Reasoning — Ranking and Order

To find the total number of students in the row when a person's position from both ends is known, use the formula: Total students = (Position from left) + (Position from right) - 1 Total students = 15 + 10 - 1 Total students = 25 - 1 Total students = 24 Therefore, there are 24 students in the row.

Let's represent the height relationships: 1. B is taller than C: B > C 2. B is shorter than A: A > B Combining (1) and (2): A > B > C 3. D is shorter than C: C > D 4. D is taller than E: D > E Combining (3) and (4): C > D > E Now, combine all relationships: A > B > C > D > E From this, it is clear that A is the tallest.

To find the rank from the bottom, use the formula: Rank from bottom = (Total students) - (Rank from top) + 1 Rank from bottom = 40 - 18 + 1 Rank from bottom = 22 + 1 Rank from bottom = 23rd Therefore, Sonal's rank from the bottom is 23rd.

Total students = 60 Rakesh's position from the left = 25th Suresh's position from the right = 30th First, check if there is an overlap or not. Sum of positions from opposite ends: 25 + 30 = 55. Since 55 < 60 (Total students), there is no overlap. The students are arranged without crossing each other. To find Suresh's position from the left end: Suresh's position from left = Total students - (Suresh's position from right) + 1 Suresh's position from left = 60 - 30 + 1 = 31st Now, we have Rakesh at 25th from the left and Suresh at 31st from the left. Number of students between Rakesh and Suresh = (Suresh's position from left) - (Rakesh's position from left) - 1 Number of students between them = 31 - 25 - 1 = 6 - 1 = 5 Therefore, there are 5 students between Rakesh and Suresh.

Let's deduce the arrangement step by step: 1. B is immediately above C: (B-C) 2. A is immediately below D: (D-A) 3. There are two boxes between D and B. Combining (D-A) and (B-C) with the condition of two boxes between D and B, there are two possible main structures: Case 1: D _ _ B If D is above B with two boxes in between, then with (D-A) and (B-C), the arrangement would be: D, A, (Box1), B, C. This is 5 boxes. The missing box is E. So, D, A, E, B, C. Let's check other conditions: - E is not at the top. (D is at top, so this is fine). - C is not at the bottom. (Here C is at the bottom, which contradicts this condition). So, Case 1 is invalid. Case 2: B _ _ D If B is above D with two boxes in between, then with (D-A) and (B-C), the arrangement would be: B, C, (Box1), D, A. This is 5 boxes. The missing box is E. So, B, C, E, D, A. Let's check all conditions for this arrangement (from top to bottom): - B is immediately above C. (B, C) - Yes. - A is immediately below D. (D, A) - Yes. - E is not at the top. (B is at the top) - Yes. - C is not at the bottom. (A is at the bottom) - Yes. - There are two boxes between D and B. (C and E are between B and D) - Yes. All conditions are satisfied by the arrangement: B, C, E, D, A (from top to bottom). Therefore, the box at the bottom is A.

There are two possible cases for this type of problem: non-overlapping and overlapping. Case 1: Non-overlapping (P and Q do not cross each other) In this case, the total number of boys is the sum of P's position from the left, Q's position from the right, and the number of boys between them. Total boys = (P's position from left) + (Q's position from right) + (Number of boys between P and Q) Total boys = 12 + 18 + 6 Total boys = 36 Case 2: Overlapping (P and Q cross each other) In this case, the total number of boys is calculated as: Total boys = (P's position from left) + (Q's position from right) - (Number of boys between P and Q) - 2 Total boys = 12 + 18 - 6 - 2 Total boys = 30 - 8 Total boys = 22 Since the question asks for 'the total number of boys' without specifying the arrangement, and both 36 and 22 are mathematically possible, in competitive exams, if not explicitly stated, the non-overlapping case (which usually yields a larger total) is often the intended answer unless the options only support the overlapping case. Given the options, 36 is present. Therefore, considering the non-overlapping arrangement as the primary interpretation, the total number of boys is 36.

Let's represent the marks scored using inequalities: 1. J scored more than L: J > L 2. J scored less than N: N > J Combining (1) and (2), we get: N > J > L 3. K scored more than M: K > M 4. K scored less than L: L > K Combining (3) and (4), we get: L > K > M Now, combine the two sequences: N > J > L and L > K > M This gives the complete order: N > J > L > K > M From this order, N scored the highest marks.

Let's break down the problem: Initial positions: - Rina: 10th from the left - Tina: 15th from the right After interchanging positions: - Rina moves to Tina's original position. - Tina moves to Rina's original position. - Rina's new position from the left is 16th. Step 1: Find the total number of girls in the row. When Rina moves to Tina's original position, Rina is 16th from the left. This means Tina's original position from the left was 16th. Now we know Tina's original position from both ends: - Tina's original position from left = 16th - Tina's original position from right = 15th Total number of girls = (Position from left) + (Position from right) - 1 Total girls = 16 + 15 - 1 = 31 - 1 = 30. Step 2: Find Tina's new position from the right. Tina moves to Rina's original position. Rina's original position was 10th from the left. So, Tina's new position from the left is 10th. Now, use the total number of girls to find Tina's new position from the right: Tina's new position from right = (Total girls) - (Tina's new position from left) + 1 Tina's new position from right = 30 - 10 + 1 = 20 + 1 = 21st. Therefore, Tina's new position from the right is 21st.

Let A's position from the left be L_A = 15. Let B's position from the right be R_B = 20. Let 'x' be the number of students between A and B. We are given x >= 5. We need to find the minimum possible number of students in the row. This requires considering two cases: non-overlapping and overlapping. Case 1: Non-overlapping arrangement (A is to the left of B, or vice-versa, without crossing) In this case, the total number of students = L_A + R_B + x Total = 15 + 20 + x = 35 + x To find the minimum total in this case, we use the minimum value of x, which is 5. Minimum Total (non-overlapping) = 35 + 5 = 40. Case 2: Overlapping arrangement (A and B cross each other) In this case, the total number of students = L_A + R_B - (x + 2) Total = 15 + 20 - (x + 2) = 35 - x - 2 = 33 - x For an overlapping arrangement to be valid, the total number of students must be at least the maximum of the individual positions from one end. Here, max(L_A, R_B from left) or max(R_B, L_A from right). Let's find the range of x for overlapping: If B is to the left of A, then B's position from the left (L_B) must be less than A's position from the left (L_A). L_B = L_A - x - 1 = 15 - x - 1 = 14 - x. Also, L_B must be at least 1. So, 14 - x >= 1 => x = 5, the possible values for x are 5, 6, ..., 13. To minimize Total = 33 - x, we need to maximize x. The maximum value for x is 13. Minimum Total (overlapping) = 33 - 13 = 20. Comparing both cases, the absolute minimum possible number of students is 20. However, 20 is not available in the given options. This implies we should look for the smallest valid total among the options. Let's check the given options against the condition 'at least 5 students between A and B': - Option 1: Total = 30 A is 15th from left. B's position from left = Total - R_B + 1 = 30 - 20 + 1 = 11th. Number of students between A (15th) and B (11th) = 15 - 11 - 1 = 3. This (3) is not 'at least 5'. So, 30 is not valid. - Option 2: Total = 35 A is 15th from left. B's position from left = Total - R_B + 1 = 35 - 20 + 1 = 16th. Number of students between A (15th) and B (16th) = 16 - 15 - 1 = 0. This (0) is not 'at least 5'. So, 35 is not valid. - Option 3: Total = 38 A is 15th from left. B's position from left = Total - R_B + 1 = 38 - 20 + 1 = 19th. Number of students between A (15th) and B (19th) = 19 - 15 - 1 = 3. This (3) is not 'at least 5'. So, 38 is not valid. - Option 4: Total = 40 A is 15th from left. B's position from left = Total - R_B + 1 = 40 - 20 + 1 = 21st. Number of students between A (15th) and B (21st) = 21 - 15 - 1 = 5. This (5) is 'at least 5'. So, 40 is a valid total. Since 40 is the only option that satisfies the condition, it is the minimum possible number of students among the given choices.

To find the total number of people in the queue, we use the formula: Total = (Position from front) + (Position from back) - 1. Given: Position from front = 12th Position from back = 18th Total = 12 + 18 - 1 = 30 - 1 = 29. So, there are 29 people in the queue.

To find Amit's rank from the bottom, we use the formula: Total = (Rank from top) + (Rank from bottom) - 1. Given: Total students = 50 Rank from top = 23rd 50 = 23 + (Rank from bottom) - 1 50 = 22 + (Rank from bottom) Rank from bottom = 50 - 22 = 28th. So, Amit's rank from the bottom is 28th.

1. First, find the total number of students in the row. After interchanging, A's new position from the left is 18th. A's old position from the right (which was B's position) is 15th. Total students = (A's new position from left) + (A's old position from right) - 1 Total students = 18 + 15 - 1 = 33 - 1 = 32. 2. Now, find B's new position from the right end. B's new position from the right = Total students - (B's old position from left) + 1. B's old position from the left was A's old position from the left, which is 10th. B's new position from the right = 32 - 10 + 1 = 22 + 1 = 23rd. So, B's new position from the right end is 23rd.

1. Total number of people in the row = 70. 2. Rakesh's position from the left end = 25th. 3. Suresh's position from the right end = 35th. 4. Sum of their positions from opposite ends = 25 + 35 = 60. 5. Since the sum of their positions (60) is less than the total number of people (70), there is no overlap. 6. Number of people between them = Total number of people - (Rakesh's position from left + Suresh's position from right) Number of people between them = 70 - (25 + 35) = 70 - 60 = 10. So, there are 10 people sitting between Rakesh and Suresh.

Let's denote '>' for 'heavier than'. 1. "P is heavier than Q": P > Q 2. "P is lighter than R": R > P Combining (1) and (2), we get: R > P > Q 3. "S is lighter than T": T > S 4. "S is heavier than R": S > R Combining (3) and (4), we get: T > S > R Now, combine the two sequences: We have R > P > Q and T > S > R. Putting them together, the complete order from heaviest to lightest is: T > S > R > P > Q. Therefore, T is the heaviest among them.

To find the minimum possible number of students in the row when there is an overlap (i.e., people between them), we use the formula: Minimum Total = (Position from Left) + (Position from Right) - (Number of people between) - 2. Given: Position of A from left = 18th Position of B from right = 22nd Number of students between A and B = 8 Minimum Total = 18 + 22 - 8 - 2 Minimum Total = 40 - 10 = 30. So, the minimum possible number of students in the row is 30.

1. P's position from the left end = 35th. 2. Q is 8 places to the right of P. So, Q's position from the left end = P's position from left + 8 = 35 + 8 = 43rd. 3. R is 15 places to the left of Q. So, R's position from the left end = Q's position from left - 15 = 43 - 15 = 28th. 4. Total number of students = 50. 5. To find R's position from the right end, use the formula: Position from right = Total - Position from left + 1. R's position from the right end = 50 - 28 + 1 = 22 + 1 = 23rd. So, R's position from the right end is 23rd.

Let's deduce the arrangement step-by-step: 1. "C is to the left of D but to the right of B": This implies the order B C D. 2. "A is to the right of D": So far: B C D A. 3. "F is between D and A": This means D F A. So far: B C D F A. 4. "E is to the right of A": So far: B C D F A E. 5. "G is to the left of B": This places G at the very beginning. So, the final arrangement is G B C D F A E. There are 7 friends in total. The middle position in a row of 7 is the 4th position from either end. The arrangement is: G (1st) B (2nd) C (3rd) D (4th) F (5th) A (6th) E (7th). The person sitting exactly in the middle (4th position) is D.