Ranking and Order

Ranking and Order — Study Notes

Overview

Ranking and Order problems test your ability to determine the position of a person or object in a linear arrangement based on given conditions.

The core skill here is translating worded statements into numerical positions and then performing simple arithmetic to find unknowns—total count, position from either end, or the number of people between two positions. Mastering this topic requires understanding the relationship between "position from left" and "position from right" and handling scenarios where positions change (like interchange of seats).

Students who build a systematic approach score full marks here, as these are essentially arithmetic problems dressed in verbal reasoning. The key is careful reading and avoiding off-by-one errors.

Key Concepts

• Rank from Left vs Right: Position from left counts starting from the leftmost as 1; position from right counts starting from the rightmost as 1. These are complementary measures.
• The Fundamental Formula: If a person is at position L from left and position R from right in a row of N people, then L + R = N + 1. This single relationship solves most problems.
• Persons Between Two Positions: If two people are at positions P₁ and P₂ (where P₂ > P₁), the number of persons between them = P₂ − P₁ − 1.
• Interchange of Positions: When two people swap seats, their individual ranks change but the total count remains same. Track each person's new position carefully.
• Same Person, Two Ranks Given: When both left and right ranks of the same person are given, directly apply L + R = N + 1 to find total.
• Overlapping vs Non-Overlapping: When two different people's positions are given, check if they could be the same person. If definitely different, their positions help establish boundaries.
• Position After Shifting: If a person moves K places to the right, new position = old position + K. If moving left, subtract K.

Formulas / Key Facts

Key Facts to Remember:

• "From the left" and "from the beginning" mean the same thing
• "From the right" and "from the end" mean the same thing
• When counting "between," exclude both endpoints
• Position is always ≥ 1 (no zero or negative positions)

Worked Examples

Example 1: Finding Total Number

Ravi ranks 7th from the top and 26th from the bottom in a class. How many students are in the class?

Solution:

• Position from top (L) = 7
• Position from bottom (R) = 26
• Total students N = L + R − 1
• N = 7 + 26 − 1 = 32 students

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Example 2: Finding Position from Other End

In a row of 40 students, Meera is 15th from the left. What is her position from the right?

Solution:

• Total N = 40, Position from left L = 15
• Position from right R = N − L + 1
• R = 40 − 15 + 1 = 26th from right

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Example 3: Persons Between Two People

In a row, A is 9th from the left and B is 16th from the left. How many persons are sitting between A and B?

Solution:

• A's position = 9, B's position = 16
• Persons between = 16 − 9 − 1 = 6 persons

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Example 4: Interchange Problem

In a row of 25 students, Amit is 14th from the left. Amit and Sumit interchange positions. Now Sumit is 10th from the right. What was Sumit's original position from the left?

Solution:

• After interchange, Sumit takes Amit's old position
• Amit's old position from left = 14
• After interchange, Sumit is 10th from right
• Check: In 25 students, 10th from right = 25 − 10 + 1 = 16th from left
• But Sumit is now at Amit's old position = 14th from left
• This means Sumit's position from right = 25 − 14 + 1 = 12th from right (not 10th)

Wait—re-reading: "Now Sumit is 10th from right" means his NEW position.

• Sumit's new position from right = 10
• Sumit's new position from left = 25 − 10 + 1 = 16
• This is Amit's original position, so Amit was at 16th from left
• Amit moved to Sumit's original position = 14th from left
• Sumit's original position from left = 16th (where Amit is now)

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Example 5: Combined Condition

Some students are standing in a row. Priya is 12th from the left and Neha is 8th from the right. If there are 5 students between them, find the total number of students.

Solution:

• Case 1: Priya is to the left of Neha
  • Neha's position from left = 12 + 5 + 1 = 18
  • Total N = Neha's left position + Neha's right position − 1 = 18 + 8 − 1 = 25
• Case 2: Neha is to the left of Priya
  • Neha's position from left = 12 − 5 − 1 = 6
  • Total N = 6 + 8 − 1 = 13

Both answers are valid unless additional constraints given.

Common Mistakes

• Forgetting the "−1" in the total formula: Students write N = L + R instead of N = L + R − 1. The person themselves gets counted twice if you just add, so subtract 1.
• Confusing "between" with "including endpoints": Between 9th and 16th position means positions 10 to 15, which is 6 people—not 8. Don't count the two endpoint positions.
• Not considering both cases in ambiguous problems: When two different people's positions are given, they could be arranged in two ways (one left of other, or vice versa). Check if both cases yield valid positive answers.
• Misreading interchange problems: After interchange, Person A takes Person B's old position. Track whose position becomes whose carefully before applying formulas.
• Applying wrong formula for "position from other end": The formula is R = N − L + 1, not R = N − L. Missing the "+1" gives an off-by-one error.

Quick Reference

• Total = Left position + Right position − 1
• Other end position = Total − Given position + 1
• Between two positions = Larger − Smaller − 1
• Interchange swaps positions, not ranks—recalculate both positions after swap
• Two cases possible when two different persons' partial positions given
• Always verify: every position must be ≥ 1 and ≤ Total