Question 1

A class of boys stands in a single line. One particular boy is 19th in order from both the ends of the line. How many boys are there in the class altogether?

Accepted Answer

Introduction / Context:This question is about a special case in ranking, where one boy has the same position when counted from both ends of the line. Recognising this pattern allows for a quick calculation of the total number of boys. Such questions often appear in reasoning exams as straightforward applications of the basic rank relationship. Given Data / Assumptions:- A single boy is 19th from the left end.- The same boy is also 19th from the right end.- The class forms a single straight line, and there are no ties. Concept / Approach:In general, for any person, rank from left + rank from right = total number of persons + 1. Here, both ranks are equal (19th from each end), which makes the calculation especially simple. We can plug the known ranks into the formula and solve for the total number of boys in the line. Step-by-Step Solution:Step 1: Let the total number of boys be N.Step 2: The boy's rank from the left is 19.Step 3: The boy's rank from the right is also 19.Step 4: Apply the relation: rank from left + rank from right = N + 1.Step 5: Substitute the values: 19 + 19 = N + 1.Step 6: Compute 19 + 19 = 38, so 38 = N + 1.Step 7: Therefore, N = 38 - 1 = 37. Verification / Alternative check:We can interpret the situation visually. If the boy is 19th from both ends, there must be 18 boys on his left and 18 boys on his right. Adding them together with the boy himself gives 18 + 1 + 18 = 37. This intuitive check matches the formula-based answer and confirms that 37 is correct. Why Other Options Are Wrong:Option 38 would imply 19 + 19 = 39, which contradicts the rank relationship. Option 39 gives N + 1 = 40, again inconsistent with 19 + 19 = 38. Option 27 is far too small and does not allow for 18 boys on each side of the central boy. Only 37 fits both the formula and the intuitive count from each side. Common Pitfalls:A common source of error is misremembering the formula and using N instead of N + 1 on the right-hand side. Some students also forget to include the boy himself when adding those on either side. Carefully applying the correct relation avoids these mistakes. Final Answer:The number of boys in the class is 37.

Question 2

Three persons A, B and C are standing in a queue. There are five persons between A and B and eight persons between B and C. If there are three persons ahead of C and 21 persons behind A, what is the minimum possible number of persons in the queue?

Accepted Answer

Introduction / Context:This is a more challenging queue and ranking question involving three people and several conditions about how many people stand between them and at the ends of the queue. You must translate the verbal description into positions on a line, then deduce the minimum total number of persons that can satisfy all constraints. Such problems test systematic reasoning and careful handling of distances. Given Data / Assumptions:- There are five persons between A and B.- There are eight persons between B and C.- There are three persons ahead of C in the queue.- There are 21 persons behind A.- We want the minimum possible number of persons in the queue that satisfies all these conditions. Concept / Approach:First, convert the statements about persons between A, B and C into equations relating their positions. If there are k persons between two people, their position numbers differ by k + 1. We then place C using the information about three persons ahead of C. After that, we place A using the information about persons behind A. By trying both possible orders of A and B around C and satisfying the distance constraints, we determine the total number and select the minimum that works. Step-by-Step Solution:Step 1: Let the queue be numbered from front to back starting at 1.Step 2: Three persons ahead of C means C is at position 4.Step 3: There are 21 persons behind A, so if the total number of persons is N, then A is at position N - 21.Step 4: There are five persons between A and B, so the positions of A and B differ by 6. So either B = A + 6 or B = A - 6.Step 5: There are eight persons between B and C, so the positions of B and C differ by 9. Since C is at position 4, B must be at 4 + 9 = 13 or 4 - 9, which is negative and impossible. Thus, B is at position 13.Step 6: With B at 13 and five persons between A and B, A can be at position 13 - 6 = 7 or 13 + 6 = 19.Step 7: A must also satisfy A = N - 21. If A is at 7, then N = 7 + 21 = 28. If A is at 19, then N = 19 + 21 = 40.Step 8: Both N = 28 and N = 40 satisfy the positional relationships, but the question asks for the minimum possible number of persons, so we choose N = 28. Verification / Alternative check:For N = 28, positions are: C at 4, B at 13 and A at 7. There are three persons ahead of C (positions 1, 2 and 3), five persons between A and B (positions 8 to 12), eight persons between B and C (positions 5 to 12), and 21 persons behind A (positions 8 to 28). All conditions are satisfied with N = 28, confirming it is valid and minimal. Why Other Options Are Wrong:Options 40 and 41 involve higher possible totals. Although N = 40 also works, the question explicitly asks for the minimum possible number, so 40 and 41 are too large. Option 27 is too small to accommodate the required gaps between A, B and C along with the persons ahead of C and behind A. Thus, only 28 is both feasible and minimal. Common Pitfalls:Students often forget to check both possible positions for A around B, or they ignore the requirement to find the minimum total. Some also misinterpret phrases like persons ahead and persons behind, confusing them with positions from the given end. Drawing a rough sketch of the queue can greatly reduce such errors. Final Answer:The minimum possible number of persons in the queue is 28.

Question 3

How many such digits are there in the number 7346285 which are at the same position from the beginning of the number as they would be when the digits are arranged in ascending order?

Accepted Answer

Introduction / Context:This question asks you to compare the positions of digits in a given number with their positions when the digits are sorted in ascending order. You must count how many digits remain in the same position under both arrangements. This tests your understanding of order and careful comparison of positions. Given Data / Assumptions:- The original number is 7346285.- We rearrange its digits in ascending order to compare positions.- We assume all digits are considered independently and positions are counted from the left starting at 1. Concept / Approach:First, list the digits of the number with their original positions. Then sort the digits in ascending order and list them again with new positions. A digit counts as qualifying if the digit at a particular position in the original number is the same as the digit at the same position in the sorted list. Since all digits here are distinct, we simply match by value and position directly. Step-by-Step Solution:Step 1: Write the original digits with positions: 7(1), 3(2), 4(3), 6(4), 2(5), 8(6), 5(7).Step 2: Arrange these digits in ascending order: 2, 3, 4, 5, 6, 7, 8.Step 3: In the ascending list, positions are: 2(1), 3(2), 4(3), 5(4), 6(5), 7(6), 8(7).Step 4: Compare position by position. At position 1, original has 7, ascending has 2, so no match.Step 5: At position 2, original has 3 and ascending has 3, so this is a match.Step 6: At position 3, original has 4 and ascending has 4, so this is another match.Step 7: At positions 4, 5, 6 and 7, the digits do not match the ascending list. Therefore, only positions 2 and 3 qualify. Verification / Alternative check:List the two matching positions explicitly: position 2 with digit 3, and position 3 with digit 4. Since no other positions have the same digit in both the original and the ascending arrangement, the total number of such digits is exactly two. A second scan confirms that no other accidental matches are being missed. Why Other Options Are Wrong:Option None is incorrect because we have found two valid matches. Option one undercounts the correct answer, and option Three overestimates, as there is no third position where the digit remains unchanged after sorting. Thus, only option Two accurately reflects the count. Common Pitfalls:Errors usually occur when candidates confuse the notion of digit value with position or when they fail to list all positions carefully. Some may also misinterpret the question as counting identical digits in both arrangements without regard to position. The key is to focus specifically on digits that occupy the same position number in both versions of the number. Final Answer:The number of such digits is two.

Question 4

If 30th January 2003 was a Thursday, then what day of the week fell on 2nd March 2003?

Accepted Answer

Introduction / Context:This calendar-based reasoning question asks you to determine the day of the week on a target date, given the day on a reference date. You must carefully count the number of days between the two dates, taking into account the lengths of the months, and then use modular arithmetic to find the resulting day of the week. Given Data / Assumptions:- 30th January 2003 was a Thursday.- We need the day of the week on 2nd March 2003.- The year 2003 is not a leap year, so February has 28 days. Concept / Approach:The basic idea is to count how many days lie between 30th January and 2nd March, then move forward that many steps in the weekly cycle starting from Thursday. Since a week has 7 days, we only need the remainder when the total number of days is divided by 7. This remainder tells us how many days ahead the target day is from the reference day. Step-by-Step Solution:Step 1: From 30th January to 31st January, there is 1 day.Step 2: February 2003 has 28 days because 2003 is not divisible by 4.Step 3: From 1st February to 28th February, there are 28 days.Step 4: From 1st March to 2nd March, there are 2 days.Step 5: Total days between 30th January and 2nd March (exclusive of the starting date, inclusive of the ending date) equals 1 + 28 + 2 = 31 days.Step 6: Now find 31 divided by 7. We get 31 = 7 * 4 + 3, so the remainder is 3.Step 7: A remainder of 3 means 2nd March is 3 days ahead of Thursday in the weekly cycle.Step 8: Count forward: Thursday to Friday (1), Friday to Saturday (2), Saturday to Sunday (3). So 2nd March 2003 falls on Sunday. Verification / Alternative check:We can cross-check by listing a shorter chain of dates. 30th January is Thursday, so 31st January is Friday. Then 7th February is also Friday, 14th February is Friday, 21st February is Friday and 28th February is Friday. That means 1st March is Saturday and 2nd March is Sunday. This alternative method confirms the same result without using formal modular arithmetic. Why Other Options Are Wrong:Thursday would imply that the number of days between the dates is a multiple of 7, which is not true for 31 days. Tuesday and Saturday correspond to remainders of different sizes when dividing by 7 and do not match the actual count of 31 days. Only Sunday is three days ahead of Thursday and therefore correct. Common Pitfalls:Errors often arise from miscounting the days in February or including the starting date incorrectly. Some students also forget to calculate the remainder when dividing by 7 and instead try to track every day, which is more time-consuming and error prone. Being systematic about month lengths and remainders greatly reduces such mistakes. Final Answer:The day of the week on 2nd March 2003 was Sunday.

Question 5

In a class of 180 students, the number of girls is twice the number of boys. Rupesh, who is a boy, ranks 34th from the top. If there are 18 girls ahead of Rupesh in the ranking, how many boys are there after him in rank?

Accepted Answer

Introduction / Context:This is a combined arithmetic and ranking question involving two groups, boys and girls, within a class. You are given the ratio of girls to boys, the total strength, the rank of one boy and the number of girls ahead of him. The task is to determine how many boys are ranked after him. Such questions test your ability to translate word statements into equations and then analyse the composition of the group above and below a particular rank. Given Data / Assumptions:- Total number of students in the class is 180.- Girls are twice the number of boys.- Rupesh is a boy and ranks 34th from the top.- There are 18 girls ahead of Rupesh. Concept / Approach:First, use the ratio of girls to boys to determine the exact number of boys and girls. Next, use the information about Rupesh's position and the number of girls ahead of him to deduce how many boys are ahead of him. Since he himself is a boy, the remaining boys must lie after him in rank. Subtracting the number of boys ahead plus Rupesh from the total number of boys gives the required answer. Step-by-Step Solution:Step 1: Let the number of boys be B and the number of girls be G.Step 2: Given that girls are twice the boys, so G = 2B.Step 3: Total students are 180, so B + G = 180.Step 4: Substitute G = 2B into B + G = 180 to get B + 2B = 180.Step 5: Simplify to 3B = 180, so B = 180 / 3 = 60.Step 6: Then G = 2B = 2 * 60 = 120.Step 7: Rupesh ranks 34th from the top and there are 18 girls ahead of him. So the number of students ahead of him is 33, and among these 33, 18 are girls.Step 8: Therefore, the number of boys ahead of Rupesh is 33 - 18 = 15.Step 9: Since Rupesh himself is a boy, the total number of boys who are not ahead of him is 60 - 15 = 45.Step 10: This group of 45 boys includes Rupesh. So the number of boys after him in rank is 45 - 1 = 44. Verification / Alternative check:We can cross-check by counting boys in all three segments: boys ahead of Rupesh (15), Rupesh himself (1) and boys after him (44). The total is 15 + 1 + 44 = 60, which matches the calculated number of boys. This consistency confirms that the reasoning is correct. Why Other Options Are Wrong:Option 45 would incorrectly count Rupesh as being after himself. Option 60 is the total number of boys in the class and not the number after him. Option cannot be determined is incorrect because the information given is sufficient to compute the exact figure. Only 44 matches both the ratio and the ranking constraints. Common Pitfalls:Students sometimes forget to subtract Rupesh when counting boys after him, or they miscount the number of boys ahead by not excluding Rupesh from the first 34 ranks. Another source of error is incorrectly solving the ratio equation for boys and girls. Keeping track of each segment clearly prevents these mistakes. Final Answer:The number of boys who are ranked after Rupesh is 44.

Question 6

In a queue of 50 persons, Amrita is 10th from the front and Mukul is 25th from the back. Mamta is standing exactly in the middle of these two positions. What position does Mamta occupy from the front of the queue?

Accepted Answer

Introduction / Context: This ranking question is about positions in a single queue. Such problems often appear in reasoning tests to check comfort with counting from the front, from the back, and finding a person who stands between two known positions. Understanding how to convert positions from the back into positions from the front is the key skill that is being tested here. Given Data / Assumptions: There are exactly 50 persons standing in a single queue. Amrita is 10th from the front. Mukul is 25th from the back. Mamta is standing exactly in the middle of the two positions of Amrita and Mukul. Everybody is standing in one straight line facing the same direction. Concept / Approach: To solve ranking questions of this type, we usually convert all positions to a common reference, either from the front or from the back. Once both positions are expressed from the same end, the middle position can be found by taking the average. Position from the back can be converted to position from the front by using the formula: position from front = total persons - position from back + 1. Step-by-Step Solution: Step 1: Let positions be counted from the front as 1, 2, 3, ..., 50. Step 2: Amrita is already given as 10th from the front, so her position is 10. Step 3: Mukul is 25th from the back. Convert this to position from the front: position of Mukul from front = 50 - 25 + 1 = 26. Step 4: Now Amrita is at position 10 and Mukul is at position 26 from the front. Step 5: Mamta is exactly in the middle of these two positions, so her position is the average of 10 and 26. Step 6: Average = (10 + 26) / 2 = 36 / 2 = 18. Step 7: Therefore, Mamta occupies the 18th position from the front. Verification / Alternative check: The distance between position 10 and position 26 is 26 - 10 = 16 places. The middle point must be 8 steps from each end, because 16 / 2 = 8. Counting 8 places forward from 10 gives 10 + 8 = 18. Counting 8 places backward from 26 gives 26 - 8 = 18. Both checks confirm that Mamta is in the 18th position from the front. Why Other Options Are Wrong: Option 14th is too close to Amrita and not exactly midway. Option 16th is not equally distant from positions 10 and 26. Option 20th is closer to Mukul and does not give equal distance on both sides. Common Pitfalls: Students sometimes forget to convert the position from the back to a position from the front using the correct formula. Others may miscalculate the average and instead count the number of persons between the two without adjusting correctly. Always convert all ranks to the same side and then use the idea of midpoint to avoid mistakes. Final Answer: Mamta occupies the 18th position from the front of the queue.

Question 7

In a row of boys, Rajan is sixth from the left end and Vinay is tenth from the right end. If there are exactly eight boys sitting between Rajan and Vinay, how many boys are there in the row in total?

Accepted Answer

Introduction / Context: This problem is a typical row ranking question involving positions from the left and from the right, along with a known number of persons between two given persons. Such questions are very common in competitive examinations and are designed to test the ability to combine different forms of positional information to find the total number of persons in the row. Given Data / Assumptions: All persons are boys sitting in a straight row. Rajan is 6th from the left end of the row. Vinay is 10th from the right end of the row. There are exactly 8 boys sitting between Rajan and Vinay. No other special conditions are given, and both boys are part of the same continuous row. Concept / Approach: For such questions, the standard formula when two persons are in a row with a known number of persons between them is: total number of persons = position of first person from one end + position of second person from the other end + number of persons between them. This works when the two persons are distinct and the positions are counted from opposite ends of the same row. The formula effectively accounts for the segments of the row on both sides and in between the two persons. Step-by-Step Solution: Step 1: Note that Rajan is 6th from the left. Step 2: Vinay is 10th from the right. Step 3: The number of boys between them is 8. Step 4: Apply the standard formula for total number of boys in the row. Step 5: Total boys = 6 (Rajan from left) + 10 (Vinay from right) + 8 (between them). Step 6: Total boys = 6 + 10 + 8 = 24. Step 7: Therefore, there are 24 boys in the row. Verification / Alternative check: Imagine the row as a line of 24 positions. Rajan at 6th from left means his position is 6. Vinay at 10th from right in a row of 24 means his position from left is 24 - 10 + 1 = 15. The number of boys between positions 6 and 15 is 15 - 6 - 1 = 8, which matches the given condition, so the total of 24 is consistent and correct. Why Other Options Are Wrong: Option 26 gives too many positions and would increase the number of boys between Rajan and Vinay beyond 8. Option 23 or 25 both break the relation between the given ranks and the stated number of boys between them. Common Pitfalls: Many students incorrectly add only one of the ranks to the number of students between or forget that ranks from opposite sides both include the persons themselves. Using the standard formula avoids forgetting to include any segment of the row and keeps the calculation straightforward. Final Answer: There are 24 boys in the row in total.

Question 8

If Tuesday falls on the 4th day of a month, then which day of the week will fall three days after the 24th day of that same month?

Accepted Answer

Introduction / Context: Calendar problems like this one test the ability to move forward and backward through days of the week using modular arithmetic. Once the day for one date is known, you can calculate the day for any other date in the same month by counting the number of days between them and using the 7 day cycle of the week. Given Data / Assumptions: The 4th day of the month is a Tuesday. We want to know the day that falls three days after the 24th of the same month. A week has 7 days and the pattern repeats every 7 days. The month is assumed to be normal with no missing days or irregularities. Concept / Approach: The central idea is that the day of the week advances by one for each calendar day. To find the day for the 24th, count how many days lie between the 4th and the 24th and then move that many steps forward from Tuesday. Once the day for the 24th is known, move three more days ahead to get the required day. Because the week repeats every 7 days, we use remainders after dividing by 7 to simplify the counting. Step-by-Step Solution: Step 1: The 4th is Tuesday. Step 2: The 24th is 20 days after the 4th, because 24 - 4 = 20. Step 3: Compute 20 mod 7 to find how many net steps forward in the weekly cycle this represents. 20 divided by 7 leaves a remainder of 6. Step 4: Move 6 days ahead from Tuesday: Tuesday + 6 days = Monday (because Tuesday + 5 days is Sunday, and one more day is Monday). Step 5: So, the 24th is a Monday. Step 6: We now need the day three days after the 24th, that is three days after Monday. Step 7: Counting forward: Monday (start), Tuesday (1), Wednesday (2), Thursday (3). Therefore, three days after the 24th is Thursday. Verification / Alternative check: You can also move forward directly from the 4th to the 27th, since three days after the 24th is the 27th. The difference between the 4th and the 27th is 23 days. Compute 23 mod 7, which is 2. Moving 2 days ahead from Tuesday gives Thursday, which confirms the result from the detailed step calculation. Why Other Options Are Wrong: Monday corresponds to the 24th itself, not three days after. Tuesday and Friday do not match the calculated shifts of 20 days and then 3 more days from the known Tuesday on the 4th. Common Pitfalls: A common mistake is to forget that the first day is already counted or to miscount the difference between the two dates. Another frequent error is to not use the remainder by 7 and instead try to count all days manually, which can introduce off by one errors. Working with mod 7 arithmetic keeps the computation structured and reliable. Final Answer: The day that falls three days after the 24th is Thursday.

Question 9

In a class, Sanjeev ranks seventh from the top and twenty eighth from the bottom. How many students are there in the class altogether?

Accepted Answer

Introduction / Context: This ranking question checks understanding of how positions from the top and bottom of a list relate to the total number of students. It is a classic type of reasoning problem where one student's two ranks, from opposite ends, are used to calculate the total class strength. Given Data / Assumptions: Sanjeev is 7th from the top of the class. Sanjeev is 28th from the bottom of the same class. All students are arranged in one linear order (for example by marks). There are no ties in rank and Sanjeev is counted only once. Concept / Approach: When the position of the same person from the top and from the bottom is known, the total number of students can be obtained by adding the two positions and subtracting 1. The subtraction of 1 is necessary because the student in question has been counted once in each rank and must not be double counted when calculating the total strength. Step-by-Step Solution: Step 1: Note Sanjeev's rank from the top: 7. Step 2: Note Sanjeev's rank from the bottom: 28. Step 3: Apply the formula: total students = rank from top + rank from bottom - 1. Step 4: Substitute the values: total students = 7 + 28 - 1. Step 5: Compute 7 + 28 = 35. Step 6: Now subtract 1: 35 - 1 = 34. Step 7: Therefore, there are 34 students in the class. Verification / Alternative check: Imagine writing all the students in a line and numbering them from 1 at the top to 34 at the bottom. If Sanjeev is 7th from the top, he stands at position 7. Counting from the bottom, positions would be 34 at the bottom, then 33, 32, and so on. The position 7 from the top corresponds to 34 - 7 + 1 = 28 from the bottom, which matches the given value, so the total of 34 is consistent. Why Other Options Are Wrong: Options 33, 35, and 36 are inconsistent with the formula and would not produce matching ranks of 7 from the top and 28 from the bottom for the same student. Common Pitfalls: Students sometimes forget to subtract 1 and instead use only the sum of the two ranks, which double counts the student in question. Others misinterpret ranks as distances between students. Remember that ranks include the person and refer to positions, so the formula with minus 1 is essential. Final Answer: There are 34 students in the class.

Question 10

Some boys are sitting in a row. P is 14th from the left end and Q is 7th from the right end. If there are exactly four boys between P and Q and Q is to the right of P, how many boys are there in the row?

Accepted Answer

Introduction / Context: This ranking problem involves persons in a row where two positions are known from opposite ends and the number of persons between them is specified. Such questions test the ability to visualise a linear arrangement and apply consistent counting from both the left and the right ends of the row to determine the total number of persons. Given Data / Assumptions: P is 14th from the left end of the row. Q is 7th from the right end of the row. There are exactly 4 boys between P and Q. We take the usual exam convention that Q is on the right side of P. The row is continuous with no gaps and all persons are counted once. Concept / Approach: When two persons in a row are at known positions from opposite ends and the number of persons between them is known, we can use a direct formula. Assuming that Q is to the right of P, the total number of boys in the row is given by: total boys = position of P from left + position of Q from right + number of persons between P and Q. This formula effectively adds the segment from the left end to P, from P to Q including those in between, and from Q to the right end. Step-by-Step Solution: Step 1: Position of P from left = 14. Step 2: Position of Q from right = 7. Step 3: Number of boys between P and Q = 4. Step 4: Apply the formula: total boys = 14 + 7 + 4. Step 5: Compute 14 + 7 = 21. Step 6: Add the boys between: 21 + 4 = 25. Step 7: Therefore, there are 25 boys in the row. Verification / Alternative check: Assume there are 25 positions in the row. Q is 7th from the right, so Q is at position 25 - 7 + 1 = 19 from the left. P is at position 14 from the left. The number of boys between P (position 14) and Q (position 19) is 19 - 14 - 1 = 4, which matches the given condition. Thus a total of 25 boys is consistent with all the information provided. Why Other Options Are Wrong: Options 24, 26, and 22 do not satisfy all three conditions simultaneously. For these totals, either the distance between P and Q or the given ranks from the ends would not match the stated four boys between them. Common Pitfalls: A common difficulty is that, without a clear assumption on which side each person stands, two mathematical totals may appear possible. Standard exam convention assumes the second person described is to the right when their relative order is not explicitly stated. Another mistake is forgetting to include the boys between P and Q when adding the positions from left and right. Always carefully apply the appropriate formula and, if needed, cross check by constructing the row mentally. Final Answer: There are 25 boys in the row.

Question 11

Each of the following three digit numbers has its first and third digits interchanged: 987, 514, 658, 487, 404 and 269. After this interchange, which digit will be the third digit (units place) of the second lowest number obtained?

Accepted Answer

Introduction / Context: This problem involves manipulating three digit numbers by swapping their first and third digits and then comparing the resulting numbers. It tests the understanding of place value in decimal numbers and the ability to order a list of numbers correctly after a specified transformation. Finally, it asks for a particular digit in the second smallest transformed number. Given Data / Assumptions: Original numbers: 987, 514, 658, 487, 404, 269. For each number, the first and third digits are interchanged. We then form a new list of six transformed numbers. We must arrange these new numbers in ascending order and find the units digit (third digit) of the second lowest number in this new arrangement. Concept / Approach: Three digit numbers can be written as ABC, where A is the hundreds digit, B is the tens digit and C is the units digit. Swapping the first and third digits converts ABC into CBA. For each given original number, we apply this rule to get the new number. After calculating all transformed numbers, we compare them to determine their order from smallest to largest. Finally, we pick the second smallest transformed number and note its third digit, which is simply its units digit. Step-by-Step Solution: Step 1: Transform each number by swapping its first and third digits. 987 becomes 789. 514 becomes 415. 658 becomes 856. 487 becomes 784. 404 remains 404 because the first and third digits are both 4. 269 becomes 962. Step 2: List all transformed numbers: 789, 415, 856, 784, 404, 962. Step 3: Arrange these numbers in ascending order: 404, 415, 784, 789, 856, 962. Step 4: The second lowest number in this ordered list is 415. Step 5: The third digit (units place) of 415 is 5. Verification / Alternative check: One can quickly recheck by confirming that 404 is indeed the smallest, since its hundreds digit is 4 and all others begin with 7, 8 or 9. Among the remaining numbers, 415 has hundreds digit 4 and tens digit 1, making it larger than 404 but smaller than any number starting with 7 or 8. Therefore it must be the second lowest, and its units digit is clearly 5, confirming the result. Why Other Options Are Wrong: Digit 4 corresponds to the units of 404 or another number, but not to the second smallest transformed number. Digits 8 and 9 appear in larger transformed numbers like 784, 789, 856 or 962 and cannot be the units digit of the second lowest number 415. Common Pitfalls: Some students mistakenly interchange the middle digit instead of the first and third digits, or treat the numbers as if only one swap were required among the list. Others forget to re order all the transformed numbers and instead try to read off digits directly from the initial list. Carefully transforming each number and then fully sorting them avoids these errors. Final Answer: The third digit of the second lowest transformed number is 5.

Question 12

What is the 13th letter of the English alphabet when the letters are written in their usual order from A to Z?

Accepted Answer

Introduction / Context: This is a basic sequence question on the English alphabet. It tests simple but important recall of the order of letters from A to Z and asks the candidate to identify the letter that lies at a specific position, namely the 13th position. Such questions check attention to detail and familiarity with alphabetical order. Given Data / Assumptions: The English alphabet is taken in its standard sequence from A to Z. The question asks for the 13th letter in that sequence. We assume no letters are omitted or rearranged. Concept / Approach: The approach is straightforward. We list or mentally recall the positions of letters in order. A is 1st, B is 2nd, C is 3rd, and so on, up to Z as 26th. The 13th letter is exactly at the halfway point of the 26 letter alphabet. By counting carefully or recalling standard position facts, we identify which letter occupies position 13. Step-by-Step Solution: Step 1: Recall that the English alphabet has 26 letters, from A to Z. Step 2: Count positions sequentially: A is 1, B is 2, C is 3, D is 4, E is 5, F is 6, G is 7, H is 8, I is 9, J is 10, K is 11, L is 12, M is 13. Step 3: Therefore, the 13th letter is M. Step 4: No further adjustments are needed, since there are no skipped letters. Verification / Alternative check: A quick way to verify is to remember that the alphabet has 26 letters, so the 13th letter is exactly in the middle. Many learners memorise that the first half ends at M and the second half starts at N. Knowing this directly confirms that M is indeed the 13th letter without recounting each letter individually. Why Other Options Are Wrong: N is the 14th letter, not the 13th. L is the 12th letter, one position before the required one. K is the 11th letter, two positions before the required one. Common Pitfalls: Some students miscount if they rush, especially when counting without writing the letters down. Others confuse which letter is the midpoint of the alphabet. Doing a quick mental or written count, or recalling that M is the 13th and N is the 14th, helps avoid these errors. Final Answer: The 13th letter of the English alphabet is M.

Question 13

From the digits 8, 5, 2, 1, 7 and 6, how many different two digit numbers can be formed which contain the digit 8 (in either the tens place or the units place), if no digit is repeated within a number?

Accepted Answer

Introduction / Context: This is a basic permutations and combinations problem involving the formation of two digit numbers using a given set of digits. The numbers must contain the digit 8 and no digit may be repeated within a number. Such problems test understanding of positional choices in number formation and counting valid combinations systematically. Given Data / Assumptions: Available digits are: 8, 5, 2, 1, 7 and 6. We need to form two digit numbers. Each required number must contain the digit 8. No digit can be repeated in the same number. Both the tens place and the units place are allowed for the digit 8. Concept / Approach: To form a two digit number with no repetition and with 8 included, we must choose a partner digit for 8 from the remaining digits and then decide in which position 8 will appear. The remaining digits available to pair with 8 are 5, 2, 1, 7 and 6. For each choice of partner digit, there are two possible arrangements: 8 in the tens place or 8 in the units place. The total number of valid two digit numbers is therefore the number of choices of the partner digit multiplied by the number of possible positions for 8. Step-by-Step Solution: Step 1: Count the digits that can pair with 8. These are 5, 2, 1, 7 and 6, which gives 5 possible partner digits. Step 2: For each partner digit, form two numbers: one with 8 in the tens place and the partner in the units place, and one with the partner in the tens place and 8 in the units place. Step 3: For example, with partner digit 5, possible numbers are 85 and 58. With partner digit 2, numbers are 82 and 28, and similarly for 1, 7 and 6. Step 4: Thus, for each of the 5 partner digits, there are 2 distinct two digit numbers. Step 5: Total required numbers = 5 * 2 = 10. Verification / Alternative check: We can list all numbers explicitly: 85, 58, 82, 28, 81, 18, 87, 78, 86, 68. All of these contain the digit 8 exactly once and no digit is repeated within any number. There are no other two digit numbers satisfying the conditions, and counting them confirms that there are exactly 10 such numbers. Why Other Options Are Wrong: 6 and 9 underestimate the total count because they do not account for both positions of 8 for every partner digit. 11 would overcount and would require an additional arrangement that is not possible under the non repetition restriction. Common Pitfalls: Learners sometimes forget that 8 can appear in both the tens and units positions and count only one arrangement per partner digit. Others may mistakenly include numbers like 88, which are not allowed because digits cannot be repeated within the same number. Being systematic about partner selection and position choices helps keep the count accurate. Final Answer: There are 10 different two digit numbers containing the digit 8.

Question 14

The positions of the first and sixth digits in the number 5109238674 are interchanged, the positions of the second and seventh digits are interchanged, the positions of the third and eighth digits are interchanged, and so on. After this rearrangemen…

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Introduction / Context: This question involves a specific pattern of interchanging digits in a long number and then identifying the digit at a particular position after the rearrangement. It tests attention to detail and the ability to follow a structured swap of positions correctly across the entire number. Given Data / Assumptions: Original number: 5109238674. Digits are numbered from left to right as positions 1 to 10. Position 1 is swapped with position 6, position 2 with position 7, position 3 with position 8, position 4 with position 9, and position 5 with position 10. We must determine the third digit from the right end after all these swaps are done. Concept / Approach: The procedure is to write the positions and digits clearly, perform each indicated swap exactly once, and then read off the resulting number. Once the new number is known, counting from the right to find the third digit is straightforward. Careful bookkeeping of each interchange is important to avoid confusion or double swapping the same pair. Step-by-Step Solution: Step 1: Write the digits with their positions from left to right: 1:5, 2:1, 3:0, 4:9, 5:2, 6:3, 7:8, 8:6, 9:7, 10:4. Step 2: Swap positions 1 and 6: digits 5 and 3 are interchanged. Now positions are 1:3, 2:1, 3:0, 4:9, 5:2, 6:5, 7:8, 8:6, 9:7, 10:4. Step 3: Swap positions 2 and 7: digits 1 and 8 are interchanged. New positions: 1:3, 2:8, 3:0, 4:9, 5:2, 6:5, 7:1, 8:6, 9:7, 10:4. Step 4: Swap positions 3 and 8: digits 0 and 6 are interchanged. New positions: 1:3, 2:8, 3:6, 4:9, 5:2, 6:5, 7:1, 8:0, 9:7, 10:4. Step 5: Swap positions 4 and 9: digits 9 and 7 are interchanged. New positions: 1:3, 2:8, 3:6, 4:7, 5:2, 6:5, 7:1, 8:0, 9:9, 10:4. Step 6: Swap positions 5 and 10: digits 2 and 4 are interchanged. Final arrangement: 1:4, 2:8, 3:6, 4:7, 5:4, 6:5, 7:1, 8:0, 9:9, 10:2. Step 7: The new number is 4867451092. Step 8: Count from the right: from rightmost, 1st digit is 2, 2nd digit is 9, 3rd digit is 0. Step 9: Therefore, the third digit from the right end is 0. Verification / Alternative check: You can verify by writing the final number again and labeling the positions from the right. From the right, positions are 1:2, 2:9, 3:0, 4:1, 5:5, 6:4, 7:7, 8:6, 9:8, 10:4. The third position from the right is indeed 0, confirming that no swap was misapplied. Why Other Options Are Wrong: 9 and 6 appear close to the right end but are the second and fourth digits from the right, not the third. 3 does not appear near the relevant position in the final arrangement. Common Pitfalls: Mistakes usually occur when a pair is swapped twice or when positions are misnumbered. Another error is to forget that the question asks for the third digit from the right, not from the left. Drawing a simple positional table and carefully marking each swap helps avoid these common errors. Final Answer: The third digit from the right end after the rearrangement is 0.

Question 15

Consider the following sequence of letters, digits and symbols: K$23DBE8HM4@5JF4%K1+WR#AA*415. How many such symbols are there which are not immediately preceded by a letter but are immediately followed by a number?

Accepted Answer

Introduction / Context: This problem involves scanning a mixed sequence of letters, digits and special symbols to count symbols that satisfy a specific positional condition. It tests the ability to distinguish character types (letter, digit, symbol) and to apply a local condition involving the immediate neighbours of each symbol in the sequence. Given Data / Assumptions: The sequence is: K23DBE8HM4@5JF4%K1+WR#AA*415. We consider a character to be a symbol if it is neither a letter nor a digit (for example @, %, +, #, *, ). We must count those symbols that are not immediately preceded by a letter but are immediately followed by a digit. Positions are checked only with respect to immediate left and immediate right neighbours. Concept / Approach: The approach is to scan the sequence from left to right, identify each symbol, and then check two conditions: the character on its immediate left must either be a digit, another symbol or not exist (at the very start), and the character on its immediate right must be a digit. Only symbols that satisfy both conditions are counted. Careful classification of each character as letter, digit or symbol is necessary. Step-by-Step Solution: Step 1: List the sequence with each character clearly: K 2 3 D B E 8 H M 4 @ 5 J F 4 % K 1 + W R A A * 4 1 5.: Step 2: Identify symbols in the sequence: , @, %, +, #, *. Step 3: Check each symbol one by one. For : previous character is K (a letter) and next character is 2 (a digit). It is immediately followed by a digit but it is immediately preceded by a letter, so it does not satisfy the given condition. For @: previous character is 4 (a digit) and next character is 5 (a digit). It is not immediately preceded by a letter and is immediately followed by a digit, so it satisfies the condition. For %: previous character is 4 (a digit) and next character is K (a letter). It is not preceded by a letter but is not followed by a digit, so it fails the condition. For +: previous character is 1 (a digit) and next character is W (a letter). It is not followed by a digit, so it fails. For #: previous character is R (a letter), so it fails immediately. For *: previous character is A (a letter), so it also fails. Step 4: Only the symbol @ satisfies both parts of the condition. Step 5: Therefore, there is exactly 1 such symbol. Verification / Alternative check: We can quickly recheck by highlighting each symbol in the sequence and marking L for letter, D for digit and S for symbol around it. Only the pattern D S D with S as the symbol and the left neighbour not a letter, appears once at 4 @ 5. This reconfirms that the count is one. Why Other Options Are Wrong: Counts of 2 or 3 would require other symbols to meet both conditions, but careful examination shows that none of , %, +, or * qualify.: A count of 0 contradicts the valid @ case that clearly fits both conditions. Common Pitfalls: A frequent mistake is to misclassify digits as letters or vice versa or to read the condition only partially, for example checking only that the symbol is followed by a number but ignoring whether it is preceded by a letter. Another pitfall is to overlook one of the symbols when scanning a long mixed sequence. A methodical left to right scan helps prevent such errors. Final Answer: There is exactly 1 such symbol in the given sequence.

Question 16

In a class of 50 students, M ranks 8th from the top and H ranks 20th from the bottom. How many students are there between M and H in the rank list?

Accepted Answer

Introduction / Context: This question is another example of ranking problems in a class. It requires converting a rank from the bottom into a corresponding rank from the top and then computing how many students lie strictly between two given students in the ordered list. Such questions help strengthen understanding of rank conversion and interval counting. Given Data / Assumptions: There are 50 students in the class. Student M is 8th from the top. Student H is 20th from the bottom. We must find the number of students between M and H. No ties in rank are assumed. Concept / Approach: To compare positions, both ranks should be expressed from the same end. The rank from the bottom can be converted to a rank from the top using the formula: rank from top = total students - rank from bottom + 1. Once both positions are known from the top, the number of students strictly between them is simply the difference in their positions minus 1. Step-by-Step Solution: Step 1: Position of M from the top is already given as 8. Step 2: Position of H from the bottom is 20. Convert this to position from the top. Step 3: Rank of H from top = 50 - 20 + 1 = 31. Step 4: Therefore, H is 31st from the top. Step 5: To find the number of students between M and H, subtract the lower rank from the higher rank and then subtract 1. Step 6: Difference in ranks = 31 - 8 = 23. Step 7: Students between them = 23 - 1 = 22. Step 8: Thus, 22 students are between M and H in the rank list. Verification / Alternative check: You can visualise the ranking as a vertical list. M is at position 8 and H is at position 31. If you count positions after 8 up to 30, you get 30 - 8 + 1 = 23 positions including both ends, but since we do not count M and H themselves, subtract 2 to get 21, and then verify carefully. The simpler difference minus one formula directly gives 22 and is consistent with careful manual counting between the two ranks. Why Other Options Are Wrong: Options 23 and 24 would correspond to incorrect use of the difference or would involve counting one or both of the endpoints M and H themselves. The option that says it cannot be determined is incorrect because the total class size and both rankings are clearly given. Common Pitfalls: Students sometimes forget to subtract 1 when finding the count of students between two positions, or they mistakenly add the ranks instead of converting one of them first. Another error is to misapply the rank conversion formula. Keeping the conversion and the subtraction steps clear helps avoid these mistakes. Final Answer: There are 22 students between M and H.

Question 17

P, Q, R, S, T, U, V and W are sitting around a circle facing the centre. P is second to the right of T, who is a neighbour of both R and V. V is a neighbour of U. S is not a neighbour of P. Q is not between S and W. W is not between U and S. Who is …

Accepted Answer

Introduction / Context: This is a circular seating arrangement puzzle involving eight persons P, Q, R, S, T, U, V and W who are all facing the centre. Several conditions about neighbours and relative positions are given. The goal is to determine who sits directly opposite U. Such puzzles are common in reasoning tests and require systematic deduction of positions using all conditions together. Given Data / Assumptions: Eight persons P, Q, R, S, T, U, V and W sit around a circle facing the centre. P is second to the right of T. T has R and V as neighbours (one on each side). V is a neighbour of U. S is not a neighbour of P. Q is not between S and W. W is not between U and S. We must find which person sits opposite U. Concept / Approach: Circular seating questions are best handled by fixing one person's position (for example T) and then placing others step by step according to the conditions. Because all face the centre, the right side means clockwise. Once a consistent arrangement is found, the position directly opposite any person is four seats away in a circle of eight. It is important to check that all conditions are satisfied and that no contradictions arise. Step-by-Step Solution: Step 1: Fix T at any convenient position, say at the top. Since everyone faces the centre, second to the right of T means move two seats clockwise to place P. Step 2: Place P two seats to the right of T. This fixes relative positions of T and P in the circle. Step 3: T must have R and V as immediate neighbours, so place R and V on the two sides of T. The order of R and V on left and right is tested in separate possibilities until one arrangement satisfies all conditions. Step 4: V must also be a neighbour of U, so place U adjacent to V on one of its free sides. Step 5: S must not be a neighbour of P, so S cannot be directly adjacent to P. This restricts possible seats for S. Step 6: Q is not between S and W, so the clockwise order S Q W or W Q S is forbidden. Similarly, W is not between U and S, so neither U W S nor S W U can appear clockwise. Step 7: By trying all consistent placements of R, V, U, S, Q and W around T and P while respecting the neighbour conditions, we find that only arrangements where P lies exactly opposite U satisfy all conditions without conflict. Step 8: In these valid arrangements, the seat directly opposite U is always occupied by P. Verification / Alternative check: A more formal method is to list all possible permutations of the eight positions with T and P fixed relative to each other and then discard those that violate any condition. When this is done carefully, all valid seating patterns agree that P is opposite U. This confirms that the answer is unique and consistent with every given statement in the puzzle. Why Other Options Are Wrong: If Q, R or W were opposite U, at least one of the conditions about who sits between whom or who is a neighbour of whom would be violated in every attempted arrangement. For example, placing W opposite U tends to force W into a position that makes W between U and S or Q between S and W, both of which are forbidden by the conditions. Common Pitfalls: Learners often try to solve such puzzles mentally without systematically drawing the circle, leading to missed conditions and contradictions. Another frequent pitfall is misunderstanding the direction of "right" in a circular arrangement where everyone faces the centre. Drawing a clear diagram, fixing one person, and methodically applying each condition one by one greatly reduces these errors. Final Answer: The person sitting directly opposite U is P.

Question 18

How many even numbers are there in the following sequence which are immediately followed by an odd number and also immediately preceded by an even number? 8 5 8 6 7 6 8 9 3 2 7 5 3 4 2 2 3 5 5 2 2 8 1 1 9 3 1 7 5 1

Accepted Answer

Introduction / Context: This problem involves scanning a sequence of numbers and identifying elements that satisfy a specific condition based on their neighbours. In this case we must count all even numbers that are immediately preceded by an even number and immediately followed by an odd number. Such questions test detailed observation and a clear understanding of even and odd parity. Given Data / Assumptions: The sequence is: 8 5 8 6 7 6 8 9 3 2 7 5 3 4 2 2 3 5 5 2 2 8 1 1 9 3 1 7 5 1. We are concerned only with positions that have both a previous and a next neighbour (so the first and last terms cannot be counted). A number is even if it is divisible by 2, and odd otherwise. We must find numbers that are even and have an even neighbour on the left and an odd neighbour on the right. Concept / Approach: The strategy is to scan through the sequence from the second element to the second last element and check three conditions for each position i: the element at i is even, the element at i - 1 is even, and the element at i + 1 is odd. Every index that meets all three conditions contributes 1 to the count. It is important to check all three conditions together rather than partially. Step-by-Step Solution: Step 1: Mark each term as E (even) or O (odd). Sequence with parity: 8(E), 5(O), 8(E), 6(E), 7(O), 6(E), 8(E), 9(O), 3(O), 2(E), 7(O), 5(O), 3(O), 4(E), 2(E), 2(E), 3(O), 5(O), 5(O), 2(E), 2(E), 8(E), 1(O), 1(O), 9(O), 3(O), 1(O), 7(O), 5(O), 1(O). Step 2: Move through the sequence from the second to the second last element and test each even element. At position 4: the number is 6 (E), preceded by 8 (E) and followed by 7 (O). This satisfies all three conditions. At position 7: the number is 8 (E), preceded by 6 (E) and followed by 9 (O). This also satisfies the conditions. At position 16: the number is 2 (E), preceded by 2 (E) and followed by 3 (O). This qualifies. At position 22: the number is 8 (E), preceded by 2 (E) and followed by 1 (O). This also qualifies. Step 3: No other even number in the sequence has both an even neighbour on the left and an odd neighbour on the right. Step 4: Therefore, exactly 4 numbers satisfy the required condition. Verification / Alternative check: We can double check by listing only the triplets (previous, current, next) where the current is even. The four valid triplets are (8, 6, 7), (6, 8, 9), (2, 2, 3) and (2, 8, 1). Each of these clearly has an even previous term, an even current term and an odd next term. Counting these triplets again gives 4, confirming the result. Why Other Options Are Wrong: Counts of 2 or 3 arise if some qualifying positions are missed, while a count of 5 would require one more triplet satisfying the condition, which does not exist in the sequence. Common Pitfalls: Students often forget to check both neighbours or mistakenly treat the first or last element as valid even though they lack one neighbour. Others may confuse the parity of certain numbers when scanning quickly. Marking E and O clearly and then checking each candidate systematically helps to avoid such mistakes. Final Answer: There are 4 such even numbers in the sequence.

Question 19

In the number 2796543018, the positions of the first and sixth digits are interchanged, the positions of the second and seventh digits are interchanged, and so on in the same pattern. After this rearrangement, which of the following digits will be t…

Accepted Answer

Introduction / Context: This problem is similar to the previous digit swapping question but with a different original number. It involves a structured interchange of digit positions in a ten digit number and then asks for the digit at a specific position after the rearrangement. The question tests careful application of the described swapping pattern and accurate reading of positions from the left. Given Data / Assumptions: Original number: 2796543018. Digits are initially in positions 1 to 10 from left to right. Positions swapped are (1,6), (2,7), (3,8), (4,9) and (5,10). We must find the digit that becomes the seventh digit from the left end after all swaps. Concept / Approach: To solve such digit interchange problems, it is important to label each position and digit clearly, then apply each swap exactly once. After all swaps are completed, the final arrangement can be written down, and the requested digit can be identified by counting from the left. Systematic recording of each step avoids confusion and prevents accidental double swapping. Step-by-Step Solution: Step 1: Label the original digits with positions from left to right: 1:2, 2:7, 3:9, 4:6, 5:5, 6:4, 7:3, 8:0, 9:1, 10:8. Step 2: Swap positions 1 and 6. The digits 2 and 4 interchange. New positions: 1:4, 2:7, 3:9, 4:6, 5:5, 6:2, 7:3, 8:0, 9:1, 10:8. Step 3: Swap positions 2 and 7. Digits 7 and 3 interchange. New positions: 1:4, 2:3, 3:9, 4:6, 5:5, 6:2, 7:7, 8:0, 9:1, 10:8. Step 4: Swap positions 3 and 8. Digits 9 and 0 interchange. New positions: 1:4, 2:3, 3:0, 4:6, 5:5, 6:2, 7:7, 8:9, 9:1, 10:8. Step 5: Swap positions 4 and 9. Digits 6 and 1 interchange. New positions: 1:4, 2:3, 3:0, 4:1, 5:5, 6:2, 7:7, 8:9, 9:6, 10:8. Step 6: Swap positions 5 and 10. Digits 5 and 8 interchange. Final positions: 1:4, 2:3, 3:0, 4:1, 5:8, 6:2, 7:7, 8:9, 9:6, 10:5. Step 7: The final number is 4301827965. Step 8: Count digits from the left: positions are 1:4, 2:3, 3:0, 4:1, 5:8, 6:2, 7:7, 8:9, 9:6, 10:5. Step 9: The seventh digit from the left is 7. Verification / Alternative check: Rewriting the final sequence and checking the position again confirms that after all swaps the digit at position 7 is 7. None of the swaps affect the fact that the seventh position is reached after counting exactly six digits to its left, and the only digit at that position in the final arrangement is 7. Why Other Options Are Wrong: Digits 1, 0 and 8 all appear in the final number but not at the seventh position. They occupy other positions from the left. Common Pitfalls: Errors commonly arise from mislabeling the initial positions or performing the swaps out of order. Some students may also misread the question and look for the seventh digit from the right instead of from the left. A clear table of positions and careful counting from the left helps avoid these mistakes. Final Answer: The seventh digit from the left after the rearrangement is 7.

Question 20

In a row of boys, Haran is 11th from the left and Manoj is 17th from the right. When they exchange their positions, Haran becomes 13th from the left. What will be Manoj's new position from the right?

Accepted Answer

Introduction / Context: This ranking puzzle combines positions from the left and from the right, and introduces an exchange of positions between two boys. After swapping, the rank of one of them is given, and we must determine the new rank of the other from the right. This type of problem tests understanding of how ranks change under position exchanges and how to deduce the total number of persons in the row. Given Data / Assumptions: In the initial arrangement, Haran is 11th from the left. In the initial arrangement, Manoj is 17th from the right. After they exchange positions, Haran becomes 13th from the left. We must find Manoj's new position from the right after the exchange. All boys are in a single straight row with fixed total length. Concept / Approach: The key idea is that exchanging positions does not change the total number of boys. First use the information after the exchange to calculate the total number of boys in the row. Then use that total to compute Manoj's new position from the right when he stands at Haran's original position. The conversion between ranks from left and right is given by: rank from right = total boys - rank from left + 1. Step-by-Step Solution: Step 1: Let the total number of boys be N. Step 2: Initially, Manoj is 17th from the right, so his position from the left is N - 17 + 1 = N - 16. Step 3: After the exchange, Haran takes Manoj's original seat. We are told that Haran then becomes 13th from the left. Step 4: This means N - 16 = 13, because Manoj's original position from the left is now 13 for Haran. Step 5: Solve for N: N - 16 = 13 implies N = 13 + 16 = 29. Step 6: The total number of boys in the row is therefore 29. Step 7: After exchanging positions, Manoj takes Haran's original position, which is 11th from the left. Step 8: Manoj's new position from the right is given by: rank from right = N - rank from left + 1 = 29 - 11 + 1. Step 9: Compute 29 - 11 + 1 = 19. Step 10: Hence Manoj is now 19th from the right. Verification / Alternative check: We can do a quick consistency check. With 29 boys, the 13th from the left is 17th from the right because 29 - 13 + 1 = 17. This matches Manoj's original right rank, confirming that Haran's new position after the exchange is indeed Manoj's initial position. Similarly, Manoj at 11th from the left is 29 - 11 + 1 = 19th from the right, confirming our final answer. Why Other Options Are Wrong: Options 11 and 13 correspond to left side positions and do not satisfy the right side logic for a total of 29 boys. Option 17 would be the rank from the right for a different left side position and does not align with Haran's original rank or Manoj's new left side rank. Common Pitfalls: Some students attempt to guess the total number of boys rather than deducing it algebraically, which can lead to errors. Another common mistake is to forget to include the plus one in conversion formulas, leading to off by one results. Writing the formula carefully and solving step by step avoids these issues. Final Answer: After the exchange, Manoj is 19th from the right.

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