Order & Ranking — Study Notes

Overview

Order & Ranking is a staple Logical Reasoning topic in UPSSSC PET, appearing in 2–4 questions every year. The problems test your ability to sequence entities (people, objects, positions) based on given clues and compute ranks from the top, bottom, or both ends of a line or queue. Unlike complex seating arrangements, these problems are linear (straight line, single row, or vertical rank list) and require quick arithmetic combined with logical deduction.

Mastering this topic means understanding how to count positions from either end, handle overlapping information (e.g., "A is 5th from top, B is 3rd from A"), and avoid off-by-one errors. Most UPSSSC PET questions are direct—rank of one person given conditions, or total number in the row. Speed and accuracy in small calculations (addition, subtraction within 50) are critical. This topic overlaps slightly with seating arrangement but remains simpler and faster to solve.

Students must practice translating word problems into number-line diagrams, double-checking whether ranks are inclusive or exclusive of endpoints, and using the standard formulas to find total persons or unknown ranks.

Key Concepts

• **Linear Arrangement**: Entities arranged in a straight line—horizontal (left to right) or vertical (top to bottom). Each entity occupies exactly one position.
• **Rank from Top/Left**: Position counted starting from the first entity on the top or left end. The first person is rank 1, second is rank 2, etc.
• **Rank from Bottom/Right**: Position counted starting from the last entity. The last person is rank 1 from bottom, second-last is rank 2 from bottom, and so on.
• **Total Entities Formula**: If a person is r₁ from one end and r₂ from the other end, total entities = r₁ + r₂ − 1. This is the most-used formula in Order & Ranking.
• **Gap/Overlap Between Two**: If person A is at position p and person B is at position q (both from same end, p < q), the number of people between A and B is q − p − 1.
• **Common Trap**: Forgetting to subtract 1 in the total formula, or miscounting "between" (does it include the endpoints?).

Formulas / Key Facts

1. **Total = Rank from Top + Rank from Bottom − 1** Person X is 7th from top and 12th from bottom → Total people = 7 + 12 − 1 = 18.

2. **Rank from Bottom = Total − Rank from Top + 1** In a queue of 25, if someone is 8th from top, they are 25 − 8 + 1 = 18th from bottom.

3. **Rank from Top = Total − Rank from Bottom + 1** If someone is 5th from bottom in a row of 20, they are 20 − 5 + 1 = 16th from top.

4. **People Between A and B (same end)**: If A is pth and B is qth from the same end (p < q), people between them = q − p − 1. A is 3rd from top, B is 9th from top → Between them = 9 − 3 − 1 = 5 people.

5. **Swapping Positions**: If two people exchange positions and their ranks from one end change by ±k, they were k positions apart.

6. **When "including/excluding" is mentioned**: "Between" typically excludes both endpoints. "From A to B" may include both. Read the question carefully.

Worked Examples

**Example 1: Basic Total Calculation** **Question**: In a row of students, Amit is 9th from the left and 15th from the right. How many students are there in the row?

**Solution**: Use formula: Total = Rank from Left + Rank from Right − 1 Total = 9 + 15 − 1 = 23. **Answer**: 23 students.

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**Example 2: Finding Rank from Other End** **Question**: In a class of 40 students, Priya ranks 12th from the top. What is her rank from the bottom?

**Solution**: Rank from Bottom = Total − Rank from Top + 1 = 40 − 12 + 1 = 29. **Answer**: 29th from bottom.

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**Example 3: People Between Two Persons** **Question**: In a queue, Ravi is 7th from the front and Sita is 14th from the front. How many people are standing between Ravi and Sita?

**Solution**: Both ranks are from the same end (front). Use formula: Between = 14 − 7 − 1 = 6. **Answer**: 6 people between them.

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**Example 4: Position Exchange** **Question**: In a row of 30 persons, A is 10th from the left. B is 18th from the right. If A and B interchange positions, what is A's new position from the left?

**Solution**: B's position from left = Total − Rank from Right + 1 = 30 − 18 + 1 = 13th from left. When A and B swap, A takes B's position → A is now 13th from left. **Answer**: 13th from left.

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**Example 5: Overlapping Information** **Question**: In a row, X is 11th from the top and Y is 8th from the bottom. If there are 3 people between X and Y, how many total people are in the row?

**Solution**: Let total = N. Y's rank from top = N − 8 + 1 = N − 7. People between X and Y = 3, so |Y's rank − X's rank| − 1 = 3 → |Y − X| = 4. Case 1: Y is below X → (N − 7) − 11 = 4 → N − 18 = 4 → N = 22. Case 2: Y is above X → 11 − (N − 7) = 4 → 18 − N = 4 → N = 14. Check which case is consistent: If N = 22, X = 11th, Y = 15th → between = 15 − 11 − 1 = 3. ✓ If N = 14, X = 11th, Y = 7th → between = 11 − 7 − 1 = 3. ✓ Typically exam will have unique answer; re-check question conditions. Assume N = 22 if Y is stated to be below X contextually, else clarify. (In real exam, additional info will disambiguate.) **Answer**: 22 (assuming Y below X) or check conditions again.

Common Mistakes

1. **Forgetting the −1 in Total Formula** Wrong: Total = Rank from Top + Rank from Bottom = 7 + 12 = 19. Correct: Total = 7 + 12 − 1 = 18. The person is counted in both ranks, so subtract 1.

2. **Confusing "between" with "including"** "How many between A and B?" excludes A and B. "How many from A to B?" may include A and B. Always read carefully; default is "between" = exclude endpoints.

3. **Mixing up Left/Right or Top/Bottom** Converting rank from right to left: Wrong formula = Total − Rank from Right. Correct: Rank from Left = Total − Rank from Right + 1. Don't forget the +1.

4. **Off-by-One in "Between" Calculation** A is 5th, B is 10th from same end. Wrong: 10 − 5 = 5 between. Correct: 10 − 5 − 1 = 4 between. The positions 6, 7, 8, 9 are between 5 and 10.

5. **Not Drawing a Quick Diagram** For tricky problems (swaps, overlaps), a small number-line sketch avoids mistakes. Write positions 1, 2, 3, … and mark persons to visualise.

Quick Reference

• **Total = Rank from One End + Rank from Other End − 1** (Most common formula)
• **Rank from Bottom = Total − Rank from Top + 1** (and vice versa)
• **Between A and B (same end) = |Rank_B − Rank_A| − 1** (excludes endpoints)
• **Position Swap**: New rank of A = Old rank of B (from the same end)
• **Always subtract 1** when a person's position is counted from both ends to find total.
• **Draw a line/diagram** if the question involves multiple people or swaps—avoids confusion and off-by-one errors.

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**Practice Tip**: Solve 10–15 problems daily, focusing on variants—interchanges, multiple persons, overlapping ranks. Time yourself: aim for under 30 seconds per straightforward question. This topic rewards speed and formula recall.