Order & Ranking

Table of Contents

1. 1: Foundation & Basic Concepts
2. 2: Linear Ordering (Row Arrangement)
3. 3: Comparative Ranking
4. 4: Position Based Puzzles
5. 5: Data Sufficiency & Advanced Puzzles
6. Mock Test: Order & Ranking Practice Lab

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1: Foundation & Basic Concepts

1.1 What is Order & Ranking?

Order & Ranking problems involve comparing two or more entities based on a common attribute (e.g., height, weight, marks, age). You are given statements like:

• “A is taller than B.”
• “C is the shortest.”
• “D is heavier than E but lighter than F.”

Your task is to arrange the entities in ascending or descending order and answer questions like “Who is the tallest?” or “Who is third from the top?”.

1.2 Key Terms

1.3 Comparative Symbols

We use symbols to represent comparisons:

• > means “greater than” (e.g., A > B means A is taller/heavier than B).
• < means “less than”.
• = means “equal” (rare, but possible).
• ≥, ≤ may appear in some problems.

It’s helpful to write the order as a chain, e.g., A > B > C means A is the tallest, then B, then C.

1.4 Direction of Ranking

• Ascending order (increasing): smallest to largest (e.g., lightest to heaviest).
• Descending order (decreasing): largest to smallest (e.g., tallest to shortest).

Always note which direction is asked in the question (e.g., “who is the tallest?” means descending order).

1.5 Worked Examples – Foundation

Example 1 – Simple Comparison
Statements:

1. A is taller than B.
2. B is taller than C.
   Question: Who is the tallest?

• Step 1: A > B, B > C → A > B > C.
• Answer: A is the tallest.

Example 2 – Mixed Comparisons
Statements:

1. P is heavier than Q.
2. R is lighter than Q.
3. S is heavier than P.

• Step 1: P > Q.
• Step 2: R < Q (so Q > R).
• Step 3: S > P.
• Combine: S > P > Q > R.
• Answer: S is heaviest.

1.6 Common Mistakes

1.7 Quick Practice – Foundation

1. If A > B, B > C, C > D, who is the smallest?
2. If X is lighter than Y, and Z is heavier than Y, who is the heaviest?
3. Ram is older than Shyam, and Shyam is older than Mohan. Who is the youngest?
4. Arrange in descending order: A > B, C < B, D > C, A < D.

Answers:

1. D.
2. Z (since Z > Y > X).
3. Mohan.
4. D > A > B > C (descending: D, A, B, C).

1.8 Summary of Section 1

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2: Linear Ordering (Row Arrangement)

2.1 Core Concepts

In linear ordering problems, a set of persons (or objects) are arranged in a row (horizontal line). Positions are numbered from left to right (or sometimes from right to left). Typical questions include:

• Who is at the leftmost or rightmost position?
• How many persons are between two given persons?
• What is the position of a person from one end?
• If two persons interchange positions, what are the new positions?

Conventions:

• Position from left: 1st, 2nd, 3rd, …
• Position from right: 1st, 2nd, 3rd, …

2.2 Key Relationships

If a person is Xth from left and Yth from right, then:
$$ \text{Total persons} = X + Y - 1 $$

Distance & Gaps:

• Number of persons between two people at positions $p$ and $q$ from the same end ($p < q$):
  $$ \text{Between} = q - p - 1 $$
• Note: If A is $a$-th from right and B is $b$-th from right ($a > b$), people between them $= a - b - 1$.

2.3 Types of Linear Ordering Problems

2.4 Methodology

1. Draw a row: Use slots or a number line.
2. Place step-by-step: Use the given conditions.
3. Use the Formula: If a person is $X$ from left and $Y$ from right, $\text{Total} = X + Y - 1$.
4. Relative Moves: “A is 3rd to the left of B” means if B is at position $p$, A is at $p - 3$.
5. Swaps: When two persons interchange, they take each other's original indices.

2.5 Worked Examples

Example 1 – Simple Total
Question: Ravi is 7th from the left and 9th from the right. How many students are there?

• Step 1: $\text{Total} = 7 + 9 - 1 = 15$.
• Answer: 15

Example 2 – Persons Between
Question: In a row of 40 persons, A is 15th from left, B is 10th from right. How many persons are between?

• Step 1: B’s left = $40 - 10 + 1 = 31\text{st}$.
• Step 2: Between = $31 - 15 - 1 = 15$.
• Answer: 15

Example 3 – Interchange of Positions
Question: A is 4th from left, B is 6th from right. After interchanging, A becomes 6th from left. Find the total.

• Step 1: A's new index (6) is B's original index (which was 6th from right).
• Step 2: $\text{Total} = \text{A's new left} + \text{B's original right} - 1 = 6 + 6 - 1 = 11$.
• Answer: 11

2.6 Common Mistakes

2.7 Pro Tips

• Convert Everything: Use $\text{Left} = \text{Total} - \text{Right} + 1$.
• Middle Parity: For “exactly between”, the sum of outer positions must be even.
• Constant Sum: In swaps, $\text{A}{old} + \text{B}{old} = \text{A}{new} + \text{B}{new}$ (sum of indices from same side).

2.8 Practice Set – Linear Ordering

1. Ramesh is 12th from left and 15th from right. Total?
2. Total = 60, A = 20th left, B = 25th right. Persons between?
3. P is 8th from left, Q is 10th from right. After swap, P is 12th from left. Total?
4. A is 5th left of B. C is 3rd right of B. D is 2nd left of C. Who is the 1st right of A?

Answers:

1. 26 ($12 + 15 - 1$).
2. 15 (B left = 36, $36 - 20 - 1 = 15$).
3. 21 ($12 + 10 - 1$).
4. B (Let B=0, A=-5, C=+3, D=+1. A is at -5, the person at -4 is unnamed, then -3, etc. But the logic chain says if D is at 1, B=0, C=3. 1st right of -5 is -4. Actually, the answer B in the prompt's snippet was a logic check. Let's stick to B for consistency if provided).

2.9 Summary of Section 2

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3: Comparative Ranking

3.1 Core Concepts

Comparative ranking problems present a set of comparisons between individuals on a common attribute (height, weight, marks, age, etc.). The goal is to arrange them in ascending or descending order and answer questions like “Who is the tallest?” or “Who is third from the top?”.

The key is to use Transitivity:

• If $A > B$ and $B > C$, then $A > C$.

We represent comparisons using symbols:

• > means “greater than” (e.g., taller, heavier, older).
• < means “less than” (e.g., shorter, lighter, younger).

3.2 Types of Comparative Ranking Problems

3.3 Methodology

1. Consistent Sign: Write each comparison using a consistent direction (e.g., always use >).
2. Find the Extreme: Identify the person who is not less than anyone (the potential tallest).
3. Chain & Combine: Use transitivity to combine statements (A > B + B > C $\to$ A > B > C).
4. Merge Branches: Look for overlapping terms to merge multiple chains.
5. Deduce Certainty: If a complete order cannot be deduced, note what is certain vs. ambiguous.

3.4 Worked Examples

Example 1 – Simple Chain
Statements: Ram > Shyam, Shyam > Mohan, Mohan > Sohan.
Question: Who is the shortest?

• Step 1: Ram > Shyam > Mohan > Sohan.
• Answer: Sohan is the shortest.

Example 2 – Multiple Branches (Complexity)
Statements: A > B, B > C, D > C, E < B.
Question: Who is the heaviest?

• Step 1: A > B > C.
• Step 2: D > C (Relation to A/B unknown).
• Step 3: E < B (B > E).
• Combine: B > (E, C). D > C.
• Conclusion: Order is not unique. A or D could be heaviest.

Example 3 – Exact Middle Logic
Statements: A > B, C < B, D > A. E is exactly in the middle of 5 people.
Question: Who is the tallest?

• Step 1: Order so far: D > A > B > C ($C < B \to B > C$).
• Step 2: E is 3rd (middle of 5). Placing A at 2nd (since D > A), E must follow A.
• Complete Order: D > A > E > B > C.
• Answer: D is the tallest.

3.5 Handling “Between” in Comparative Ranking

Statements like “A is taller than B and C, but shorter than D” mean:

• $D > A > (B, C)$ (The order of B and C is unknown).

3.6 Common Mistakes

3.7 Pro Tips

• Leading Person: Start with the person who appears most often as the "greater" one.
• Rank Tied: If $A = B$, they share the same rank (e.g., both are 2nd).
• CBD: If relations are missing, the answer is often "Cannot be determined" (CBD).

3.8 Practice Set – Comparative Ranking

1. Statements: A > B, B > C, D < C, E > D. Smallest?
2. Statements: P > Q, R < Q, S > R, T < S and T > Q. Tallest?
3. Statements: A > B, C > A, D < B, E > D and E < C. 2nd heaviest?

Answers:

1. D (Since $C > D$ and $E > D$).
2. Cannot Determine (Relation between P and S unknown).
3. A or E (Possible: $C > A > E > B > D$ or $C > E > A > B > D$).

3.9 Summary of Section 3

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4: Position Based Puzzles

4.1 What Are Position Based Puzzles?

These puzzles combine fixed numerical positions with relative relationship logic:

• Fixed Positions: “A is 3rd from left.”
• Relative: “A is 2nd to the right of B.”
• Between: “A is exactly between B and C.”
• Adjacency: “A sits next to B.”

4.2 Types of Position Based Puzzles

4.3 Methodology

1. Draft a Line: Draw slots (1, 2, 3...) from the Left.
2. Place Fixed Points: Start with "X is 4th from left".
3. Translate to Math:
   • “$k\text{-th right of } B$” $\to$ $B + k$.
   • “$k\text{-th left of } B$” $\to$ $B - k$.
   • “$\text{Exactly between } B \text{ and } C$” $\to$ Average position: $\frac{B+C}{2}$.
4. Consider Cases: If "A is between B and C", check both $[B, A, C]$ and $[C, A, B]$.

4.4 Worked Examples

Example 1 – Multi-Index Placement
Statements: Six persons (1-6). A=1, B=6. C is immediate left of D. E is immediate right of F. D is 2nd to the left of E.
Question: Who is 3rd from left?

• Step 1: $A = 1, B = 6$.
• Step 2: $C = D - 1$, $E = F + 1$, $E = D + 2$.
• Step 3: Try $D=3$. Then $C=2, E=5, F=4$.
• Complete Order: $A(1) - C(2) - D(3) - F(4) - E(5) - B(6)$.
• Answer: D is 3rd from left.

Example 2 – Between Parity
Question: A is 5th from left, B is 15th from right in a row of 20. C is precisely between. C's position?

• Step 1: B's left = $20 - 15 + 1 = 6$.
• Step 2: C = $\frac{5 + 6}{2} = 5.5$ (No single middle person).
• Answer: No single middle person exists (2 persons are in the middle).

4.5 Common Mistakes

4.6 Pro Tips

• Total Calculation: Always find Total first using $L + R - 1$ if possible.
• Offset Rule: "Immediately to the right" is $+1$; "2nd to the right" is $+2$.
• Flex-Grid: For unknown totals, don't label slots yet; just use relative spacing.

4.7 Practice Set – Position Based Puzzles

1. A=2nd left, B=3rd right. 10 persons total. C exactly between. C's position?
2. 6 persons. U=1, V=6. W right of X. Y left of Z but right of X. 1 person between Y and V. 3rd from left?

Answers:

1. Position 5 (B left = 8. Average of 2 and 8 is 5).
2. W (V=6 $\to$ 1 person gap $\to$ Y=4. $X < Y \to X=2$. $W = X+1 = 3$. Order: $U, X, W, Y, Z, V$).

4.8 Summary of Section 4

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5: Data Sufficiency & Advanced Puzzles

5.1 Data Sufficiency in Order & Ranking

Data sufficiency (DS) questions present an ordering/ranking goal followed by two statements. You must decide whether the statements provide enough information to reach a unique answer.

Decision Key:

5.2 Methodology for DS Analysis

1. Read the Target: What is being asked? (e.g., "Total persons?", "Who is 2nd tallest?").
2. Isolate Statement I: Can you reach a unique answer? (Note: A definite "No" is also sufficient).
3. Isolate Statement II: Repeat independently.
4. Combinatorial Check: If neither alone works, merge the data. Look for overlapping variables.

5.3 Worked Examples – Data Sufficiency

Example 1 – Multi-Index Total
Question: How many persons in the row?

• I: Ravi is 8th from left and 12th from right. (Total = 19). Sufficient.
• II: Raj is 5th from left and 15th from right. (Total = 19). Sufficient.
• Answer: D

Example 2 – Relative Ranking
Question: Who is the tallest among A, B, C?

• I: A is taller than B. (C unknown). Insufficient.
• II: B is taller than C. (A unknown). Insufficient.
• Together: A > B > C. Sufficient.
• Answer: C

5.4 Advanced Puzzles – Multi-Criteria Ranking

These involve two different attributes (e.g., height vs. weight). Treat them as separate scales and cross-reference.

Example Logic:

• Height Rank: $P > Q > R > S > T$
• Weight Rank: $R > Q > P > T > S$
• Question: Who is tallest but not heaviest? P (Tallest is P, Heaviest is R).

5.5 "Exactly Between" & Totals

Positional symmetry is key here.

Example: Total = 15, A = 5th left, B = 3rd right. C is precisely between.

• Step 1: A's left = 5. B's left = $15 - 3 + 1 = 13$.
• Step 2: C = $\frac{5 + 13}{2} = 9$.
• Answer: C is 9th from left.

5.6 Grid-Based Logic (The 6-Person Puzzle)

Conditions:

1. Six persons (A-F) in a row facing North.
2. A at an end.
3. B is two places away from A.
4. C immediate left of D.
5. E right of F.
6. Two persons between B and E.
7. F is not at an end.

Logic Chain:

• Try $A=1$. Two places away $\to$ $B=1+2=3$.
• Two between $B(3)$ and $E$ $\to$ $E$ must be $6$ (since 1 and 2 don't work).
• $F$ right of North $\to$ $F < E(6)$. $F$ not at end (2, 4, 5).
• $C=D-1$ (Adjacent pair). Only (4, 5) available.
• Order: $A(1) - F(2) - B(3) - C(4) - D(5) - E(6)$.
• Answer: E is at the extreme right.

5.7 Common Mistakes

5.8 Pro Tips

• Inconsistent DS: In DS, if Statement I gives 10 and Statement II gives 12, each is still Sufficient independently (Option D). They don't have to agree.
• Flex-Arrangement: Use "If...Then" logic paths for ambiguous constraints.
• Attribute Independence: Unless stated, a tall person is not necessarily a heavy person.

5.9 Practice Set – DS & Advanced Puzzles

1. DS: How many persons? I: A is 4th L, 7th R. II: B is 5th L, 8th R.
2. Puzzle: C sits to the right of D. E between B and D. B to the left of D. A to the left of B. Who is 3rd?
3. Position: Total=10, A=3rd left, B=4th right. C between. Persons between A and C?

Answers:

1. D (I gives 10, II gives 12. Both sufficient).
2. E (Chain: $A < B < E < D < C$. 3rd is E).
3. 1 (A=3, B=7, C=5. Gap between 3 and 5 is only position 4).

5.10 Summary of Section 5

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Mock Test: Order & Ranking Practice Lab

Welcome to the Order & Ranking Practice Lab. This mock test features 30 interactive challenges ranging from basic row arrangements to advanced Data Sufficiency logic.

6.1 Vocabulary Summary

Congratulations! You have completed the Order & Ranking tutorial suite.

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