3.2 Order, Ranking & Comparison Tests
Key Takeaways
- The fundamental single-row ranking equation is Total = (Rank from Left + Rank from Right) - 1, where 1 is subtracted because the individual is counted twice.
- To find an individual's rank from the opposite end when total row strength T and current rank R are known, use the reciprocal formula: Opposite Rank = (T - R) + 1.
- A queue is non-overlapping when Total ≥ (Left Rank + Right Rank), whereas an overlapping queue occurs when Total < (Left Rank + Right Rank - 1), yielding the minimum queue strength formula: Total = Left + Right - Between - 2.
- When two individuals swap positions, the magnitude of shift for Person A exactly equals the shift for Person B, and the total strength equals the new rank of Person A plus the original rank of Person B minus 1.
- Vertical ranking arranges candidates along qualitative comparative attributes (sanctions, speed, deposits) by chaining definite binary inequalities into a unified transitive descending sequence.
3.2 Order, Ranking & Comparison Tests
Order and Ranking is an indispensable component of the Reasoning Ability section in the SBI Clerk examination. Questions appear in two distinct varieties: linear ranking problems involving positions of individuals in a line or queue (e.g., customers at a cash counter or candidates in a merit list), and comparison tests (vertical ranking) where entities are sequenced according to comparative metrics such as transaction processing speed, loan sanctions, or account balances.
Linear Ranking Fundamentals
Linear ranking problems depend upon precise algebraic relationships between an individual's rank from one terminal end, their rank from the opposing terminal end, and the total count of individuals in the sequence.
The Fundamental Single-Row Ranking Formula
When the position of a single individual $P$ is known from both ends of a linear row:
Why the '$- 1$' is mandatory: When you count from the left end up to Person $P$, Person $P$ is counted once. When you count from the right end down to Person $P$, Person $P$ is counted a second time. Subtracting 1 eliminates this duplicate enumeration.
Calculating Rank from Opposite Ends
When the total number of persons $T$ is fixed and an individual's rank from one end is given, the position from the opposite end is derived by transposing the core formula:
[!NOTE] Beginners frequently make the mistake of calculating $T - R$. If 30 candidates are seated in a row and Rohan is 10th from the left, his rank from the right is not $30 - 10 = 20$; it is $(30 - 10) + 1 = 21\text{st}$. There are 20 candidates seated to his right, making Rohan the 21st person from the right.
Mathematical Ranking Formula Reference Table
| Variable to Determine | Required Inputs | Exact Formula |
|---|---|---|
| Total Persons ($T$) | Ranks of one person from both ends ($L, R$) | $T = L + R - 1$ |
| Rank from Left ($L$) | Total persons ($T$) and Rank from Right ($R$) | $L = T - R + 1$ |
| Rank from Right ($R$) | Total persons ($T$) and Rank from Left ($L$) | $R = T - L + 1$ |
| Persons Between ($B$) [Simple / Non-Overlapping] | Total ($T$), Person A from Left ($L$), Person B from Right ($R$) | $B = T - (L + R)$ |
| Persons Between ($B$) [Overlapping Case] | Total ($T$), Person A from Left ($L$), Person B from Right ($R$) | $B = (L + R) - T - 2$ |
| Minimum Total ($T_{\min}$) [Overlapping Row] | Person A from Left ($L$), Person B from Right ($R$), Persons Between ($B$) | $T_{\min} = L + R - B - 2$ |
| Maximum Total ($T_{\max}$) [Non-Overlapping Row] | Person A from Left ($L$), Person B from Right ($R$), Persons Between ($B$) | $T_{\max} = L + R + B$ |
Non-Overlapping vs. Overlapping Queues
When a question specifies the positions of two distinct individuals (Person A at rank $L$ from the left, Person B at rank $R$ from the right) along with the number of persons standing between them ($B$), two mutually exclusive physical configurations are possible: non-overlapping and overlapping.
The Non-Overlapping Condition (Maximum Row Strength)
In a non-overlapping queue, Person A (from the left) and Person B (from the right) have not crossed each other. A sits strictly to the left of B:
- Condition: $T \ge L + R$
- Formula for Persons Between: $B = T - (L + R)$
- Formula for Total Persons: $T = L + R + B$
The Overlapping Condition (Minimum Row Strength)
In an overlapping queue, Person A (counted from the left) and Person B (counted from the right) have crossed paths. Person B is situated closer to the left end than Person A:
- Condition: $T < (L + R - 1)$
- Inequality Check: $(L + R) > (T + 1)$
- Formula for Persons Between: $B = (L + R) - T - 2$
- Formula for Minimum Total Persons: $T = L + R - B - 2$
Proof of the '$- 2$' Deduction in Overlap
Why does the overlapping formula subtract 2? When you sum the rank of Person A from the left ($L$) and Person B from the right ($R$):
- All persons standing strictly between Person A and Person B are counted twice (once during the left-to-right pass, and once during the right-to-left pass).
- Both Person A and Person B are themselves counted twice (each is included in their own rank, and also included as an antecedent in the opposing person's traversal).
Therefore: $L + R = T + B + 2$. Transposing for $T$ yields $T = L + R - B - 2$.
Overlap Feasibility Condition
For an overlapping queue to physically exist, the number of persons between them must satisfy:
If $B > \min(L, R) - 2$, an overlapping configuration is physically impossible, and only the non-overlapping calculation can be valid.
Worked Example: Customer Token Queue at SBI Branch
Scenario: In an SBI branch pension disbursement queue, Ananya ranks 18th from the front (left) counter, while Brijesh ranks 24th from the rear (right) exit. There are 8 pensioners standing between Ananya and Brijesh. Find (1) the maximum possible number of pensioners in the queue, and (2) the minimum possible number of pensioners in the queue.
Step-by-Step Solution:
- Check Overlap Feasibility: $\min(L, R) - 2 = \min(18, 24) - 2 = 18 - 2 = 16$. Because $B = 8 \le 16$, both non-overlapping and overlapping configurations are valid.
- Maximum Case (Non-Overlapping):
- Minimum Case (Overlapping):
Rank Interchange Problems
Rank interchange scenarios involve two individuals swapping physical seats in a row. These questions appear frequently in SBI Clerk Prelims and can be solved in under 30 seconds once their algebraic properties are understood.
Mechanics of Position Swapping
Suppose Person A is originally $L_1$ from the left, and Person B is originally $R_1$ from the right. When they swap positions, Person A occupies Person B's former seat, and Person B occupies Person A's former seat.
If the problem states that after swapping, Person A's new rank from the left becomes $L_2$:
- Calculate the Positional Shift ($\Delta$): The number of positions Person A shifted is $\Delta = |L_2 - L_1|$.
- Determine Persons Standing Between Them: The number of persons seated between their original positions is strictly:
- Calculate Total Persons in the Row ($T$): Because Person A now occupies Person B's original seat, that specific physical seat has two known ranks: $L_2$ from the left, and $R_1$ from the right. Applying the single-seat total formula:
- Calculate Person B's New Rank ($R_2$): Because the physical separation between their seats is constant, Person B must undergo the exact same numerical shift from the opposing end:
Worked Example: Newly Recruited Junior Associates in Training
Scenario: In an SBI induction hall, Ramesh is seated 14th from the left end, while Sneha is seated 21st from the right end. Upon interchanging their seats, Ramesh becomes 26th from the left end. Find: (1) Sneha's new position from the right end, and (2) the total number of associates in the row.
Step-by-Step Solution:
- Compute Ramesh's Shift: $\Delta = 26 - 14 = 12$ places to the right.
- Find Associates Between Ramesh and Sneha: $B = 12 - 1 = 11$ associates.
- Compute Total Associates: Sum Ramesh's new left rank with Sneha's original right rank:
- Compute Sneha's New Position from Right: Sneha moves 12 places from her original position: Verification: $T = L_1 + R_2 - 1 = 14 + 33 - 1 = 46$. The total is perfectly consistent.
Vertical Ranking & Comparison Deductions
Comparison test puzzles provide fragmented binary inequality statements concerning 5 to 7 individuals ranked by an attribute such as transaction volume, monthly deposit mobilization, or customer service ratings.
Sequential Attribute Chaining
To solve vertical ranking efficiently:
- Use Strict Relational Notations: Write "$A > B$" for $A$ sanctioned more than $B$. Never use ambiguous dashes.
- Anchor Extreme and Quantified Clues:
- Clues like "$P$ processed more than only $Q$" establish that $Q$ is strictly the lowest (last position) and $P$ is immediately above $Q$ (second from bottom).
- Clues like "Only two persons scored higher than $M$" establish that $M$ is fixed in exactly the 3rd position from the top.
- Form a Unified Transitive Chain: Chain overlapping segments ($X > Y$ and $Y > Z \implies X > Y > Z$).
Worked Example: Branch Manager Housing Loan Portfolios
Scenario: Six SBI Branch Managers (P, Q, R, S, T, U) sanctioned varying amounts of home loans in a quarter:
- R sanctioned more than T but less than P ($P > R > T$).
- S sanctioned more than U but less than T ($T > S > U$).
- Q sanctioned more than P ($Q > P$).
- The manager who sanctioned the second-highest amount sanctioned ₹45 Crore, while the manager who sanctioned the third-lowest amount sanctioned ₹28 Crore.
Deduction Protocol:
- Combine the inequality statements into a single chain: $Q > P > R > T > S > U$.
- Identify specific ranks:
- 1st (Highest): Q
- 2nd (Second-Highest = ₹45 Crore): P
- 3rd: R
- 4th (Third-Lowest = ₹28 Crore): T
- 5th: S
- 6th (Lowest): U
- Answering questions: If asked for the possible loan value of R, R must fall strictly between ₹45 Crore (P) and ₹28 Crore (T).
In a queue of customer tokens at an SBI Main Branch, Divya ranks 17th from the front counter and Gaurav ranks 22nd from the rear exit. If there are exactly 6 customers standing between Divya and Gaurav in an overlapping queue, what is the total number of customers present in this single line?
In a row of SBI locker holders waiting for biometric verification, Manoj is 16th from the left end and Pooja is 19th from the right end. When they swap their positions, Manoj becomes 25th from the left end. What is Pooja's new position from the right end, and what is the total number of locker holders in the row?
Six branch officers (K, L, M, N, O, P) scored marks in an internal promotional test. M scored higher than P but lower than L. N scored higher than only O. K scored higher than L. Who scored the highest mark, and who scored the third-lowest mark?