so the correct option is D.

Question 4

On which day of the last week did Satish meet Kapil at the residence of Kapil?  Consider the following statements:  I. Kapil was out of town from Monday to Wednesday and returned on Thursday morning. II. On Friday night, Satish telephoned his friend…

Accepted Answer

Introduction / Context: This data sufficiency question is about determining on which day of the week Satish met Kapil at Kapil residence. One statement gives information about when Kapil was out of town, and the other gives information about a phone call made by Satish on Friday night. The challenge is to decide which statements are sufficient to determine the exact day of the meeting. Given Data / Assumptions: Statement I: Kapil was out of town from Monday to Wednesday and returned on Thursday morning. Statement II: On Friday night, Satish informed his friend by phone that only yesterday he had obtained Kapil approval after personally explaining all details. The phrase only yesterday refers to the day immediately before Friday, that is, Thursday. Satish meeting with Kapil for approval is taken to have occurred at Kapil residence as per the main question. Concept / Approach: The core idea is to interpret temporal phrases like only yesterday correctly and connect them to named days such as Friday. If statement II alone gives a direct reference to the day of the meeting, it may be sufficient by itself. Statement I may only indicate availability of Kapil and may not be strictly necessary if the day is already fixed by the second statement. Step-by-Step Solution: Step 1: Interpret statement II. On Friday night, Satish told his friend that only yesterday he had received approval from Kapil after explaining all the details. The expression only yesterday refers to the day before Friday, which is Thursday. Step 2: Since the approval from Kapil came after Satish personally explained the details, we infer that the personal meeting between Satish and Kapil took place on Thursday. Step 3: The question is about the day on which Satish met Kapil at Kapil residence. Statement II clearly identifies Thursday as the day of the meeting by tying the approval to the phrase only yesterday used on Friday night. Step 4: Therefore, statement II alone is sufficient to answer the question. Step 5: Now examine statement I alone. It tells us that Kapil was out of town from Monday through Wednesday and returned on Thursday morning. This shows that any meeting at Kapil residence must have taken place on or after Thursday, but it does not say which day Satish actually met him. The meeting could have happened on Thursday, Friday, Saturday, or Sunday. Hence statement I alone is not sufficient. Step 6: When the two statements are combined, the information remains consistent. Statement I confirms that Kapil was available in town from Thursday onwards, and statement II gives the exact day of the meeting as Thursday. Since statement II already fixes the answer, statement I is not necessary for sufficiency. Verification / Alternative check: To check the reasoning, imagine any other possible interpretation of only yesterday that does not refer to Thursday when spoken on Friday night. Within a standard seven day week, yesterday on Friday must be Thursday. There is no alternative. Furthermore, if Kapil had still been out of town on Thursday, statement II would contradict statement I, but instead they fit together, since statement I explicitly says Kapil returned on Thursday morning. This confirms that the meeting day must have been Thursday and that statement II alone is sufficient. Why Other Options Are Wrong: Option A is incorrect because statement I alone does not narrow the meeting to a single day. Option C, which suggests that either statement alone is sufficient, is wrong since I on its own leaves multiple days possible. Option D, claiming that both statements are necessary, is not correct because statement II alone already gives the answer. Option E, stating that even both together are not sufficient, clashes with the clear interpretation of the temporal phrase used in statement II. Common Pitfalls: Learners sometimes underuse context in interpreting phrases like only yesterday and may assume that another day could be intended. Another common mistake is to overemphasise the travel schedule in statement I and forget that the second statement directly references the meeting timing. Keeping careful track of which statement actually pins down the required information is vital in data sufficiency questions. Final Answer: The data in statement II alone are sufficient to answer the question, while statement I alone is not sufficient, so the correct option is B.

Question 5

Among four friends A, B, C, and D, who is the heaviest?  Consider the following statements:  I. B is heavier than A but lighter than D. II. C is lighter than B.

Accepted Answer

Introduction / Context: This data sufficiency problem is about comparing weights of four friends A, B, C, and D. The question asks who is the heaviest among them. Two statements provide partial information about relative weights. The goal is to determine which statements, alone or together, allow us to identify the heaviest friend without ambiguity. Given Data / Assumptions: Statement I: B is heavier than A but lighter than D. Statement II: C is lighter than B. No two friends have exactly the same weight unless implied, and all inequalities are consistent. Concept / Approach: Each statement gives a set of inequalities on a weight scale. We want to know whether they jointly force one person to be at the top of this order. If one friend is heavier than all others based on the combined information, then that combination is sufficient. If any friend could still be heavier under some allowed arrangement, then the data are not sufficient. Step-by-Step Solution: Step 1: Translate statement I. B is heavier than A, so B > A. B is lighter than D, so D > B. Combining these, we get D > B > A. Step 2: Under statement I alone, we do not know the position of C in relation to D, B, or A. C could be heavier than D, lighter than A, or anywhere in between. For example, one possible order is C > D > B > A, where C is heaviest, and another is D > B > A > C, where D is heaviest. Hence statement I alone is not sufficient to identify the heaviest person. Step 3: Translate statement II. C is lighter than B, so B > C. We do not know where A or D stand relative to B or C from this statement alone, so II alone is not sufficient either. Step 4: Combine statements I and II. From I, D > B > A. From II, B > C. Combining these, we have D > B > A and D > B > C. Step 5: Put these together into a single order. We know that B is heavier than both A and C, and D is heavier than B. A and C might still be ordered either way relative to each other, but that does not affect who is heaviest. The crucial point is that D is heavier than B, and B is heavier than both A and C, so D is heavier than all three of A, B, and C. Step 6: Since there is no information suggesting any friend is heavier than D, and the inequalities show D above all others, D must be the heaviest. This conclusion relies on both statements I and II. Verification / Alternative check: To verify, attempt to construct any arrangement where someone other than D is heaviest while satisfying both statements. If we try to make B heaviest, we contradict D > B. If we try to make C heaviest, we contradict B > C and D > B. If we try to make A heaviest, we contradict B > A and D > B. Therefore, every consistent arrangement places D at the top, which confirms that the combined statements are sufficient to answer the question. Why Other Options Are Wrong: Option A, giving sufficiency to statement I alone, is wrong because C is unconstrained there and could be heavier than D. Option B, suggesting statement II alone is sufficient, is clearly incorrect since it does not involve D or A at all. Option C, claiming either statement alone is sufficient, fails for both reasons. Option E, which states that even both statements together are not sufficient, is false because the combined inequalities force D to be heaviest. Only option D correctly states that both statements together are necessary and sufficient. Common Pitfalls: Learners sometimes focus only on the explicit comparisons and overlook people not mentioned directly in a statement, leading to premature conclusions. Another frequent mistake is to fail to notice that, under statement I alone, C position is completely undetermined. Drawing a simple diagram or number line for weights when combining statements greatly reduces this kind of confusion. Final Answer: The data in both statements I and II together are necessary to answer the question, so the correct option is D.

Question 6

What is the monthly salary of Praveen?  Consider the following statements:  I. Praveen receives 15 percent more salary than Sumit, while Sumit receives 10 percent less than Lokesh. II. Lokesh has a monthly salary of Rs 2500.

Accepted Answer

Introduction / Context: This data sufficiency question involves salaries of three people: Praveen, Sumit, and Lokesh. Their salaries are related by given percentages, and the final goal is to find Praveen monthly salary. The two statements provide relative and absolute information. We need to judge whether each statement alone or their combination is enough to compute Praveen salary uniquely. Given Data / Assumptions: Statement I: Praveen receives 15 percent more salary than Sumit, and Sumit receives 10 percent less than Lokesh. Statement II: Lokesh monthly salary is Rs 2500. Salaries are positive amounts measured in rupees. Monthly salary refers to a fixed amount with no hidden components beyond what is described. Concept / Approach: Let L denote Lokesh salary, S denote Sumit salary, and P denote Praveen salary. Statement I expresses S and P as percentages of L and S respectively. Statement II gives the numerical value of L. Once L is known, S follows from the percent relation, and P follows from S. The question is whether any statement alone can provide the required numeric base or whether both are needed. Step-by-Step Solution: Step 1: Translate statement I into equations. If Sumit receives 10 percent less than Lokesh, then S = L * (1 minus 0.10) = 0.90 * L. If Praveen receives 15 percent more than Sumit, then P = S * (1 plus 0.15) = 1.15 * S. Step 2: Substitute S into P. We get P = 1.15 * (0.90 * L) = 1.035 * L. So Praveen salary is 1.035 times Lokesh salary. Step 3: Statement I alone gives only proportional relationships with L as a free variable. Without a numeric value for L, P cannot be computed exactly. Therefore statement I alone is not sufficient. Step 4: Consider statement II alone. It states that L = Rs 2500. However, without the percentage relationships from statement I, we do not know how P compares to L, so we still cannot compute P. Statement II alone is also not sufficient. Step 5: Combine statements I and II. From II, L = 2500. Using the formula P = 1.035 * L derived from I, we get P = 1.035 * 2500. Step 6: Multiply to find P. 1.035 * 2500 equals 2587.50 rupees. Thus Praveen monthly salary is Rs 2587.50, and this value is uniquely determined when both statements are used together. Verification / Alternative check: From a sufficiency perspective, note that three salaries are linked by two percentage relations in statement I. This single statement reduces the degrees of freedom but still leaves L undetermined. Statement II provides the missing numeric value for L. With both pieces, there is no remaining free variable. In contrast, each statement alone leaves also at least one variable free and therefore cannot yield a unique value for P, which confirms that both are required. Why Other Options Are Wrong: Option A is incorrect because statement I alone provides only relationships, not numbers. Option B is incorrect since statement II alone provides a number but no relationship to Praveen. Option C incorrectly claims that either statement alone is sufficient. Option E, which says that even both statements together are not sufficient, clearly conflicts with the exact computation of P. Only option D correctly indicates that both statements I and II together are necessary and sufficient. Common Pitfalls: Learners may misinterpret phrases like 10 percent less and 15 percent more, for example by subtracting or adding the same amount independently of the base. Another typical error is to assume that knowing Lokesh salary immediately gives Praveen salary without applying the intermediate relation through Sumit. Carefully writing out percentage equations avoids these mistakes. Final Answer: The data in both statements I and II together are necessary to answer the question, so the correct option is D.

Question 7

It is now 8:00 p.m. When can Hemant get the next bus for Ramnagar from Dhanpur?  Consider the following statements:  I. Buses for Ramnagar leave the bus stand after every 30 minutes until 10:00 p.m. II. Fifteen minutes ago, one bus had left the bus …

Accepted Answer

Introduction / Context: This data sufficiency problem involves bus departure times from Dhanpur to Ramnagar. The current time is 8:00 p.m., and Hemant wants to know when he can catch the next bus. Two statements give the departure frequency and the time of a recent departure. The challenge is to determine whether any one statement or both together allow us to find the exact time of the next bus. Given Data / Assumptions: The current time for Hemant is exactly 8:00 p.m. Statement I: Buses leave every 30 minutes until 10:00 p.m., but the starting time of the schedule is not specified. Statement II: A bus left for Ramnagar fifteen minutes ago. All times mentioned refer to the same day and bus route, and the every 30 minutes pattern is regular and continuous during service hours. Concept / Approach: The key is to recognise that knowing only the frequency of bus departures is not enough without an anchor time. Similarly, knowing only one recent departure time does not guarantee knowledge of the repeating pattern unless frequency is known. When both frequency and one departure time are known, we can generate the full schedule and identify the next bus after a given current time. Step-by-Step Solution: Step 1: Consider statement II alone. Fifteen minutes ago, at 7:45 p.m., a bus left Dhanpur for Ramnagar. Without any information about the frequency of departures, we cannot infer whether the next bus is in 15 minutes, 30 minutes, one hour, or some other interval. Statement II alone is therefore not sufficient. Step 2: Consider statement I alone. Buses leave every 30 minutes until 10:00 p.m., but the statement does not specify from which exact time this pattern starts. For example, the bus times might be 7:30, 8:00, 8:30, and so on, or they might be 7:45, 8:15, 8:45, and so on. Without the starting point, we cannot determine whether a bus is due at 8:00 p.m., 8:15 p.m., or some other time, so statement I alone is not sufficient. Step 3: Combine statements I and II. From statement II, we know a bus left at 7:45 p.m. From statement I, we know that buses leave every 30 minutes. Therefore, the departure times are 7:45 p.m., 8:15 p.m., 8:45 p.m., and so on, continuing until service stops at 10:00 p.m. Step 4: Determine the next bus after 8:00 p.m. Given that the departures are at 7:45 p.m. and then 8:15 p.m., the next departure after 8:00 p.m. is at 8:15 p.m. Step 5: Since the combined information from both statements uniquely fixes the schedule and gives Hemant the next bus time as 8:15 p.m., we conclude that both statements are necessary and sufficient. Verification / Alternative check: To verify, imagine trying different starting times for the 30 minute schedule without statement II. Many different patterns satisfy statement I, so the next bus after 8:00 p.m. cannot be uniquely determined. Conversely, using only statement II, we can construct many different repeating patterns that include a departure at 7:45 p.m. but differ in interval length. Only when we enforce both the 30 minute interval and the specific departure at 7:45 p.m. do we get a fixed schedule that answers the question. Why Other Options Are Wrong: Option A is wrong because statement I alone lacks an initial time. Option B is wrong because statement II alone does not specify the timetable interval. Option C incorrectly claims that either statement alone is sufficient. Option E, which states that even both statements together are not sufficient, is contradicted by the clear derivation of the next bus at 8:15 p.m. only option D correctly states that both statements I and II together are necessary and sufficient. Common Pitfalls: Learners may assume that a 30 minute schedule must start on the hour or the half hour, which is not given. Another common error is to assume that the last bus departure is always at the closing time of service, rather than up to that time. Carefully reading the exact wording and combining the anchor departure time with the frequency avoids such mistakes. Final Answer: The data in both statements I and II together are necessary to answer the question, so the correct option is D.

Question 8

On which date is the birthday of Manohar?  Consider the following statements:  I. The father of Manohar was born on 27 May 1948. II. Manohar is 25 years younger than his mother.

Accepted Answer

Introduction / Context: This data sufficiency question is about birthdays and age differences within a family. The aim is to identify the exact birthday of Manohar. Two statements give information about the father birth date and Manohar age difference with his mother. The main issue is whether these pieces of information are enough to fix Manohar date of birth without further assumptions. Given Data / Assumptions: Statement I: The father of Manohar was born on 27 May 1948. Statement II: Manohar is 25 years younger than his mother. No explicit relationship is given between the ages of Manohar parents. No information is provided about the mother date of birth or any calendar alignment between parents and child beyond the age gap in statement II. Concept / Approach: To know a person date of birth, we need both the year and the day and month. Knowing the birth date of one parent gives complete information for that parent only. Knowing an age difference between Manohar and his mother can at best give the difference between their years of birth, and even then only if one birth year is known. The question is whether these statements together or individually allow us to derive Manohar full birthday. Step-by-Step Solution: Step 1: Consider statement I alone. It tells us that Manohar father was born on 27 May 1948. However, there is no information about the age difference between the father and Manohar or between the father and the mother. Without such data, we cannot infer any specific year or date for Manohar birth. Step 2: Therefore, statement I alone is not sufficient to answer the question. Step 3: Consider statement II alone. It states that Manohar is 25 years younger than his mother. This information only gives the age gap between Manohar and his mother. The mother could have been born in many possible years and on many possible dates. Without the mother birth date, Manohar date of birth cannot be known. Step 4: Hence, statement II alone is also not sufficient. Step 5: Now consider both statements together. Statement I fixes the father date of birth, but no relation is given between the ages of the father and the mother. The mother could be younger than the father, older than the father, or even the same age, as long as no explicit contradiction arises. Step 6: Because the mother date of birth is completely free, the pair of statements allows many possible combinations of mother and child birthdays that satisfy Manohar being 25 years younger than his mother. For example, if we choose one possible birth year for the mother, we get a corresponding year for Manohar, but different choices of the mother birth date still produce different possible dates for Manohar. Step 7: Since there are multiple possible birth dates for Manohar consistent with both statements, the data are not sufficient to determine his birthday uniquely. Verification / Alternative check: To verify, suppose the mother was born on 1 January 1960. Then Manohar might be born on 1 January 1985. If instead the mother was born on 10 October 1965, then Manohar could be born on 10 October 1990. Both scenarios fit statement II and do not contradict statement I, because statement I concerns only the father. The fact that multiple valid birth dates for Manohar exist proves that the data are insufficient. Why Other Options Are Wrong: Options A and B claim sufficiency of individual statements, which is impossible because each statement deals with a different person without linking them numerically. Option C states that either statement alone is enough, which is clearly not correct. Option D suggests that both statements together are necessary and sufficient, but as seen, even the combination leaves many possibilities open. Only option E correctly states that even both statements together are not sufficient to answer the question. Common Pitfalls: A frequent error is to assume that parents are of a standard age when children are born, such as a fixed age difference between parents and children. Such assumptions are not given and cannot be introduced in data sufficiency problems. Another pitfall is to treat the known birth date of the father as if it constrains the mother age, which the problem does not state. Avoiding extra assumptions is crucial in this type of reasoning. Final Answer: The data even in both statements together are not sufficient to answer the question, so the correct option is E.

Question 9

A, B, C, D, and E are sitting in a single row of five chairs facing the same direction. B is seated between A and C.  Who among them occupies the middle position in the row?  Consider the following statements:  I. A is to the left of B and to the ri…

Accepted Answer

Introduction / Context: This data sufficiency question concerns seating arrangement in a row. Five people A, B, C, D, and E sit in a straight line, and some relative positions are known. The goal is to identify who sits in the exact middle position of the row. Two statements provide partial information about the order, and we need to judge which of them, alone or together, fix the middle seat. Given Data / Assumptions: B is between A and C in the row, which normally implies that A, B, and C occupy three consecutive seats in the order A B C or C B A. Statement I: A is to the left of B and to the right of D. Statement II: C is at the right end of the row. All five seats are in a single row with distinct positions from left to right, and no two people share a seat. Concept / Approach: We first understand the base constraint B between A and C, then apply each statement to see how the arrangement tightens. If a statement alone leaves multiple possible people in the middle, it is not sufficient. If combining the statements yields a single valid arrangement, then both are necessary and sufficient. Step-by-Step Solution: Step 1: Use the base condition. Since B is between A and C and they sit in a row, the possible patterns for A, B, and C on consecutive seats are A B C (from left to right) or C B A. No other arrangement keeps B strictly between A and C. Step 2: Interpret statement I. A is to the left of B and to the right of D. This implies the order D, A, B in that sequence from left to right. Combining this with the base pattern, A must be to the left of B, so the correct pattern among A, B, C is A B C, not C B A. Step 3: From the base condition and statement I, we get the partial sequence D, A, B, C as four consecutive positions in that order. The person E is still free to sit either at the extreme left or at the extreme right, as long as he does not break the constraints on A, B, C, and D. Step 4: Consider statement I alone. If E sits to the left of D, the row is E, D, A, B, C and the middle seat is A. If E sits to the right of C, the row is D, A, B, C, E and the middle seat is B. Since both arrangements satisfy statement I and the base condition but give different people in the middle, statement I alone is not sufficient. Step 5: Now consider statement II alone. C is at the right end. Knowing only that B is between A and C and that C is at the extreme right leaves many possibilities for the exact positions of A, B, and the other two persons. The middle position may be occupied by different people depending on the placement of D and E. Thus statement II alone is also not sufficient. Step 6: Combine statements I and II. From earlier, D, A, B, C must be in order, and now statement II requires C to be at the right end. Therefore the only possible arrangement is E, D, A, B, C, where E is at the left end, D next, then A, then B, and C at the right end. Step 7: In this unique arrangement, the third seat from the left is occupied by A, so A is in the middle position. Since no other arrangement fits all conditions, the combination of both statements is necessary and sufficient. Verification / Alternative check: To verify, try to place C at the right end while keeping B between A and C and A left of B but right of D. The sequence D, A, B, C is forced, leaving only one seat for E. Any attempt to swap E with another person violates either the base condition or one of the statements. Therefore, there is exactly one valid seating order, and it yields A as the middle person. This confirms sufficiency of the combined statements. Why Other Options Are Wrong: Option A is wrong because statement I alone admits at least two different middle occupants, A and B, depending on E position. Option B is wrong because statement II alone leaves the entire left side of the row underdetermined. Option C, which claims either statement alone is sufficient, fails for both reasons. Option E, saying that even both together are not sufficient, contradicts the unique arrangement we found. Only option D correctly states that both statements together are necessary and sufficient. Common Pitfalls: Learners sometimes misinterpret the phrase B is between A and C as allowing non consecutive seating, which leads to extra invalid patterns. Others may forget to check multiple placements of the unused person E when judging statement I alone. Drawing the row and trying all possibilities systematically is an effective way to avoid these mistakes in arrangement based data sufficiency questions. Final Answer: The data in both statements I and II together are necessary to answer the question, so the correct option is D.

Question 10

The chairman of a large company visits exactly one department on Monday of every week, except that he does not visit any department on the Monday of the third week of every month.  When did he visit the Purchase department?  Consider the following s…

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Introduction / Context: This data sufficiency question involves a recurring schedule of visits by a company chairman to various departments. The chairman visits exactly one department each Monday except on the Monday of the third week of every month. The question asks when he visited the Purchase department. Two statements refer to visits to Accounts, Stores, and Purchase, but do not explicitly fix calendar dates beyond a reference to September. We must decide whether these statements uniquely determine the week or date of the Purchase visit. Given Data / Assumptions: The chairman visits one department on each Monday of every week, except that he skips the Monday of the third week of every month. Statement I: He visited the Accounts department in the second week of September after having visited the Purchase department on an earlier occasion. Statement II: He visited the Purchase department immediately after visiting the Stores department but before visiting the Accounts department. Departments can be revisited in different weeks or months; nothing states that each department is visited exactly once. Earlier occasion refers to some previous Monday visit, but not necessarily the immediately preceding Monday unless so stated. Concept / Approach: We interpret the recurring Monday visits as a sequence of time slots. Each Monday that is not the third Monday of a month has exactly one department visited. The question is whether the references to Accounts in the second week of September and the order involving Stores and Purchase are enough to pin down the exact week number or calendar date for the Purchase visit. If multiple Monday schedules fit the statements, then the data are not sufficient. Step-by-Step Solution: Step 1: From statement I, we know that the chairman visited the Accounts department on the Monday of the second week of September. We also know that the Purchase department was visited on an earlier occasion, but this earlier occasion could be the Monday of the first week of September, or any suitable Monday in previous months. Step 2: Statement I alone does not specify whether that earlier occasion was the immediately previous Monday or some Monday many weeks earlier. Therefore, statement I alone cannot fix the date or even the exact week number of the Purchase visit. Step 3: From statement II, we know that in the sequence of visits there is a pattern Stores then Purchase then Accounts, with Purchase immediately after Stores and before Accounts. There is no information about which month or week this sequence occurs, nor whether Accounts in this sequence is the same Accounts visit mentioned in September in statement I. Step 4: Combining statements I and II, we might naturally guess that the Accounts visit in the order Stores, Purchase, Accounts is the same Accounts visit in the second week of September. However, even if we assume this, the data still do not force the exact week of the Purchase visit. Step 5: Suppose the Accounts visit in the second week of September is the one referred to in statement II. Then the Purchase visit occurred at some earlier Monday visit between a Stores visit and that Accounts visit. The Stores and Purchase visits could have occurred in late August or in the first week of September, or even earlier, depending on how the sequence of Monday visits and the third week gaps line up in the calendar. Multiple schedules can satisfy both statements. Step 6: Additionally, nothing prevents the possibility that the chairman visits Accounts more than once. If there are multiple Accounts visits, then the sequence Stores, Purchase, Accounts may correspond to a different run of Mondays than the September Accounts visit, which makes the date of the Purchase visit even more ambiguous. Step 7: Because there are several possible ways to place the Stores, Purchase, and Accounts visits in the calendar while satisfying all the given conditions, the specific time of the Purchase department visit cannot be uniquely determined from these statements. Verification / Alternative check: To verify, imagine two different timelines that both satisfy all given information. In the first, the sequence Stores, Purchase, Accounts might be placed so that Purchase falls on the first Monday of September and Accounts on the second Monday. In another, the same three visits could occur entirely in August, with a different Accounts visit taking place in the second week of September. Since both timelines match statements I and II, yet they assign different dates to the Purchase visit, the answer is not uniquely determined. This confirms that the data are insufficient. Why Other Options Are Wrong: Option A claims that statement I alone is sufficient, but as seen, it only says that the Purchase visit occurred sometime before the September Accounts visit, which is too vague. Option B claims that statement II alone is sufficient, which is clearly false because II contains no calendar reference. Option C suggests that either statement alone could be sufficient, which is also incorrect. Option D claims that both statements together are necessary and sufficient, which fails because the combined data still allow many possible placements for the Purchase visit. Only option E correctly states that even both statements together are not sufficient. Common Pitfalls: Learners often assume that phrases like earlier occasion and immediately after force a very tight schedule without checking whether multiple months and repeated visits are allowed. Another common mistake is to treat the September Accounts visit as the only such visit, although the problem never makes that explicit. In data sufficiency, we must avoid adding extra assumptions beyond what is stated clearly. Final Answer: The data even in both statements together are not sufficient to answer the question, so the correct option is E.

Question 11

What is the area of a rectangular plot?  Consider the following statements:  I. The perimeter of the plot is 208 metres. II. The length of the plot is greater than its breadth by 4 metres.

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Introduction / Context: This data sufficiency question is about finding the area of a rectangular plot. We are given the perimeter and a relation between length and breadth in two separate statements. The key is to determine whether either statement alone or both together allow us to compute a unique area. The question checks understanding of basic geometry formulas and how many conditions are needed to fix a rectangle dimensions. Given Data / Assumptions: The plot is rectangular with length L and breadth B. Statement I: The perimeter of the plot is 208 metres. Statement II: The length is 4 metres more than the breadth, that is, L = B + 4. Area of a rectangle is L * B, and perimeter is 2 * (L + B). Concept / Approach: A rectangle is determined by two independent measurements, such as length and breadth. Perimeter alone gives only one equation relating L and B, and a simple difference relation alone also gives one equation. To compute the area uniquely, we must determine L and B. Therefore we check if each statement alone is enough to fix both dimensions, and then whether the combination is enough. Step-by-Step Solution: Step 1: From statement I, perimeter P is 208 metres. The formula for the perimeter of a rectangle is P = 2 * (L + B). So we have 2 * (L + B) = 208, or L + B = 104. Step 2: With statement I alone, we only know that L + B = 104. There are many pairs (L, B) that satisfy this, such as (50, 54), (40, 64), and so on. Each pair gives a different area. Therefore statement I alone is not sufficient. Step 3: From statement II, L = B + 4. This gives a difference relation between length and breadth, but no information about their sum or any absolute size. Many pairs of dimensions satisfy this relation, such as (10, 6), (20, 16), and so on. Each would result in a different area. Hence statement II alone is also not sufficient. Step 4: Combine statements I and II. We have L + B = 104 and L = B + 4. Substitute L into the sum equation: (B + 4) + B = 104. Step 5: Simplify the equation. This gives 2B + 4 = 104. Subtract 4 from both sides to obtain 2B = 100, so B = 50. Then L = B + 4 = 54. Step 6: Compute the area. Area = L * B = 54 * 50 = 2700 square metres. Because the dimensions are now uniquely determined, the area is unique as well. Verification / Alternative check: To verify sufficiency, note that we started with two unknowns L and B. Statement I reduced them to one free variable by giving a relation between L and B, but not a unique solution. Statement II also reduced them to one free variable by another relation. When both relations are used together, they form a system of two equations in two unknowns, which yields a unique solution. Without both equations, we always have infinitely many possible rectangles and therefore multiple possible areas. Why Other Options Are Wrong: Option A is wrong because perimeter alone does not fully determine the shape dimensions. Option B is wrong because knowing only the difference between length and breadth is not enough to fix their values. Option C claims that either statement alone is sufficient, which is not true. Option E claims that even both statements together are not sufficient, which contradicts the clear solution we obtained. Only option D correctly recognises that both statements together are necessary and sufficient. Common Pitfalls: A typical mistake is to think that a known perimeter always fixes a unique rectangle, forgetting that length and breadth can vary while keeping the same sum. Another error is to overlook basic algebra when dealing with the difference relation. Writing down the two equations explicitly and solving them step by step prevents these misunderstandings. Final Answer: The data in both statements I and II together are necessary to answer the question, so the correct option is D.

Question 12

How is B related to A?  Consider the following statements:  I. A is the sister of B. II. C is the father of both A and B.

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Introduction / Context: This data sufficiency question asks about the relationship of B to A in a family. Two statements describe A as the sister of B and introduce a father C of both A and B. The aim is to decide whether these statements uniquely determine whether B is a brother, a sister, or some other relation to A. The challenge lies in noticing which details are specified and which remain unknown, especially regarding gender. Given Data / Assumptions: Statement I: A is the sister of B. This tells us that A is female and that A and B are siblings. Statement II: C is the father of both A and B. This confirms that A and B share the same father and are siblings. Brother means a male sibling, and sister means a female sibling. No explicit information is given about the gender of B. Concept / Approach: The question asks How is B related to A, which usually requires us to say whether B is the brother of A, the sister of A, or some other clearly defined relation. To answer this precisely, we must know both the basic sibling relationship and the gender of B. Data sufficiency analysis checks whether the statements force B to be male or female, or whether both possibilities remain open. Step-by-Step Solution: Step 1: From statement I, A is the sister of B. This establishes that A is female and that A and B are siblings. However, it does not state whether B is male or female. Therefore, B could be a brother or a sister of A under this statement alone. Step 2: Because B gender is not specified in statement I, we cannot answer the question How is B related to A with a unique description. Statement I alone is not sufficient. Step 3: From statement II, C is the father of both A and B. This tells us that A and B are children of C and therefore siblings. Again, there is no mention of gender for either A or B, so we cannot say whether B is a brother or a sister from statement II alone. Step 4: Therefore, statement II alone is also not sufficient to answer the question. Step 5: Now combine statements I and II. Together, they still give only that A and B are siblings, that A is female, and that C is their father. The combination does not add any new information about the gender of B beyond what was already missing. Step 6: Consider possible scenarios. In one scenario, B is a male child of C, in which case B is the brother of A. In another scenario, B is a female child of C, in which case B is the sister of A. Both scenarios satisfy statements I and II and are compatible with A being described as the sister of B. Step 7: Since at least two different possibilities for B relation exist and both fit all the given information, the data are not sufficient to decide uniquely how B is related to A. Verification / Alternative check: A quick sufficiency check is to ask whether we can fix the gender of B from the statements. Because neither statement ever tells us that B is a son or a daughter, or uses words like brother or sister for B, the gender of B remains undetermined even when both statements are used. Whenever a relationship label depends on gender, and gender is not fixed, the data cannot be sufficient to answer the relationship question exactly. Why Other Options Are Wrong: Option A incorrectly claims that statement I alone is sufficient. Although it gives a sibling relation, it does not specify whether B is male or female. Option B is also incorrect because statement II alone provides only a common parent, not gender. Option C, which claims that either statement alone is sufficient, is doubly wrong. Option D suggests that both statements together are necessary and sufficient, but in fact they still leave B gender open. Only option E correctly states that even both statements together are not sufficient. Common Pitfalls: A common error is to assume that if A is the sister of B, then B must be male, but this is not implied by the language. The phrase A is the sister of B only fixes A gender and the sibling relationship, not B gender. Another mistake is to think that giving a father automatically implies that one child must be a son. Careful attention to what is explicitly stated, especially about gender, is vital in family relationship questions. Final Answer: The data even in both statements together are not sufficient to answer the question, so the correct option is E.

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