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  1. 3 TIPS to crack Reasoning Questions on Statements and Assumptions Tip #1: Assumptions should be drawn only from the given statement, not from your real life experiences Statement: "Please put more people on the job to make up for the delay." Assumptions: (I) Delay is inevitable in most jobs. (II) Output will increase with more number of people on the job. A. Only assumption I is implicit B. Only assumption II is implicit C. Either I or II is implicit D. Neither I nor II is implicit E. Both I and II are implicit Solution: Assumption I is not implicit, even if based on your real life experiences, you may believe that delay is inevitable in most jobs. As per the statement, increasing the number of people will make up for the delay by increasing the output. So II is implicit. Thus, correct option is B. Statement: If you are an engineer, we have a challenging job for you. Assumptions: (I) We need an engineer. (II) You are an engineer. A. Only assumption I is implicit B. Only assumption II is implicit C. Either I or II is implicit D. Neither I nor II is implicit E. Both I and II are implicit Solution: An engineer is needed, hence the advertisement. So assumption I is implicit. Assumption II is not implicit, even if the reader is an engineer. Hence the correct option is A. Tip #2: Don’t extrapolate beyond what is given in the statement to arrive at false assumptions Statement: Everybody loves reading adventure stories. Assumptions: (I) Adventures stories are the only reading material. (II) Nobody loves reading any other material. A. Only assumption I is implicit B. Only assumption II is implicit C. Either I or II is implicit D. Neither I nor II is implicit E. Both I and II are implicit Solution: Neither (I) nor (II) can be reasonably drawn from the given statement. Thus, correct answer will be D. Statement: “You can't solve syllogism question without constructing a diagram. Difficult questions on syllogism need Venn diagram solving,” a teacher tells his students. Assumption: (I) The students are not intelligent. (II) Problem cannot be solved. A. Only assumption I is implicit B. Only assumption II is implicit C. Either I or II is implicit D. Neither I nor II is implicit E. Both I and II are implicit Solution: Assumption I is false because nothing is said about the students. Assumption II is false since the statement does not say that the problem cannot be solved. So, correct answer is D. Tip #3: If a notice/ advertisement is put up, the implicit assumption is that it will have impact Statement: Do not copy our software without our permission - A notice. Assumptions: (I) It is possible to copy the software. (II) Such warning will have some effect. A. Only assumption I is implicit B. Only assumption II is implicit C. Either I or II is implicit D. Neither I nor II is implicit E. Both I and II are implicit Solution: If it were not possible to copy the software, there would be no need for the notice. Hence, (I) is implicitly true. Similarly, if the warning would have no impact, the notice would not be posted. Hence, (II) is also implicit. So, the correct answer is E. Statement: "A rare opportunity to be a professional while you are at home." - An advertisement for computer literate housewives by a computer company. Assumptions: (I) Some housewives simultaneously desire to become professional. (II) Computer industry is growing at a fast pace. (III) It is possible to be a professional as well as a housewife. A. Only I and II are implicit B. Only II and III are implicit C. Only I and III are implicit D. Only II is implicit E. None of these Solution: If no housewives desired to become professionals, there would be no reason to make the advertisement. Hence, (I) is implicit. (III) is also implicit, otherwise the advertisement would not be made in the first place. Thus, the correct answer is C. 3 TIPS on cracking Reasoning Questions on Statements and Arguments Classification of Arguments 1. Strong Argument: Strong arguments are those which are directly related to the statement, do not make extra assumptions, are valid and hold true. 2. Weak Argument: Weak arguments are those which not related to the statement directly, may make unjustifiable assumptions, are false, vague or invalid. Example: Statement: Are nuclear families better than joint families? Arguments: I. No. Joint families ensure security and also reduce the burden of work. II. Yes. Nuclear families ensure greater freedom. With so many people around in a joint family, there is more security. Also, work is shared. Again, in nuclear families, there are lesser number of people and so lesser responsibilities and more freedom. Both arguments are Strong. Tip #1: Arguments with unjustifiable assumptions are weak Statement: Should the education at all levels be offered only in vernacular medium? Arguments: I. Yes. This is the only way to enhance performance of the students. II. No. This will severely affect acquiring knowledge for want of good text books in vernacular medium. A. Only argument I is strong B. Only argument II is strong C. Either I or II is strong D. Neither I nor II is strong E. Both I and II are strong Solution: In (I) it is assumed that offering education in vernacular is the only way to enhance performance of the students. But, this assumption is not justifiable. Hence (I) is weak. Argument (II) is strong since offering books in English or other languages would open up more avenues for Students. So the correct answer is B. Statement: Should girls learn arts like judo and karate? Arguments: (I) Yes. It will enable them to defend themselves from rogues and ruffians. (II) No. They will lose their feminine grace. Solution: Learning martial arts is necessary for girls for self-defense. So, argument (I) is strong. In argument (II), it is assumed without justification that martial arts will impact feminine grace. Thus, the answer will be A. Tip #2: If an argument is not scientific or contradicts your general knowledge or goes against research by reputed organizations, it is weak Statement: Should books by only deserving authors be published? Arguments: (I) Yes. It will save a lot of paper which is in short supply. (II) No. It is not possible to draw a line between the deserving and the undeserving. A. Only argument I is strong B. Only argument II is strong C. Either I or II is strong D. Neither I nor II is strong E. Both I and II are strong Solution: Argument (I) says that paper is in short supply. But, this is not something we have ever heard in the news. It is contradictory to our general knowledge and is therefore weak. Argument (II) is a strong argument because establishing committees to control the publishing of books would be against freedom of speech. Such committees would also be partial and have their own agendas. So the correct answer is B. Tip #3: Arguments that are illogical and unrelated to the statement are weak Statement: Should all the school teachers be debarred from giving private tuitions? Arguments: (I) No. The needy students will be deprived of the expertise of these teachers. (II) Yes. This is an injustice to the unemployed educated people who can earn their living by giving tuitions. (III) Yes. Only then the quality of teaching in schools will improve. (IV) Yes. Now salary of these teachers is reasonable. A. Only I and III are strong B. Only I, II and III are strong C. Only III and IV are strong D. Only II, III and IV are strong E. None of these Solution: Argument (I) is weak because needy students will not be deprived of their education in schools, if those teachers also offer private tuitions on the side. Argument (II) is weak because other unemployed individuals can also hold private tuitions if they are capable of doing so. Argument (III) is strong. There could be a tendency for teachers to reduce their efforts in schools to encourage more students to join their private tuition classes. Argument (IV) is weak. By what standards can anyone decide to stop a worker from seeking further employment by deciding that their salary is already reasonable? Thus, the correct answer is E. 5 TIPS on cracking Reasoning Questions on Conclusions and Course of Action Tip #1: Conclusion drawn on the basis of data not provided in the statement(s) does not hold Statements: In a one-day cricket match, the total runs made by a team were 200. Out of these 160 runs were made by spinners. Conclusions: (I) 80% of the team consists of spinners. (II) The opening batsmen were spinners. A. Only conclusion I follows B. Only conclusion II follows C. Either I or II follows D. Neither I nor II follows E. Both I and II follow Solution: Nothing has been said in the statements about the number of spinners or the batsmen. So both conclusions do not follow. Hence the correct option is D. Statement: Any student who does not behave properly while in the school brings bad name to himself and also for the school. Conclusions: (I) Such student should be removed from the school. (II) Stricter discipline does not improve behavior of the students. A. Only conclusion I follows B. Only conclusion II follows C. Either I or II follows D. Neither I nor II follows E. Both I and II follow Solution: The statement mentions nothing about disciplinary action against misbehaving students, so none of the conclusions follows. The correct answer is D. Tip #2: Conclusions with definitive qualifiers (all, no one, at least) require stringent support to be valid Statements: A forest has as many sandal trees as it has Ashoka trees. Three-fourth of the trees are old ones and half of the trees are at the flowering stage. A. All Ashoka trees are at the flowering stage. B. All sandal trees are at the flowering stage. C. At least one-half of the Ashoka trees are old. D. One-half of the sandal trees are at the flowering stage. E. None of these Solution: Conclusions A, B, C do not hold because of the presence of definitive words ‘all’ and ‘at least’ when the statements do not provide such definite information. D does not follow because nothing has been said about the fraction of sandal trees in flowering stage. Thus, the answer is E. Statement: Money plays a vital role in politics. Conclusion: (I) The poor cannot become politicians. (II) All rich men take part in politics. A. Only conclusion I follows B. Only conclusion II follows C. Either I or II follows D. Neither I nor II follows E. Both I and II follow Solution: None of the conclusions follows since they are definitive whereas the given statement is not. Hence the correct answer is D. Tip #3: Conclusion should not contradict your common sense and general knowledge Statements: Our securities investments carry market risk. Consult your investment advisor or agent before investing. Conclusions: (I) One should not invest in securities. (II) The investment advisor can reasonably estimate the market risk A. Only conclusion I follows B. Only conclusion II follows C. Either I or II follows D. Neither I nor II follows E. Both I and II follow Solution: Investment in securities involves risk but this does not mean that one should not invest in securities. So I does not follow. Since the statement advises one to consult investment advisor before investing, so II follows. Thus the correct answer is B. Statements: Soldiers serve their country. A. Men generally serve their country. B. Those who serve their country are soldiers. C. Some men who are soldiers serve their country. D. Women do not serve their country because they are not soldiers. Solution: Conclusion A does not follow since it is ambiguous. Again, B does not hold since soldiers are not the only people serving their country. Conclusion D does not hold because it contradicts the fact that women can also be soldiers. C is the answer. Tip #4: There must be no extremity involved in the course of action Statement: The members belonging to two local societies occasionally fight with each other on the main highway and traffic is jammed always. Courses of Action: I. The local police station should immediately deploy policemen round the clock on the main highway. II. Those involved in fighting should be identified and put behind bars. III. Local authority should cease the management of the two societies with immediate effect. A. Only I and II B. Only I and III C. Only II and II D. All of I, II, III E. None Solution: To stop the fighting, the police must deploy troops and the culprits must be put behind bars to deter them from causing such scenes in the future, so I and II follow. The 3rd course of action would be extreme. So the answer is A. Tip #5: The course of action should be justified based on the statement Statement: The Company X has rejected the first load of valves supplied by Company A and has cancelled its entire huge order quoting use of inferior quality material and poor craftsmanship. Courses of Action: I. The Company A needs to investigate functioning of its purchase, production and quality control departments. II. The Company A should inspect all the valves rejected by Company X. III. The Company A should inform Company X that steps have been taken for improvement and renegotiate schedule of supply. A. Only I and II B. Only I and III C. Only II and II D. All of I, II, III E. None Solution: First of all, company A should inspect the rejected valves to ensure if they are really substandard. If so, it should scrutinize its working thoroughly and fix its quality issues. III is not justified from the statement, because the quality errors have not been fixed and issuing a statement to Company X to that effect would be a lie. The answer is A. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  2. 9 TIPS on cracking Aptitude Questions on Visual-Spatial Problem Solving Tip 1: For pattern construction questions, remember that the resultant pattern must be formed using only the given figures Question: Find out which of the figures (1), (2), (3), (4) can be formed from the pieces given in figure (X). Solution: The figure must consist of a right triangle, a trapezium and an irregular pentagon, as shown above. Looking at the options one by one, we see that option (1) consists of the given figures rotated by 90⁰, the triangle and trapezium, clockwise, and the pentagon, anti-clockwise. (2) is not made of any of (X) except the trapezium. Similarly, (3) does not contain the pentagon and (4) consists of all shapes other than those in (X). Thus, the correct option is (1). Question: Find out which of the figures (1), (2), (3), (4) can be formed from the pieces given in figure (X). Solution: (X) Consists of 2 isosceles triangles and 2 right angled triangles. Now, (1), (2) and (4) cannot be formed using only these figures. Only (3) can be made using them, as follows: Thus, the answer is (3). Tip 2: Sometimes, it is quicker to just eliminate the invalid options Question: Find out how the key figure (X) look will like after rotation. Solution: We analyze the options one by one. (1) Can be eliminated since it contains more sections than the given figure. Similarly, (2) and (3) are also different from (X). Hence the answer is (4). Question: Select the alternative in which the specified components of the key figure (X) are found. Solution: Analyzing the options one by one, we see that (1) and (2) do not consist of ( ̶ ), while (3) consists of (ᴓ), which is not present in (X) at all. Thus, all these options are eliminated, and the correct answer is (4), which, we may cross-check as containing all components of (X) only. Tip 3: For problems on Paper Cutting, imagine the process of folding, cutting and opening, and draw indicative diagrams if required Question: Choose a figure which would most closely resemble the unfolded form of Figure (Z). A. (1) B. (2) C. (3) D. (4) Solution: The paper has been folded at 2 corners and 1 hole has been made on each fold. Since the paper was folded from either corner, when they are unfolded, there will be 4 holes, all on the diagonal joining them. Thus, figure (3) is the correct figure, and the answer is (C). Question: Choose a figure which would most closely resemble the unfolded form of Figure (Z). A. (1) B. (2) C. (3) D. (4) Solution: The paper has been folded equally thrice. Now, when it is unfolded, it will consist of 4 symmetrical parts, that look exactly like the cutout in form (Z). Thus, one of (2) and (3) is correct. Now, figure (3) is incorrect since there is a cut along the base of the (Z) which will result in a joint shape. So the answer is (B). Tip 4: In Dot Situation problems, simply find the figure that contains all the sections as the dots in the original figure Question: Select the figure which satisfies the same conditions of placement of the dots as in Figure-X. Solution: In (X), one of the dots lies in the region common to circle and square only, another dot lies in the region common to square, triangle and rectangle only and the 3rd dot lies in the region common to triangle and rectangle only. In each of the figures (1), (2) and (3) there is no region common to square, triangle and rectangle only. Only fig. (4) Consists of all the 3 types of regions. Tip 5: For problems on Embedded Images, find the figure from the given options in which the original figure can be completely traced Question: Find out the alternative figure which contains figure (X) as its part. (A) 1 (B) 2 (C) 3 (D) 4 Solution: Looking at the answer figures one by one: (X) Cannot be traced anywhere in (1). Again, in (2), the figure is the lateral inversion of (X). Similarly, in (4), the figure is broken. Thus, all of these can be eliminated. Only in fig. (3) can fig. (X) be traced. Therefore, the answer is (C). Question: Find out the alternative figure which contains figure (X) as its part. (A) 1 (B) 2 (C) 3 (D) 4 Solution: From the figures, it is clear that fig. (X) cannot be traced in fig. (1), (2) and (3), but can be traced only in fig. (4). Thus, the answer is (D). Tip 6: Mirror Image of a figure has all parts of the figure laterally inverted but not vertically Some points to note are: 1. Mirror images of alphabets A, H, I, M, O, T, U, V, W, X, Y, and digits 0, 8, are the same as the original ones. 2. The mirror image of a sequence of words and/or numbers is the mirror image of each, written in the reverse order. 3. The figure should remain the same vertically. Question: Choose the alternative which is closely resembles the mirror image of the given combination. Solution: Only the 2nd option can be correct since in all the other options the 1st and last characters are incorrect, i.e. not beginning with the mirror image of 2 and ending with A. Thus, the answer is (2). Question: Choose the correct mirror image of the figure (X) from amongst the alternatives Solution: Options (1) and (2) may be eliminated since they are vertically inverted as well. Now, since the small square appears on the left side of the line in (X), it should appear on the right side of the line in the mirror image. So the answer is (3). Tip 7: Water Image of a figure has all parts of the figure vertically inverted Thus, here: 1. The water images of alphabets B, C, D, E, H, I, K, O, X, and of digits 0, 8, remain the same as the original. 2. The water image of a sequence of words and/or numbers is the water image of each, written in the same order. 3. The figure should remain the same vertically. Question: Choose the alternative which closely resembles the water-image of the given combination. Solution: Option (1) can be eliminated since it is laterally inverted as well. Again, in (2), L and in (3), E is laterally inverted as well. Only (4) is the correct water image of (X). Tip 8: While counting the number of figures, count smaller ones first followed by larger ones etc. Note: Remember to count all the figures. For example, if you are asked to count the no. of triangles in a figure, count the simplest triangles, triangles made by 2 simpler components, those made by 3 simpler components, etc. until no triangle can be formed any longer. Question: Count the number of triangles and squares in the given figure. (A) 36 triangles, 7 squares (B) 38 triangles, 9 squares (C) 40 triangles, 7 squares (D) 42 triangles, 9 squares Solution: The figure may be labeled as follows: Triangles: Simplest triangles: BGM, GHM, HAM, ABM, GIN, UN, JHN, HGN, IKO, KLO, LJO, JIO, KDP, DEP, ELP, LKP, BCD and AFE i.e. 18 in number. Triangles composed of 2 components each: ABG, BGH, GHA, HAB, HGI, GIJ, IJH, JHG, JIK, IKL, KLJ, LJI, LKD, KDE, DEL and ELK i.e. 16 in number. Triangles composed of 4 components each: BHI, GJK, ILD, AGJ, HIL and JKE i.e. 6 in number. Total number of triangles in the figure = 18 +16 + 6 = 40. Squares: Squares composed of 2 components each: MGNH, NIOJ and OKPL i.e. 3 in number. Squares composed of 4 components each: BGHA, GIJH, IKLJ and KDEL i.e. 4 in number. Total number of squares in the figure = 3 + 4 = 7. Therefore, the answer is (A). Tip 9: For problems on Paper Folding, draw mirror/water images as required for the folded section and merge with the other section Question: Find out from amongst the four alternatives as to how the pattern would appear when the transparent sheet is folded at the dotted line. Solution: The mirror image of the right side of the image is given by (C). Hence, it is the correct answer. Question: Find out from amongst the four alternatives as to how the pattern would appear when the transparent sheet is folded at the dotted line. Solution: Option (A) is clearly not the correct answer. Again, options (2) and (3) may be eliminated since in fig. (X), the two straight lines on the left side are vertical, and those on the right are horizontal. Thus, the answer is (D). (contd..) Tips on Non-Verbal Reasoning Test on Visual Spatial Problems - https://learningpundits.com/module-view/84-visual-spatial-problems/1-tips-on-visual-spatial-problems/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  3. 3 TIPS on cracking Reasoning Questions on Venn Diagrams Tip #1: Start working from the outermost Venn diagram and gradually move inwards Question: Which of the following diagrams indicates the best relation between Women, Mothers and Engineers? Solution: Some women are mothers and some are engineers. All mothers are women, and some mothers are engineers. However, not all engineers are women. So, the answer is A. Question: In an organization of pollution control board, engineers are represented by a circle, legal experts by a square and environmentalist by a triangle. Who is most represented in the board as shown in the following figure? (a) Environmentalists with Legal background (b) Legal Experts (c) Engineers with legal background (d) Environmentalists with engineering background Solution: We have to assume that the shape with the largest area is the most represented. Among the given options, D (i.e.) Environmentalists with Engineering background has the largest area. Thus, the correct answer is D. Tip #2: Reduce the question as the overlapping of different portions of the Venn diagram In the following figure Upper Square represents the persons who know English, triangle those who know Marathi, Lower Square to those who know Telugu and circle those who know Hindi. In the different regions of the diagram, the counts are given. Question: How many persons can speak English and Hindi both the languages only? Solution: The people who can speak English and Hindi is represented by the portion of intersection of the circle and the upper square only. The no. here is 5. So 5 people can speak Hindi and English both. Question: How many persons can speak all languages? Solution: The people speaking all languages is represented by the portion of the Venn diagram that has all the figures overlapping. Since there is no such area where all the figures are overlapping, it means that no person can speak all the languages. Tip #3: The portion outside a figure represents the complement of the quantity inside it In the following diagram rectangle represents men, Triangle represents educated, Circle represents urban and square represents government employees. Question: How many men are educated but not urban? (a) 9 (b) 5 (c) 4 (d) 11 Solution: Educated but non-urban men is represented by the portion of the triangle that overlaps with the rectangle but not the circle. Thus, 11 men are educated but not urban. The correct answer is D. Question: Which one of the following represents a woman who is urban as well as government employee? (a) 7 (b) 13 (c) 10 (d) 6 Solution: Since the rectangle represents men, the portions of the other shapes that do not overlap with the rectangle represent women. The portion of the circle that overlaps with the square but not with the rectangle represents urban women who are government employees. Thus the answer is 10, i.e., C. (contd..) Tips on Logical Reasoning Test Questions on Venn Diagrams - https://learningpundits.com/module-view/72-venn-diagrams/1-tips-on-venn-diagrams/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  4. TIP on cracking Reasoning Questions on Seating Arrangement Draw a diagram to illustrate the problem: Question: A, P, R, X, S and Z are sitting in a row. S and Z are in the center. A and P are at the ends. R is sitting to the left of A. Who is to the right of P? A. A B. X C. S D. Z Solution: A has to be at the right end otherwise there would be no place to A’s left for R to occupy. S and Z are seated at the central positions, although the specific order in which they are seated is not mentioned. This leaves only 1 space vacant, i.e., next to P and only 1 person, i.e., X. Hence the answer is B. Question: 6 friends are sitting in a circle and facing the center of the circle. Deepa is between Prakash and Pankaj. Priti is between Mukesh and Lalit. Prakash and Mukesh are opposite to each other. Who is sitting right to Prakash? A. Mukesh B. Pankaj C. Pankaj D. Lalit Solution: Fixing Prakash’s position, we find Mukesh’s position. Deepa is between Pankaj and Prakash, and this can be on either side of Prakash. Now, Priti is between Mukesh and Lalit. This can happen when they are on the opposite side of Prakash. So, either Lalit or Deepa will be to the right of Prakash. So the correct option is D. (contd..) Tips on Logical Reasoning Test Questions on Seating Arrangement - https://learningpundits.com/module-view/76-seating-arrangement/1-tips-on-seating-arrangement/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  5. 3 Tips on Solving Reasoning Questions on Number Series Tip #1: Arithmetic Series, Geometric Series, Patterns in Differences (1) Arithmetic Series: When the differences between the successive numbers given in the series is the same. For example: 2, 5, 8, 11, 14... (Here the difference between the numbers is 3, hence the next number will be 17) (2) Geometric Series: When each successive number in the series is obtained by multiplying or dividing the previous one by a fixed number. For example: 2, 6, 18, 72,… (3) Patterns in differences: Calculate the differences between the numbers given in the series provided in the question. Then try to observe the pattern in the new set of numbers that you have obtained after taking out the difference. Question: Look at this series: V, VIII, XI, XIV, __, XX, ... What number should fill the blank? Solution: This is an arithmetic series in Roman numerals; each no. is 3 more than the previous one. Thus, the missing number will be the Roman equivalent of 20 – 3 = 17, i.e., XVII. Question: Look at this series: 201, 202, 204, 207, ... What number should come next? Solution: The difference between the successive numbers is 1, 2, 3 respectively. Thus, the difference keeps on increasing by 1. Thus, the next number in the series will be 207 + 4 = 211. Tip #2: Pattern in Alternate/Adjacent numbers 1. Pattern in Alternate numbers: When there is a pattern between every alternate or third number in the series. For example: 2, 9, 5, 12, 8 , 15, 11.... 2. (2) Pattern in adjacent number: When adjacent numbers in the series changes based on a logical pattern. For example: 2, 4, 12, 48... Here the numbers are being multiplied by 2, then by 3, then by 4 etc. Question: Look at this series: F2, __, D8, C16, B32, ... What number should fill the blank? Solution: This is a complex series in which the successive letters decrease by 1 and the successive numbers are multiplied by 2. Thus, the number in the blank will be E4. Question: Look carefully at the following series and choose the pair that comes next. 42 40 38 35 33 31 28 A. 25, 22 B. 26, 23 C. 26, 24 D. 25, 23 Solution: This is an alternating subtraction series in which 2 is subtracted twice, then 3 is subtracted once, then 2 is subtracted twice, and so on. So the next terms in the following series will be 26 and 24. Thus the correct answer is C. Tip #3: Prime Numbers, Squares/Cubes, Alternate primes/exponents (1) Squares/Cubes: When numbers are a series of perfect squares. For example: 81, 100, 121, 144, 169... (2) Cube/Square roots: When the numbers are a series of perfect cubes. For example: 512, 729, 1000... (3) Alternate Primes: Here the series is framed by taking the alternative prime numbers. For example: 2, 5, 11, 17, 23, 31… Question: Look at this series: 4, 7, 16, 13, __, 19, 64, 29, ... What number should fill the blank? Solution: There are two alternating series here. The first one is the square of the multiples of 2 {22, 42, 62, 82 …} and the second one is a series of alternating prime numbers. The missing number is a part of the first series so the missing number is 62 = 36. Question: Look carefully at the following series and choose the pair that comes next. 4 7 26 10 13 20 16 A. 14, 17 B. 18, 14 C. 19, 13 D. 19, 14 Solution: Two patterns alternate here, with every third number following the alternate pattern. In the main series, beginning with 4, 3 is added to each number to arrive at the next. In the alternating series, beginning with 26, 6 is subtracted from each number to arrive at the next. So the next numbers will be 16 + 3 = 19, 20 – 6 = 14. Thus, the correct answer is D. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  6. 3 Tips on Cracking Reasoning Questions on Letter and Symbol Series Tip #1: Identify the changes happening in the successive patterns of a symbol series. Question: Look carefully at the sequence of symbols to find the missing pattern. Solution: In each segment, the shapes are the same, but the one in the middle is larger than the two on either side. Also, one of the figures is shaded and that this shading alternates between 1st right and then left. Thus in the 3rd segment, there will be two small squares with a big square between them and the right small square will be blackened. Thus, the correct option is (2). Question: Look carefully at the sequence of symbols to find the missing pattern. Solution: In each of the segments, the figures alternate between one-half and one-fourth shaded on the opposite sides. Thus, to complete the last segment, there should be one square that is one-fourth shaded and another that is half-shaded on the left side. Thus, the correct option is (4). Tip #2: Break the complex letter series into simpler ones and solve them individually Question: Look at the following series and fill in the blank. SCD, TEF, UGH, ____, WKL Solution: There are two alphabetical series here. The first series is with the first letters only, and consists of the consecutive letters of the alphabet starting from S, i.e., S, T, U, V, W. The second series involves the remaining two letters and involving 2 consecutive letters of the alphabet starting from C, i.e., CD, EF, GH, IJ, KL. So the missing term will be VIJ. Question: Look at the following series and fill in the blank. DEF, DEF2, DE2F2, _____, D2E2F3 Solution: In this series, the letters remain the same: DEF. The subscript numbers follow the following series: 111, 112, 122, 222, 223, 233, 333, ... Thus, the missing term is D2E2F2. Question: Look at the following series and fill in the blank. JAK, KBL, LCM, MDN, ____. Solution: This series may be broken up into 3 component series. The 1st one is the series J, K, L, M and N. The 2nd one follows the pattern A, B, C, D, E, and the 3rd one: K, L, M, N and O. Thus the missing term is NEO. Tip #3: Eliminate the incorrect choices Question: Look carefully at the sequence of symbols to find the missing pattern. Solution: Since no 2 figures in the same segment are the same, Options (1) and (2) are eliminated. Now, the shading inside the circles gradually increases or decreases. Thus, the correct option is (3). Question: Look at the following series and fill in the blank. JAK, KBL, LCM, MDN, _____ A. OEP B. NEO C. MEN D.PFQ Solution: The first letter of each term follows the series J, K, L, M and N. This eliminates choices A, C, and D. Thus, the correct answer is B. Question: Look carefully at the sequence of symbols to find the missing pattern. Solution: In all these segments, the count of dots in the bottom box of the previous set equals the count of dots in the top box of the current set. This eliminates options (2), (3) and (4). Thus, the correct option is (1). (contd..) Tips on Letter & Symbol Series - https://learningpundits.com/module-view/71-letter-and-symbol-series/1-tips-on-letter-and-symbol-series/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  7. 2 TIPS on cracking Reasoning Questions on Direction Sense Tip #1: Make a sketch of the data provided Question: A man walks 5 km toward south and then turns to the right. After walking 3 km he turns to the left and walks 5 km. Now in which direction is he from the starting place? A. West B. South C. North-East D. South-west Solution: As we may see from the figure, the man is finally South-West from the starting point. Thus, the correct answer is D. Question: Ravi left home and cycled 10 km towards South, then turned right and cycled 5 km and then again turned right and cycled 10 km. After this he turned left and cycled 10 km to reach home. How many kilometers will he have to cycle to reach his home straight? A. 10 km B. 15 km C. 20 km D. 25 km Solution: Required distance = (5 + 10) km = 15 km. Thus, the answer is B. Tip #2: Draw a grid in questions when multiple possibilities can occur Question: 8 trees: mango, guava, papaya, pomegranate, lemon, banana, raspberry and apple are in two rows of 4 in each facing North and South. Lemon is between mango and apple but just opposite to guava. Banana is at one end of a line and is to the right of guava tree. Raspberry tree which at one end of a line, is just diagonally opposite to mango tree. Which tree is just opposite to banana tree? A. Mango B. Pomegranate C. Papaya D. Data is inadequate. Solution: Filling Banana first at one end, and Guava to its left, we can find the corresponding position for Lemon, i.e., opposite to Guava. On the sides of Lemon are Apple and Mango. Raspberry is at an end, being diagonally opposite to Mango. Thus, Mango is at the end next to lemon. This, determines the position of Apple and Raspberry as well. The empty positions can be occupied by Papaya or Pomegranate. We see that Mango is just opposite to Mango. So the answer is A. Question: Based on the data given above, which tree is just opposite to raspberry tree? A. Papaya B. Pomegranate C. Papaya or Pomegranate D. Data is inadequate Solution: We can see that either papaya or pomegranate is next to Raspberry. The answer is C. (contd..) Tips on Direction Sense - https://learningpundits.com/module-view/77-direction-sense/1-tips-on-direction-sense/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  8. 3 Tips on Cracking Reasoning Questions on Dice, Cubes and Cuboids Tip #1: Mentally rotate the dice to see which faces are adjacent and which are opposite 1.Question: Two positions of a dice are shown below. Which number will appear on the face opposite to the face with the number 4? Solution: 2, 3, 5 and 6 are adjacent to 4 and cannot be opposite to 4. The current answer is 1. 2. Question: Two positions of a dice are shown below. Which number will appear on the face opposite to the face with the number 2? Solution: From the given figures, it is clear that figure (ii) is obtained by rotating the cube in figure (i) one face to the left while holding the top and bottom constant. Hence, the opposite face of 2 has 4. 3. Question: Two positions of a dice are shown below. Which number will appear on the face opposite to the face with the number 4? Solution: Figure (ii) is obtained by making two rotations from (i) while holding the front (face with 3) and back constant. Hence, the opposite of 4 is 2. 4. Question: Two positions of a dice are shown below. Which number will appear on the face opposite to the face with the number 3? Solution: If you mentally rotate cube in figure (i) such that face 1 goes to the right and 5 and 3 are hidden from view with 5 at the bottom and 3 at the back, you will arrive at (ii). Hence, the opposite of 3 is 2. Tip #2: While folding a plus into a cube, the square at the longer end always forms the top of the cube and the middle square at the intersection will be the base of the cube Question: The figure given on the left hand side in each of the following questions is folded to form a box. Choose from the alternatives (1), (2), (3) and (4) the boxes that is similar to the box formed A. 1, 2 and 4 only B. 3 and 4 only C. 1 and 2 only D. 1, 2 and 3 only Solution: The square at the longer end will form the top while the middle square will form the base. The rest 4 sides that consist of 2 unshaded squares and 2 partially shaded squares will form the lateral edges. By rotating the dice formed, we would be able to visualize (1), (2) and (4). So, the correct option will be A. Tip #3: Draw indicative diagrams for cubes and cuboids to simplify the problem A cuboid shaped wooden block has 4 cm length, 3 cm breadth and 5 cm height. Two sides measuring 5 cm x 4 cm are colored in red. Two faces measuring 4 cm x 3 cm are colored in blue. Two faces measuring 5 cm x 3 cm are colored in green. Now the block is divided into small cubes of side 1 cm each. Question: How many small cubes will have two faces colored with red and green colors? Other faces of the small cube could be colored or blank. A. 12 B. 8 C. 16 D. 20 Solution: Cubes painted red and green on two sides will be along the left and right sides (5 cm length each) of the top and the bottom face. Thus, no. of such cubes = 5 x 4 = 20. Answer is D. Question: How many small cubes will have no faces colored? A. None B. 2 C. 4 D. 6 Solution: The cubes that will not be part of the faces of the cuboid will have no color. Thus, no. of such cubes = 2 x 3 x 1 = 6. So the correct answer will be D. (contd..) Tips on Logical Reasoning Test on Dice, Cube & Cuboid - https://learningpundits.com/module-view/75-dice,-cube-and-cuboid/1-tips-on-dice,-cube-and-cuboid/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  9. 5 TIPS on cracking Aptitude Questions of Data Sufficiency type Tip #1: Do not solve the problems Analyze the information provided: Do not solve the question using the information at hand. Question: Two towns are connected by railway. Can you find the distance between them? I. The speed of the mail train is 12 km/hr more than that of an express train. II. A mail train takes 40 minutes less than an express train to cover the distance. A. I alone sufficient while II alone not sufficient to answer. B. II alone sufficient while I alone not sufficient to answer. C. Either I or II alone sufficient to answer. D. Both I and II are not sufficient to answer. E. Both I and II are necessary to answer Solution: Let the distance between the two towns be D km. Let the speed of the express train be S km/hr. The speed of the mail train is S+12 km/hr. Creating an equation using I and II: D/(S+12) = (D/S) – (2/3) [because 40 minutes = 2/3 hrs] Don’t try to solve this equation. There is one equation with two variables and cannot be solved. Hence, the correct answer is D. Note: Questions of this type do not require you to actually solve them- you just need to interpret the information provided to you in the statements. Also, working out the problems may mislead you into hasty assumptions. So, avoid trying to solve them. Tip #2: Represent the given information visually on paper to easily process it Question: How many children does M have? I. H is the only daughter of X who is wife of M. II. K and J are brothers of M. Options: A. I alone sufficient while II alone not sufficient to answer. B. II alone sufficient while I alone not sufficient to answer. C. Either I or II alone sufficient to answer. D. Both I and II are not sufficient to answer. E. Both I and II are necessary to answer. Solution: From I, we have that H is the only daughter of M. But that does not mean that M has no son. Thus, the information is not enough to answer the question. II does not give tell us anything about M’s children. Thus, the correct answer is D. Tip #3: Do not make assumptions that cannot be justified by the given statements Question: How many ewes (female sheep) in a flock of 50 sheep are black? I. There are 10 rams (male sheep) in the flock. II. Forty percent of the animals are black. Options: A. I alone sufficient while II alone not sufficient to answer. B. II alone sufficient while I alone not sufficient to answer. C. Either I or II alone sufficient to answer. D. Both I and II are not sufficient to answer. E. Both I and II are necessary to answer Solution: Do not assume that the proportion of white: black sheep is uniform across rams and ewes. We know from (I) that there are 40 ewes but we do not know that 40% of ewes are black. Hence the right answer is D. Tip #4: Use Venn Diagrams when possible Question: Of the 70 children that visited a certain doctor last week, how many had neither caught cough nor cold? I. 40 of 70 had cough but not cold. II. 20 of 70 had both cough and cold. Options A. I alone sufficient while II alone not sufficient to answer. B. II alone sufficient while I alone not sufficient to answer. C. Either I or II alone sufficient to answer. D. Both I and II are not sufficient to answer. E. Both I and II are necessary to answer. Solution As we see in the Venn diagram, neither is sufficient but together, they are enough to solve the question. Hence, the answer is E. Note: Try representing Venn diagrams when possible. That way you do not have to rattle your brain on the problem unnecessarily. Also, this might be the fastest way to figuring out the answer in some questions. Tip #5: Typically the choices A through E tend to be the same We have already seen from the above examples that the choices are always given as follows: A. I alone sufficient while II alone not sufficient to answer. B. II alone sufficient while I alone not sufficient to answer. C. Either I or II alone sufficient to answer. D. Both I and II are not sufficient to answer. E. Both I and II are necessary to answer. So you can save time by just skimming over the choices superficially... (contd..) Tips on Data Sufficiency - https://learningpundits.com/module-view/74-data-sufficiency/1-tips-on-data-sufficiency/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  10. 4 TIPS on cracking Reasoning Questions on Cause and Effect Tip #1: Generally statements that happened in the past are the cause while statements in the present are the effect Statements: 1. The farmers have decided against selling their kharif crops to the Government agencies. 2. The Government has reduced the procurement price of kharif crops starting from last month to the next six months. Options: A. Statement I is the cause and statement II is its effect B. Statement II is the cause and statement I is its effect C. Both statements are independent causes D. Both statements are effects of independent causes E. Both statements are effects of some common cause Solution: I happened after II and thus I cannot be a cause. This eliminates options A and C. The events are inter-related, which eliminates D. The answer can be either B or E. Now, it seems that the reduction in procurement price of crops must have instigated the farmers not to sell their produce to Government agencies. So, the correct answer must be B. Tip #2: Read each statement carefully to understand the nature of the cause-effect relationship Options: A. Statement I is the cause and statement II is its effect B. Statement II is the cause and statement I is its effect C. Both statements are independent causes D. Both statements are effects of independent causes E. Both statements are effects of some common cause Statements: I. The life today is too fast, demanding and full of variety in all aspects which at times leads to stressful situations. II. Number of suicide cases among teenagers is on increase. Solution: Life has become too stressful and this has a major impact on the teenagers, leading to them committing suicide. Thus, the correct answer is A. Statements: I. The police authority has recently caught a group of house breakers. II. The citizens group in the locality have started night vigil in the area. Solution: Both the events are effects of the cause that there has been an increase in thefts in the area. So the answer is E. Tip #3: Do not try to force fit the statements into a cause-effect relationship as they could be the effects of a common cause or two independent causes Statements: I. The employees of the biggest bank in the country have given an indefinite strike call starting from third of the next month. II. The employees of the Central Government have withdrawn their week long demonstrations. Options A. Statement I is the cause and statement II is its effect B. Statement II is the cause and statement I is its effect C. Both statements are independent causes D. Both statements are effects of independent causes E. Both statements are effects of some common cause Solution: The strike must have been called because of some cause. Similarly, The Central Government employees have withdrawn their demonstrations due to some cause. But the causes of these effects seem to be different and independent. So answer is D. Tip #4: Use general knowledge to link the statements Statements: I. There is increase in water level of all the water tanks supplying drinking water to the city during the last fortnight. II. Most of the trains were cancelled last week due to water-logging on the tracks. Options: A. Statement I is the cause and statement II is its effect B. Statement II is the cause and statement I is its effect C. Both statements are independent causes D. Both statements are effects of independent causes E. Both statements are effects of some common cause Solutions: Both the statements are clearly the result of heavy downpour in the area. So the correct answer is E. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  11. 2 TIPS on cracking Reasoning Questions on Blood Relations Tip #1: Sound knowledge of blood relations Father's father → Grandfather Mother's father → Maternal grandfather Father's mother → Grandmother Mother's mother → Maternal grandmother Father's brother → Uncle Mother's brother → Maternal uncle Father's sister → Aunt Mother's sister → Aunt Children of uncle → Cousin Children of maternal uncle → Cousin Wife of uncle → Aunt Wife of maternal uncle → Maternal aunt Children of aunt → Cousin Husband of aunt → Uncle Question: Pointing to a photograph of a boy Suresh said, "He is the son of the only son of my mother." How is Suresh related to that boy? A. Brother B. Uncle C. Cousin D. Father Solution: Suresh’s mother’s only son is Suresh himself. So, the boy in the photo is Suresh’s son, i.e. Suresh is the boy’s father. The answer is then D. Question: If A + B → A is the mother of B; A - B → A is the brother B; A % B → A is the father of B and A x B → A is the sister of B, which of the following shows that P is the maternal uncle of Q? A. Q - N + M x P B. P + S x N – Q C. P - M + N x Q D. Q - S % P Solution: Analyzing the options, we see, D. Q is the brother of S, S is the father of P → Q is the uncle of P. C. P is the brother of M, M is the mother of N, N is the sister of Q → P is the maternal uncle of Q. Thus, the answer is C. Tip #2: Draw flowchart when suitable Question: P is the mother of K; K is the sister of D; D is J’s father. How is P related to J? A. Mother B. Grandmother C. Aunt D. Data inadequate Solution: Thus, P is the mother of J’s aunt. Or, in other words, P is the grandmother of J. The correct option is B. (The letters beside/under the arrows indicate: m: mother, sis: sister, f: father) Question: Given, M % N means M is the son of N, M @ N means M is the sister of N and M $ N means M is the father of N. Which of the following shows the relation that C is the granddaughter of E? A. C % B $ F $ E B. B $ F $ E % C C. C @ B % F % E D. E % B $ F $ C Solution: Analyzing the options, we see, With Option D: E is the uncle of C With Option C: C is the granddaughter of E. Hence, the correct option is C. (contd..) Tips on cracking Reasoning Questions Based on Blood Relation - https://learningpundits.com/module-view/73-blood-relations/1-tips-on-blood-relations/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  12. 2 TIPS on cracking Reasoning Questions on Artificial Language Tip #1: Break the given words in the artificial language as well as English into smaller parts to find their translations Question: Here are some words translated from an artificial language. (I) migenlasan means cupboard (II) lasanpoen means boardwalk (III) cuopdansa means pullman Which word could mean "walkway“ ? A. Poenmigen B. cuopeisel C. lasandansa D. poenforc Solution: ‘Board’ is common between cupboard and boardwalk. Since ‘lasan’ is common between migenlasan and lasanpoen, we can conclude that ‘lasan’ implies ‘board’ and ‘poen’ implies ‘walk’. Both A and D start with ‘poen’ but ‘migen’ means cup and therefore A can be rejected. Thus, the correct answer is poenforc, i.e., D. Question: Here are some words translated from an artificial language. (I) apatose means first base (II) epatose means second base (III) lartabuk means ballpark Which of the following words could mean “baseball” ? A. buklarta B. oseepta C. bukose D. oselarta Solution: The word for ‘base’ could be ‘patose’ or ‘atose’ or ‘tose’ or ‘ose’ etc. From the options, since ‘ose’ is common across options B, C and D, we can conclude that the translation for ‘base’ is ‘ose’. The words in both the artificial language and English appear in the same order. Thus, option C can be eliminated. Among B and D, only D has the phrase ‘larta’ from lartabuk which corresponds to ‘ball’. So, the correct option is D. Tip #2: Eliminate options with (a) incorrect translations and/ or (b) incorrect order of appearance of the words Question: Here are some words translated from an artificial language. (I) agnoscrenia means poisonous spider (II) delanocreania means poisonous snake (III) agnosdeery means brown spider. Which word could mean ‘black widow spider’? A. deeryclostagnos B. agnosdelano C. agnosvitriblunin D. trymuttiagnos Solution: It is evident that in the artificial language, nouns appear first and then the adjectives, with the phrase ‘agnos’ meaning spider. Thus, it should appear first. This eliminates options A and D. Now, ‘delano’ means snake, so this eliminates option B. Hence, the answer is C. Question: Here are some words translated from an artificial language. (I) jalkamofti means happy birthday (II) moftihoze means birthday party (III) mentogunn means goodness. Which of the following could mean ‘happiness’? A. jalkagunn B. mentohoze C. moftihoze D. Hozemento Solution: In the artificial language, the phrase ‘mufti’ is common between ‘happy birthday’ and ‘birthday party’ and therefore, it refers to ‘birthday’. Hence, ‘jalka’ means happy. There is only one option with this phrase. So the correct answer is A. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  13. 2 TIPS on cracking Aptitude Questions on Trains Tip #1: Understand the concepts involved in a train crossing a pole or a platform 1. Time taken by a train of length L to pass a pole or standing man or a signal post is equal to the time taken by the train to cover distance L. 2. Time taken by a train of length L to pass a station of length b is the time taken by the train to cover the distance (L + b). Question: A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train? Solution: Time taken to cross a pole = time taken to cover distance equal to its own length. Speed of the train in m/s = 60 x 5 / 18 = (50/3) m/s. Length of the train = Speed of the train x Time taken to cross the pole = 50/3 x 9 = Length of the train = 150m. Question: A train passes a station platform in 36 sec and a man standing on the platform in 20 sec. If the speed of the train is 54 km/hr, what is the length of the platform? Solution: Speed of the train in m/s = 54 x 5 / 18 = 15m/s. Length of the train = 15 x 20 = 300m. Distance traveled in 36s = 15 x 36 = 540m. (This is the length of the train + platform combined) Length of the platform = (540 – 200) m = 240m. Tip #2: For problems on 2 trains, use the concept of relative velocity 1. If two trains of length a and b are moving in opposite directions at u m/s and v m/s, then time taken by the trains to cross each other = (a + / (u + v). 2. If two trains of length a and b are moving in the same direction at u m/s and v m/s, then time taken by the faster train to cross the slower train = (a + / (u - v) . Question: 2 trains of length 137 m and 163 m are running towards each other and speeds 42 km/hr and 48 km/hr respectively. In what time will the two trains cross each other? Solution: Relative velocity = 42 + 48 = 90 km/hr = (90 x 5/18) m/s = 25 m/s. (Opposite directions) Time taken to cross each other = 300 / 25 = 12 sec. Question: 2 trains running in opposite directions cross a man standing on the platform in 27s and 17s respectively and they cross each other in 23s. Find the ratio of their speeds. Solution: Let the speeds be x m/s and y m/s respectively. Then, length of 1st train = 27x and that of 2nd train = 17y. Time taken to cross each other = (27x + 17y) / (x + y) = 23. = 27x + 17y = 23x + 23y. = 4x = 6y. = x: y = 3 : 2. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  14. TIPS on cracking Aptitude Questions on Time and Work Assume that the productivity of each worker is constant 1. Question: A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for Rs.3200. With the help of C, they completed the work in 3 days. How much is to be paid to C? Solution: Let the total amount of work to be done be W units. Productivity of A, Pa = W/6 units per day. Productivity of B, Pb = W/8 units per day. 3 days x [Pa + Pb + Pc] = W => Pc = W/24 units per day Ratio of wages of A: B: C = Ratios of their productivities = (W/6): (W/8): (W/24) = 4: 3: 1. Amount to be paid to C = Rs.3200 x (1/8) = Rs.400 2. Question: It takes 6 hours for pump A, used alone, to fill a tank of water. Pump B used alone takes 8 hours to fill the same tank. We want to use three pumps: A, B and another pump C to fill the tank in 2 hours. What should be the rate of pump C? How long would it take pump C, used alone, to fill the tank? Solution: Let the total capacity of the tank be C liters. Fill rate of pump A, Fa = C/6 liters per hr Fill rate of pump B, Fb = C/8 liters per hr 2 hrs x [Fa + Fb + Fc] = C => Fc = 5C/24 liters per hr Let‘t’ be the time taken by only pump C to fill the tank. ‘t’ hrs x 5C/24 = C => t = 24/5 = 4.8 hrs LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  15. 2 TIPS on Cracking Aptitude Questions on Stocks and Shares Recapitulation of the concepts – I 1. Stock Capital: The total amount of money needed to run a company is called its stock capital. 2. Shares or Stock: The whole capital of the company is divided into equal units. Each unit is called a share or a stock. 3. Shareholder or Stockholder: Each individual who purchases one or more shares is called a shareholder (stockholder) of the company. The company issues share certificates to each of its shareholders indicating the number of shares allocated and the value of each share. 4. Dividend: The annual profit distributed among shareholders is called dividend. It is paid annually as per share or as a percentage. Dividend is always paid on the face value of a share. 5. Face Value: The value of a share or stock printed on the share-certificate is called its Face Value or Nominal Value or Par Value. The face value of a stock always remains the same. Recapitulation of the concepts – II 1. Market Value: The stocks of different companies can be traded (bought or sold) in the market through brokers at stock exchanges. The market value of a share changes from time to time. The price at which a stock is traded in the market is called its market value. A share or stock is said to be: a) At premium or above par, if its market value is more than its face value. At par, if its market value is the same as its face value. c) At discount or below par, if its market value is less than its face value. Example: Assume that the face value of a company X is Rs.10 and it is now traded at a premium of Rs.2. Then its market value now is (Rs.10 + Rs.2) = Rs.12. Similarly, if the company X having face value of Rs.10 is now traded at a discount of Rs.2, it means the market value of X now is (Rs.10 – Rs.2) is Rs.8 2. Brokerage: Stocks of different companies can be traded (bought or sold) in the market through brokers at stock exchanges. The brokers’ charge is called brokerage. Brokerage is added to the cost price when the stock is purchased and subtracted from the selling price when stock is sold. Tip #1: Interpret the question correctly Rs.100, 10% stock at 120 means: a) The face value of stock = Rs.100 Dividend= 10% of the Face Value = Rs.10 c) Market Value = Rs.120. Question: Find the cash required to buy Rs.3200, 7.5% stock at 107. Solution: Face Value = Rs.3200 => 32 shares must be purchased [Assume Face Value = Rs.100] Market Price of 32 shares = 3200 x 107 = Rs.3424 Question: In order to obtain an income of Rs.650 from 10% stock at Rs.96, what amount must one invest? Solution: Face Value = Rs.100 Dividend = 10% of Rs.100 = Rs.10 Thus, for gaining Rs.650, investment = 96 x (650 / 10) = Rs.6240. Question: Which is better investment: 11% stock at 143 or 9.75% stock at 117? Solution: Let the investment be Rs. X. Then, Income on 1st stock = X x 11 / 143 = X / 13 Income on 2nd stock = X x 9.75 / 117 = X / 12 Thus, income on 2nd stock > Income on 1st stock. Hence, 2nd stock is a better investment. Tip #2: If investment is not mentioned, choose the investment in the relevant stock as x Question: Juno invests a part of Rs.12000 in 12% stock at Rs.120 and the remainder in 15% stock at Rs.125. If her total dividend per annum is Rs.1360, how much does she invest in 12% stock at Rs.120? Solution: Let the investment in the 1st stock be X. Then, investment in 2nd stock = 12000 – X. Income on 1st stock = 12 / 120 x X = X / 10. Income on 2nd stock = 15 / 125 x (12000 – X) = 3(12000 – X) / 25 => X / 10 + 3 (12000 – X) / 25 = Rs.1360. => 5x + 72000 - 6x = 1360 x 50. => x = Rs.4000. Question: Rs.9800 are invested partly in 9% stock at 75 and 10% stock at 80 to have equal amount of incomes. Find the investment in 9% stock. Solution: Let the investment in the 1st stock be X. Then, investment in 2nd stock = 9800 – X. Income on 1st stock = 9/75 x X = 3X/25 & on 2nd stock = 10/80 x (9800 – X) = (9800 – X)/8. => 3X / 25 = (9800 – X) / 8 => 24x = 9800 x 25 - 25x => 49x = 9800 x 25 => x = Rs.5000. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  16. 3 TIPS on cracking Aptitude Questions on Distance, Speed and Time Tip #1: Use the formula Speed = Distance/ time while ensuring that units are same Question: In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. Calculate the duration of the flight. Solution: Let the average duration of the flight be ‘t’ hours. Distance = 600 km. Average Speed (km/ hr) = 600/ t Average Speed – 200 = 600 / (t + 0.5) (600/ t) – 200 = 600 / (t + 0.5) (600 – 200 t) (t + 0.5) = 600 t => 2t2 + t - 3 = 0 =>t = 1 hr Duration of the flight = 1.5 hrs Question: Robert is travelling on his cycle and has calculated to reach point A at 2 P.M. if he travels at 10 km/hr. But he will reach there at 12 noon if he travels at 15 km/hr. At what speed must he travel to reach A at 1 P.M.? Solution: According to the question, (D / 10) – (D / 15) = 2 =>D = 60 km. Time taken to reach A at 10 km/hr = 60 / 10 = 6 hours. (Starting time is 8 AM) Therefore, we need to find the speed to cover 60 km should be in 5 hours. Hence, speed = 60 / 5 =12 km/hr. Tip #2: If a person travels same distances with different speeds, then the average speed is not the arithmetic mean but the harmonic mean If a person covers a distance d first at x km/hr and then covers the same distance d at y km/hr, then the average speed is: = Total distance travelled/ Total time taken = 2d/ (d/x + d/y) = 2d/ [(yd + xd)/xy] = 2dxy/[d(x+y)] = 2xy/ x+y (Harmonic mean of x and y) Question: A travels 25km at 50 km/hr and then 25km again with 70km/hr. What is A’s average speed during the whole journey? Solution: Average speed for the whole journey = (2x50x70) / (50+70) = 58.3km/hr Tip #3: If A runs x times faster than B, A’s speed is actually 1+x the speed of B Question: A runs 1⅔ times as fast as B. If A gives B a start of 80 m, how far must the winning post be so that A and B might reach it at the same time? Solution: Speed of A, Sa = 5/3 x Sb Let the distance of the course be ‘d’ meters Time taken by A to cover distance ‘d’ = Time taken by B to cover distance ‘d-80’ d/[5/3 x Sb] = (d-80)/Sb 3d = 5d – 400 => 2d = 640 => d = 200m Question: A runs 1⅔ times faster than B. If A gives B a start of 80 m, how far must the winning post be so that A and B might reach it at the same time? Solution: Speed of A, Sa = (1 + 5/3) x Sb = 8/3 x Sb Let the distance of the course be ‘d’ meters Time taken by A to cover distance ‘d’ = Time taken by B to cover distance ‘d-80’ d/[8/3 x Sb] = (d-80)/Sb 3d = 8d – 640 => 5d = 640 => d = 128m [Note: Here, A 5/3 times faster than B, i.e., A’s speed = B’s speed + 5/3 times B’s speed = 8/3 times B’s speed.] LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  17. 3 TIPS on cracking Aptitude Questions on Simple and Compound Interest Tip #1: Understand the formulae 1. Amount to be repaid after N years if simple interest is applied = P + (P x N x R) = P (1 + N x R) 2. Simple Interest = P x N x R 3. Amount to be repaid after N years if interest is compounded = P [(1 + R)^N] 4. Compound Interest = [P x (1 + R)^N] - P Question: Simple interest on a certain sum of money for 3 years at 8% per annum is half the compound interest on Rs.4000 for 2 years at 10% per annum. Calculate the principal placed on simple interest. Solution: Let the Principal be Rs. P Then, SI = (P x R x T) = 0.24P Given CI = 4000(1 + 0.1)2 – 4000 = 4000(1.21 – 1) =4000 x 0.21 According to the question, 0.24P = 2000 x 0.21 => P = 2000 x 0.21 / 0.24 = 2000 x 7 / 8 = Rs. 1750 Tip #2: If the interest rate is applied on a half-yearly, quarterly or monthly basis, the effective annual rate is calculated by compounding the interest Question: What is the effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half-yearly? Solution: Let the Principal be Rs. P. Now, the rate is 6% per annum but the interest has to be paid twice a year (i.e.) an interest of 3% is applied every 6 months. Amount to be repaid after 1 year = P x 1.03 x 1.03 = 1.0609P => Effective Annual Rate of interest = 6.09% Tip #3: Use logarithms to find the time when compound rates are applied 1. log 2 = 0.301 2. log 3 = 0.477 3. log 4 = 0.602 4. log 5 = 0.699 5. log 6 = 0.778 6. log 7 = 0.845 Question: At 3% annual interest compounded monthly, how long will it take to double your money? Solution: Let the number of months be n and the Principal be Rs. P. Then, P(1 + 0.03)n = 2P => (1 + 0.03)n = 2 => n log ( 1.03) = log 2 => n = log 2 / log 1.03 = 0.301 / 0.128 = 23.5 Thus. It’ll take 1 year and 11.5 months. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  18. 3 TIPS on cracking Aptitude Questions on Ratios and Proportions Tip #1: Express each element in the ratio as a fraction of the total for easy calculations Question: A sum of money is to be distributed among A, B, C, D in the proportion of 2: 5: 4: 3. If C gets Rs.2000 more than D, what is B's share? Solution: A = 2/14 * T B = 5/14 * T C = 4/14 * T D = 3/14 * T C – D = 2000 4/14 * T – 3/14 * T = 2000 T = 28000 => B = 5/14 * T = Rs.10000 Tip #2: You can directly perform multiplication/ division operations on ratio elements Question: Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats? Solution: Ratio after proposal = 5/20 * T * 1.4 : 7/20 * T * 1.5 : 8/20 * T * 1.75 = 5/20 * 1.4 : 7/20 * 1.5 : 8/20 * 1.75 {dividing by T} = 5 * 1.4 : 7 * 1.5 : 8 * 1.75 {multiplying by 20} = 7 : 10.5 : 14 = 2 : 3 : 4 {dividing by 3.5} As a quick shortcut, we can also directly multiply the ratio elements to determine the new ratio. Ratio after proposal (shortcut) = 5 * 1.4 : 7 * 1.5 : 8 * 1.75 = 2 : 3 : 4 Tip #3: Ensure that the units for the numerator and denominator match across the equation Question: Ken can walk 40 dogs in 8 hours. How many dogs can Ken walk in 12 hours? Solution: We assume that the number of dogs walked and the time taken are directly proportional. Number of dogs walked αTime taken Number of dogs walked = k * Time taken (k is some constant) Number of dogs walked / Time taken = k 40 / 8 = x / 12 where x is count of dogs walked in 12 hours In the above equation, numerator unit is dogs walked and denominator unit is time taken. Ensure that units match to avoid errors. => x = 60 dogs walked LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  19. 3 TIPS on cracking Aptitude Questions on Races and Games Tip #1: Acquaint yourself with the terms Dead Heat Race: A race in which all the contestants reach the Goal at the same time. Start: If A and B are two contestants in a race, such that before the start of the race, A is at the starting point and B is ahead of A by 12 meters, then we say that ‘A gives B a start of 12 meters’. Game: A game of 100, means that the person among the contestants who scores 100 points first is the winner. If A scores 100 points while B scores only 80 points, then we say that 'A can give B 20 points’. This implies that if A actually gave B a start of 20 points, then the contest would result in a dead heat. Tip #2: Assume that the speed or the scoring rate for each player is constant Question: In a game of 100 points, A can give B 20 points and C 28 points. How many points can B give C? Solution: By the time A scores 100 points, B scores only 80 and C scores only 72 points. Let the Scoring Rate of A be Sa. (Scoring Rate = score/ time) Scoring Rate of B, Sb = 80/100 x Sa = 0.8 Sa Scoring Rate of C, Sc = 72/100 x Sa = 0.72 Sa Time taken for B to get 100 points = 100/Sb = 100/ (0.8 x Sa) Score taken by C in this time period = Sc x 100/ (0.8 x Sa) = 72/0.8 = 90 Thus, B can give C 10 points. Question: In a 200 m race A beats B by 35 m or 7 sec. Find A's time over the course. Solution: By the time A completes the race, B is 35m behind A and would take 7 more seconds to complete the race. => B can run 35 m in 7 s. Thus, B’s speed = 35 / 7 = 5 m/s. Time taken by B to finish the race = 200 / 5 = 40 s. Thus, A’s time over the course = (40 – 7)s = 33 s. Tip #3: If A runs x times faster than B, A’s speed is actually 1+x the speed of B Question: A runs 1⅔ times as fast as B. If A gives B a start of 80 m, how far must the winning post be so that A and B might reach it at the same time? Solution: Speed of A, Sa = 5/3 x Sb Let the distance of the course be ‘d’ meters Time taken by A to cover distance ‘d’ = Time taken by B to cover distance‘d-80’ d/[5/3 x Sb] = (d-80)/Sb 3d = 5d – 400 => 2d = 640 => d = 200m Question: A runs 1⅔ times faster than B. If A gives B a start of 80 m, how far must the winning post be so that A and B might reach it at the same time? Solution: Speed of A, Sa = (1 + 5/3) x Sb = 8/3 x Sb Let the distance of the course be ‘d’ meters Time taken by A to cover distance ‘d’ = Time taken by B to cover distance ‘d-80’ d/[8/3 x Sb] = (d-80)/Sb 3d = 8d – 640 => 5d = 640 => d = 128m Note: Here, A 5/3 times faster than B, i.e., A’s speed = B’s speed + 5/3 times B’s speed = 8/3 times B’s speed. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  20. Formula for Cracking Aptitude Questions on Profit and Loss Reduce the problem to one equation and use the formula: Selling Price = Cost Price (1 + 0.01 x profit percentage) Question: Alfred buys an old scooter for Rs.4700 and spends Rs.800 on its repairs. If he sells the scooter for Rs.5800, what is his gain percent? Solution: Net C.P. = Cost + Repairs= Rs.4700 + 800= Rs.5500 S.P. = Rs.5800 Now, 5800= 5500(1+0.01p) => p = 300/(550x0.01)= 5.45 % Question: By selling a Jeans for $ 432, John loses 4%. For how much should John sell it to gain 6%? Solution: For the first transaction, we have: => 432 = C.P. (1-0.04) = C.P. * 0.96 => C.P. = 432/0.96 For the second transaction, we have: => S.P. = C.P. (1+0.06) = (432/0.96) * 1.06 => S.P.= $477 Note: All the problems on profit and loss can be reduced to a single step that can be solved directly using the formula S.P. =C.P. (1±0.01x), where x is the profit/loss percent. Remember that the ’+’ve sign is used when a profit is incurred and the ‘-’ve sign is used when a loss is incurred. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  21. 3 TIPS on cracking Aptitude Questions on Probability Tip #1: Categorize the events as mutually dependent, independent or exclusive Mutually Dependent Events: 2 or more events are that are such that the occurrence of one affects the occurrence of the other. The words ‘and’, ‘together’, ‘all’, etc. indicate that the events are mutually dependent. Example: Let a card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a 2nd card is chosen. The probability of choosing any card is 1 out of 52. However, if the 1st card is not replaced, then the 2nd card is chosen from only 51 cards. Accordingly, the probability of choosing another card is 1 out of 51. Thus, these events are mutually dependent. Mutually Independent Events: 2 or more events such that the occurrence of one does not affect the occurrence of the other. If the problem is such that after each event, the sample space is restored to its original state, then the events are mutually independent. Example: In the above example, the 1st card that was drawn is replaced before drawing the 2nd card. The probability of choosing any card is 1 out of 52. Now, the 1st card is replaced, then the 2nd card is chosen from 52 cards again. Accordingly, the probability of choosing another card is 1 out of 52. Thus, these events are mutually independent. Mutually Exclusive Events: 2 or more events that cannot happen simultaneously. The indicative words are ‘or’, ‘at most’, ‘at least’, etc. Example: The events “running forward” and “running backwards” are mutually exclusive. Similarly, you can’t toss a coin and get both a heads and tails at the same time, so “tossing a heads” and “tossing a tails” are mutually exclusive. Tip #2: Probability of occurrence of mutually exclusive events is the sum of their individual probabilities Question: Three unbiased coins are tossed. What is the probability of getting at most two heads? Solution: Probability of getting at most 2 heads = Probability of getting no head + Probability of getting 1 head + Probability of getting 2 heads Probability of getting no heads = Probability of getting all tails = ( ½ )( ½ )( ½ ) = 1/8 Probability of getting 1 head = C(3,1)( ½ )( ½ )( ½ ) = 3/8 Probability of getting 2 heads = C(3,2)( ½ )( ½ )( ½ ) = 3/8 Thus, required probability = 1/8 + 3/8 + 3/8 = 7/8 Question: Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even? Solution: There might be 2 cases: both numbers are even or one is even and the other is odd. Probability that both numbers are even = (3/6) (3/6) = ¼ Probability that one is even and the other is odd = (3/6) (3/6) x 2 = ½ Probability of getting 2 numbers whose product is even = ¼ + ½ = ¾ Tip #3: Probability of occurrence of mutually dependent or independent events is the product of their individual probabilities Question: Two cards are drawn together from a pack of 52 cards. What is the probability that one is a spade and one is a heart? Solution: Probability of getting a Spade = 13 / 52 = ¼ Now, probability of getting a Heart = 13/51 Since the order of getting the cards is not mentioned, both the orders count. Probability of getting a Spade and a Heart = 2 x (¼)(13/51) = 13 / 102 Question: A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. Find the probability that all of them are red. Solution: Total no. of balls = 4+5+6 = 15 Probability of drawing a red ball at 1st draw = 5/15 = 1/3 Probability of drawing a red ball at 2nd draw = 4/14 = 2/7 Probability of drawing a red ball at 3rd draw = 3/13 Probability of drawing all 3 red cards = 1/3 x 2/7 x 3/13 = 2/91 LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  22. TIPS on cracking Aptitude Questions on Pipes and Cisterns Assume that the filling rate or the emptying rate for each pipe is constant: 1. Question: Pipe A can fill the tank in 20 hours while Pipe B alone can fill it in 30 hours and Pipe C can empty the tank in 40 hours. If all the pipes are opened together, in how long will the tank be full? Solution: Let the capacity of the tank be C liters Fill rate of A, Fa = C/20 liters per hr Fill rate of B, Fb = C/30 liters per hr Fill rate of C, Fc = - C/40 liters per hr [Negative since this pipe is emptying the tank] Let t be the time taken to fill the tank to maximum capacity. t x [Fa + Fb + Fc] = C => t = 120/7 hrs 2. Question: A pump can fill a tank with water in 2 hours. Because of a leak, it took 2.5 hours to fill the tank. If the pump is turned off, how long will it take to empty the tank? Solution: Fill rate of pipe, Fp = C/2 liters per hr Let the fill rate of the leak be FL 2.5 x [Fp + FL] = C => FL = - C/10 With the pump turned off, it would take 10 hrs for the leak to empty the tank. Note: The problems on Pipes and Cisterns are much alike those on Time and Work so it is easy to draw analogy from them and can be solved following similar procedure. Time taken to fill/empty a cistern is equivalent to time taken to do a work. Different pipes are equivalent to the different workers. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  23. Grammar Rules with 6 Tips on using Interrogative Sentences What is an Interrogative? An Interrogative is a type of sentence which usually asks a question or requests information and ends with a question mark (?). An interrogative sentence usually begins with: a) A question word such as what, who, where, which or how. Example: What is your name? An auxiliary verb such as do/does, can or would. Example: Do you speak English? Formation of Interrogative Sentences 1. From an assertive sentence in the simple present tense: Do (for I, You and Plurals)/Does (Singular) + subject + present tense form of the verb. Meera sings a song. (Assertive) Does Meera sing a song? (Interrogative) 2. From an affirmative sentence that contains the auxiliaries is, am, are, has or have, can, may, will, shall etc, the interrogative sentence will begin with these words. She is a doctor. (Affirmative) Is she a doctor? (Interrogative) 3. If the interrogative sentence is in the negative, we begin it with do not or does not. Example: Don’t you want to come with us? Types of Interrogatives: 1. Yes/No interrogatives are questions that can be answered with a yes or a no response. Example: Are you ready to go? (Yes I am ready to go) Did you go to the game Friday night? 2. Alternative interrogatives are questions that provide for two or more alternative answers. Example: Would you prefer chocolate or vanilla ice cream? Should I call or email you? 3. Wh-interrogative sentences begin with a wh-word and call for an open-ended answer. They begin with what, when, where, who, whom, which, whose, why and how. The answer can be a simple response or complex explanation. Example: What are you doing? Which songs do you like best? 4. Tag questions are questions attached or tagged onto the ending of a declarative statement. They transform a declarative sentence into an interrogative sentence. Example: You live in the city, don’t you? We need to get going now, don’t we? Tips on using Interrogatives: Tip 1: Direct & Indirect Interrogative Direct questions normally use inverted word order (verb before subject) and end with a question mark. Example: When is she coming for dinner? Indirect questions normally do not use inverted word order and do not end with a question mark. Example: I wonder when she is coming for dinner. An indirect question can form part of an interrogative sentence. Example: Can you tell me what material she likes? (Direct-question version: What material does she like?) Tip 2: Interrogative with Auxiliary Verb If the verb is an auxiliary verb, the interrogative is formed without the auxiliary do/does/did: Is Brinda in his office? (Brinda is in office) Can I talk to you? If the verb is 'normal', the interrogative is formed with the auxiliary do/does/did. After an auxiliary verb, the verb is added in the infinitive without to: Do you like that album? Did she see the movie? In both cases, the sentence is formed by inverting the first auxiliary verb: She is writing. -> Is she writing? Note: The 'normal' verb to do is also conjugated with the auxiliary do/does/did: Did you do it? Tip 3: W-H Interrogative Form Wh-questions: wh- + an auxiliary verb (be, do or have) + subject + main verb When are you leaving? wh- + a modal verb + subject + main verb: What has she done now? When what, who, which or whose is the subject or part of the subject, we do not use the auxiliary. We use the word order subject + verb: Who wants an ice cream? Who doesn’t want an ice cream? Tip 4: How ‘How’ can be used to form questions in many different ways. 1. Used by itself to mean "in what way". How do you start the car? 2. With adjectives to ask about the degree of an attribute. How old is your house? 3. With ‘much’ and ‘many’ to ask about quantity. How many people are coming to the party? (many is used with countable nouns.) How much flour do I need? (much is used with uncountable nouns) 4. With other adverbs to ask about the frequency or degree of an action. How quickly can you drive the car? Tip 5: What, Which 1. What, which: are used to ask questions about people or objects and in most cases can be replaced by each other. 2. Which: is used to ask about a fixed/limited number of things/people or when the options are visible or known to the speaker. 3. Which flavor of ice cream do you want? (The speaker knows about the choices offered or available) 4. What: is used to ask about things/people without the limitation or knowledge of the choices offered. 5. What do you want for dessert? (The speaker doesn’t know) Tip 6: Who, Whom, Whose Who: is used to refer to the subject of a sentence, i.e., subject pronoun like "he," "she" and "we" a) I see you. Who sees you? Whom: refers to the object and object pronoun like "him," "her" and "us." a) I see you. I see whom? Or whom do I see? Whose: is used to refer to possessive pronoun like "his," "her" and "our.” Whose camera is this? If the interrogative pronoun is a subject, there is no inversion: Who told you this? (she told me this) If the interrogative pronoun is an object, there is inversion: Who(m) are you talking to? (..talking to him) Spot the Errors: Each of the following questions will contain a mistake in the usage of Interrogatives. See if you can spot that mistake. #1: Rahul asked whether anybody had seen his laptop? (Incorrect) Rahul asked whether anybody had seen his laptop. (Correct) #2: Do have you seen my book? (Incorrect) Have you seen my book? (Correct) #3: Did Rajesh called to you? (Incorrect) Did Rajesh call you? (Correct) #4: Who does want a sandwich for breakfast? (Incorrect) Who wants a sandwich for breakfast? (Correct) #5: How many water should I add to the curry? (Incorrect) How much water should I add to the curry? (Correct) #6: What hand do you write with? (Incorrect) Which hand do you write with? (Correct) #7: Who you fear the most? (Incorrect) Whom do you fear the most? (Correct) #8: Whom is in the kitchen? (Incorrect) Who is in the kitchen? (Correct) #9: Who did he blame for the accident? (Incorrect) Whom did he blame for the accident? (Correct) #10: Whom cell phone keeps ringing? (Incorrect) Whose cell phone keeps ringing? (Correct) (contd..) Tips on Interrogative Sentences - https://learningpundits.com/module-view/25-interrogatives/1-tips-on-interrogatives/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  24. Tips on Solving Aptitude Type Questions on Permutation Understanding Permutation & Combination: Suppose we have 4 objects A, B, C and D and we are required to choose 3 from them and then arrange them on a shelf. This can be done in the following ways: Thus, there are 4 ways of choosing 3 objects from 4 and there are 6 ways of arranging the chosen objects. The process of selecting things is called combination and that of arranging things is called permutation. Examples of problems relating to combination: 1. Formation of a team from a number of players. 2. Formation of a particular committee from a number of players. Examples of problems relating to permutation: 1. Arrangement of books on a shelf. 2. Formation of numbers with the given digits. 3. Formation of words with the given letters. Note: Let’s be honest. There aren’t any shortcuts here. The only way to solving these types of questions with speed and ease is by understanding the topic thoroughly. However, since Permutations and Combinations are related topics, let us first revisit them both before proceeding with tips for problems on Permutations. This shows that while considering the alternatives of things or acts, we come across 2 types of problems: (a) Selection, (b) Arrangement. The Sum Rule: The Sum Rule: If A and B are 2 disjoint events (A or B can occur but they never occur together) such that A occurs in m ways and B in n ways, then A or B can occur in m + n ways. Question: How many odd numbers less than 1000 can be formed using the digits 1,2,5,7,8 if repetition of digits is allowed? Solution: Total no. of digits= 5, No. of odd digits= 3. One-digit numbers: Only 3 1-digit numbers are possible: 1,5,7 Two-digit numbers: No. of possible ways= 5 x 3= 15 Three-digit numbers: No. of possible ways= 5 x 5 x 3= 75 Thus, there are (3+15+75) = 93 ways of forming odd numbers less 1000 using the given digits. The Product Rule: The Product Rule: If an operation can be performed in m ways, and if corresponding to each of these m ways of performing this operation, there are n ways of performing a second operation, then the number of ways of performing the two operations together is m x n. Question: There are 10 buses running between 2 towns X and Y. In how many ways can a man go from X to Y using a specific bus and return by a different bus? Solution: A man can go from X to Y in 10 ways and return in (10 – 1) ways= 9 ways. Thus, number of ways of doing the journey= 10 x 9 = 90. Formula for Permutations: Question: What is the number of ways in which 5 persons can occupy 3 vacant seats? Solution: Assume that any one of the 5 persons can sit in Seat 1. That would imply that the choice for filling Seat 1 can be done in 5 ways. We now have 4 persons left that can sit in Seat 2. That implies that the choice for filling Seat 2 can be done in 4 ways. With 3 persons left after Seat 1 & 2 are filled, we can fill Seat 3 in 3 ways. Applying product rule, no of ways of filling 3 seats with 5 persons = 5 x 4 x 3 = 60. Formula for Permutations: No of ways of choosing and arranging ‘r’ elements from a total of ‘n’ given elements is: nPr or P(n, r) = n! /(n – r)! Restricted Permutations: Illustration 1: Question: In how many ways can a shelf for 4 books be formed out of 10 books such that a particular book is always (i) included (ii) left out? Solution: (i) If a book is always included, then it can come in the first, second, third or fourth positions (i.e.) it’s position can be selected in 4 ways. The other 3 books in the shelf can be selected from the remaining 9 books in P(9,3) ways Total Number of ways = P(9,3) x 4= 9! / 6! x 4 = 9 x 8 x 7 x 4 = 2016 (ii) If a book is always excluded, effectively we only have 9 books to fill in 4 positions. Total Number of ways = P(9,4)= 9! / 5!= 9 x 8 x 7 x 6= 3024 Note: Each of the different arrangements which can be made by taking some or all of a number of things at a time is called a permutation. When we have certain restrictions imposed on the arrangement or permutations of the things, we call it restricted permutations. Based on the type of restrictions imposed, these can be classified into 4 types. Illustration 2: Question: In how many ways can 6 boys and 4 girls be arranged in a straight line such that no two girls are ever together? Solution: First, fixing the positions of boys, no. of permutations= 6! Since two girls cannot sit together, a girl can sit either between two boys or at the ends resulting in 7 possible seats as shown below: Fixing the positions of girls = P(7,4) Thus, no. of ways of arranging 6 boys and 4 girls such that no two girls are together = 6! x P(7,4)= 7x 6 x 5 x 4 x 6 x 5 x 4 x 3 x 2 x 1= 604800. Illustration 3: Question: How many 5-digit numbers can be formed out of the digits 0, 1, 2,….., 9 if each number starts with 35 and no digit is repeated? Solution: Since the first two positions are defined and no digit is to be repeated, the remaining 3 positions have to be filled with digits 0,1,2,4,6,7,8,9, i.e., 8 digits. Number of ways of forming required number = P(8,3)= 8! / 5! = 8 x 7 x 6= 336. Note: If n digits are given, including 0 and the digit at the first position is not specified, remember that the digit at the first position cannot be 0, so that the first position can have (n – 1) digits. Permutations of Alike Things: The number of permutations (x) of n things taken all at a time where p things are alike of one kind, q things are alike of a second kind, r things are alike of a third king and so on, is, x= n!/p! q! r! … Question: There are 3 copies each of 4 different books. Find the number of ways of arranging them on a shelf. Solution: Total no. of books= 3 x 4= 12 Required no. of ways of arranging them= 12! / (3! 3! 3! 3!) = 369600 Question: Find the number of arrangements of the letters of the words ‘BANANA’ such that the 2 N’s do not appear together. Solution: There are 3 A’s and 2 N’s. Total no. of ways of arranging the letters= 6! / (3! 2!)= 60 No. of arrangements in which N’s appear together = 5! / 3! = 20. [We assume that the two N’s are combined to form a single character] Thus, no. of arrangements in which the 2 N’s do not appear together= 60 – 20= 40. Permutations of Repeated Things: The number of permutations of n different things taken r at a time, when each thing may occur any number of times is nr . Question: 8 different letters of the alphabet are given. Words of 4 letters are formed. Find the no. of such words with at least one letter repeated. Solution: If any letter can be used any no. of times, no. of letters that can be formed= 84 = 4096. No. of words with no letter repeated= P(8,4)= 8 x 7 x 6 x 5= 1680. No. of words with at least 1 letter repeated= 4096 – 1680= 2416. Question: How many 3-digit numbers can be formed with the digits 1,2,3,4,5 when the digits may be repeated? Solution: Required no. of 3-digit numbers that can be formed= 53 = 125. Circular Permutations: Consider A, B and C to be arranged in a circular fashion. 3 linear arrangements ABC, BCA and CAB are result in the same circular arrangement: 1. No of circular arrangements with n elements = No of linear arrangements with n elements/n = n! / n = (n – 1)! 2. The number of ways in which n objects taken r at a time can form a ring = nPr / r 3. If clock-wise and counter clock-wise arrangements are equivalent, divide the number of ways by 2 Question: In how many ways can 10 boys and 5 girls sit around a circular table such that no 2 girls sit together? Solution: 10 boys can be seated around a table in 9! Ways. There are 10 spaces between the boys which can be filled up by the 5 girls in P(10,5) ways. Thus, total no. of ways of arranging the boys and girls= 9! x 10P5 = (9! 10! )÷ 5! Question: Find the no. of ways in which 10 different flowers can form a garland so that 4 particular flowers are never separated. Solution: Let the 4 flowers be considered as a single flower. Then we have 7 flowers. These can be formed into a garland in 6! Ways. The 4 particular flowers can be arranged in 4! Ways. Thus, total no. of ways of forming the garland= (6! x 4! ) / 2 = 8640. [Dividing by 2 because garlands can be easily flipped implying that clockwise and counter clockwise arrangements are equivalent] Note: So far we have been arranging the things in a straight line. However, at times we are required to arrange the objects in the form of a circle. The method of arranging things in a circle is called circular permutation. (contd..) Tips on Permutation & Combination - https://learningpundits.com/module-view/45-permutations-and-combination/1-tips-on-permutations-and-combination/ LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
  25. 5 TIPS on cracking Aptitude Questions on Number Systems Tip #1: Factorize the expression 1. (a + b)(a - = (a2 - b2) 2. (a + b)2 = (a2 + b2 + 2ab) 3. (a - b)2 = (a2 + b2 - 2ab) 4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) 5. (a3 + b3) = (a + b)(a2 - ab + b2) 6. (a3 - b3) = (a - b)(a2 + ab + b2) 7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc – ac) Question: 297 x 297 =? Solution: 297 x 297 = (300 – 3) (300 – 3) = (300 – 3)2 = 3002 + 32 – 2x3x300 = 297 x 297 = 90000 + 9 – 1800 = 88209 Question: 753 x 753 + 247 x 247 - 753 x 247 = ? 753 x 753 x 753 + 247 x 247 x 247 Solution: Let 753 = a and 247 = b. Then the fraction can be written as: a2 + b2 – ab = a2 – ab + b2 = 1 = 1 . a3 + b3 (a + (a2 – ab + b2) a + b 1000 Tip #2: Use divisibility tests where applicable Question: How many 3-digit numbers are completely divisible 6? Solution: Smallest 3-digit number divisible by 6: 100 is not divisible by 3. 101 is not divisible by 2. 102 is divisible by both 2 and 3 and hence, by 6. Largest 3-digit number divisible by 6: 999 is not divisible by 2. 998 is not divisible by 3. 997 is not divisible by 2. 996 is divisible by both 2 and 3 and hence, divisible by 6. Thus, the numbers form an AP with a = 102, d = 6, l = 996, n = ? 102 + (n – 1)6 = 996 = 6n = 996 – 102 + 6 = 900 = n = 900 / 6 = 150 Question: If “481d673” is divisible by 9, what should be the smallest whole number in place of d? Solution: For the number to be divisible by 9, the sum of the digits should be divisible by 9. 4+8+1+6+7+3 = 29 The whole number just largest than 29 that is divisible by 9 is 36. Thus, d = 36 – 29 = 7 Tip #3: Apply the rules of Binomial Theorem n (x + y)n = Σ nCrx(n – r)yr r=0 1. (1 – x)-1 = 1 + x + x2 + x3 + … where x is between 0 and 1 2. (1 + x) -1 = 1 – x + x2 – x3 + … where x is between 0 and 1 3. xn + 1 is divisible by x + 1 for all odd values of n. Proof: Let us prove this formula using the method of induction. When n=1, xn + 1 = x+1 which is divisible by (x+1). When n=2, x2 + 1 is not divisible by (x+1). Let x2k+1 + 1 be divisible by (x+1). When n = 2k+3 (the next odd number after 2k + 1) x2k+3 + 1 = x2(x2k+1 + 1) + (1 - x2) = x2(x2k+1 + 1)+(1+x)(1-x), which is divisible by (x+1). Thus, xn + 1 is divisible by x + 1 when n is odd. xn – 1 is divisible by x + 1 only when n is even. [Proof is similar to the one above] Question: What will be remainder when (6767 + 67) is divided by 68? Solution: From Binomial Theorem, we know, xn + 1 is divisible by x + 1 only when n is odd. 6767 + 67 = (6767 + 1) + 66 Thus, remainder = 66 Question: Find the value of (0.999)3 correct to 3 decimal places. Solution: (0.999) 3 = (1 – 0.001) 3 = 13 - C(3,1)12(0.001) + C(3,2)1(0.001)2 - (0.001)3 = 1 - 0.003 = 0.997 (neglecting 3rd and 4th terms since they are « 0.001) Tip #4: Remember the rules for division Dividend = (Divisor x Quotient) + Remainder Any recurring decimal can be written as the sum of a non-recurring decimal and a fraction of the recurring portion of the decimal out of (10n – 1). Ex: 0.125125125…= 0.125 = 125/999 and 7.2341341341… = 7.2 + 341/9999 Question: Express 0.232323..... As a rational number. Solution: 0.232323… = 23/99. Question: A student mistook the divisor as 12 instead of 21 and obtained 35 as quotient and reminder as 0. What is the correct quotient? Solution: Let the dividend be x. Then, x / 12 = 35 = x = 12 x 35 = Correct quotient = (12 x 35 / 21) = 4 x 5 = 20. Question: Find the smallest 6 digit number exactly divisible by 111. Solution: Smallest 6 digit number = 100000 When 100000 is divided by 111, the quotient is 900 and the remainder is 100. The number would be divisible by 111 if the difference between 99900 (the number just lesser than 100000 that is divisible by 111) and the next divisible number is 111. Thus, the smallest 6 digit number divisible by 111 is 99900 + 111 = 100011. Tip #5: Memorize the formulae for sum of powers and apply where suitable 1. 1 + 2 + 3 + ….. + n = n (n + 1) / 2 2. 12 + 22 + 32 + ….. + n2 = n (n + 1)(2n + 1) / 6 3. 13 + 23 + 33 + …... + n3 = [ n (n + 1) / 2 ] 2 Question: (51 + 52 + 53 + ... + 100) =? Solution: (51 + 52 + 53 + ….. + 100) = (1 + 2 + 3 + ….. + 100) – (1 + 2 + 3 + ….. + 50) = 100(100 + 1) - 50(50 + 1) 2 2 = 5050 – 1275 = 3775. Question: (22 + 42 + 62 + ... + 202) =? Solution: (22 + 42 + 62 + ... + 202) = 22(12 + 22 + 32 + ….. + 102) = 22 x 10(10 + 1) (2•10 + 1) 6 = 4 x 10 x 11 x 21 / 6 = 1540. LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.
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