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Combined Rank from Two Lists

Introduction

Many ranking problems give a person's position counted from the top (or left) and from the bottom (or right). Converting between these two references quickly and accurately is a frequent aptitude requirement.

This pattern is important because a single person’s ranks from both ends immediately reveals the total size - and from that you can answer a range of follow-up questions.

Pattern: Combined Rank from Two Lists

Pattern

Key concept: Total = (Rank from top) + (Rank from bottom) - 1.

Useful derived formulas:

  • If a person is rtop from top and rbottom from bottom, then total n = rtop + rbottom - 1.
  • If total n and one-side rank are known, opposite-side rank = n - (given rank) + 1.
  • When asked how many between two people whose ranks are from opposite ends, convert both to the same reference first (people-ahead or position-from-top) and apply difference formulas.

Step-by-Step Example

Question

Anu is 7th from the top of a class and 18th from the bottom. Find the total number of students in the class.

Solution

  1. Step 1: Write the given ranks

    Rank from top = 7; Rank from bottom = 18.

  2. Step 2: Apply the combined-rank formula

    Total = 7 + 18 - 1 = 24.

  3. Step 3: Quick verification by conversion

    Anu’s position from bottom computed from total: 24 - 7 + 1 = 18 → matches given, so total = 24 is correct.

  4. Final Answer:

    24 students.

  5. Quick Check:

    Top rank 7 → 6 ahead; bottom rank 18 → 17 below; 6 ahead + 1 (Anu) + 17 below = 24 ✅

Quick Variations

1. If given one-side rank and total, compute opposite-side rank as n - given + 1.

2. If two different people have ranks from opposite ends, convert both to the same end (e.g., both to "from top") before computing between-counts.

3. When told a person’s rank changes after additions/removals at one end, adjust total first if needed and reapply conversions.

4. If asked how many between person X (rtop) and person Y (rbottom), convert Y to top reference and use |rY_top - rX| - 1.

Trick to Always Use

  • Step 1: If both ranks are given (top & bottom), use n = top + bottom - 1 immediately.
  • Step 2: If only one rank is given, convert to "people ahead" (rank - 1) for intermediate reasoning.
  • Step 3: For counts between two people, convert both to the same reference and apply the difference-minus-one rule.

Summary

Summary

  • Use n = (rank from top) + (rank from bottom) - 1 to find total immediately.
  • Convert ranks to "people ahead" (rank - 1) when it helps with intermediate steps.
  • To switch sides: opposite-rank = n - given-rank + 1.
  • For persons with mixed-end ranks, convert both to one side before computing distances or between-counts.

Example to remember:
If someone is 5th from top and 12th from bottom → total = 5 + 12 - 1 = 16 students.

Practice

(1/5)
1. In a class, Rohan is 8th from the top and 13th from the bottom. Find the total number of students in the class.
easy
A. 20
B. 19
C. 21
D. 22

Solution

  1. Step 1: Write ranks

    Rank from top = 8, Rank from bottom = 13.

  2. Step 2: Apply formula

    Total = Top + Bottom - 1 = 8 + 13 - 1 = 20.

  3. Final Answer:

    20 students → Option A

  4. Quick Check:

    20 - 8 + 1 = 13 (matches bottom rank) ✅
Hint: Total = Top + Bottom - 1.
Common Mistakes: Adding both ranks without subtracting 1.
2. In a competition, Asha is 5th from the top and 22nd from the bottom. How many participants were there in total?
easy
A. 26
B. 25
C. 27
D. 28

Solution

  1. Step 1: Use combined formula

    Total = Top + Bottom - 1 = 5 + 22 - 1 = 26.

  2. Step 2: Verify

    Opposite check → 26 - 5 + 1 = 22 ✅

  3. Final Answer:

    26 participants → Option A

  4. Quick Check:

    Top + Bottom - 1 = 26 ✅
Hint: Use top + bottom - 1 formula directly.
Common Mistakes: Forgetting to subtract 1 from the total.
3. In a list of students, Priya is 9th from the top and 7th from the bottom. What is the total number of students?
easy
A. 16
B. 17
C. 15
D. 18

Solution

  1. Step 1: Apply formula

    Total = Top + Bottom - 1 = 9 + 7 - 1 = 15.

  2. Step 2: Verify

    15 - 9 + 1 = 7 → matches bottom rank.

  3. Final Answer:

    15 students → Option C

  4. Quick Check:

    Sum - 1 = 15 ✅
Hint: Always subtract 1 after adding ranks.
Common Mistakes: Including both ends without subtracting 1.
4. In a race, Ravi’s position is 10th from the start and 16th from the end. How many runners are there?
medium
A. 24
B. 23
C. 25
D. 26

Solution

  1. Step 1: Use combined formula

    Total = Top + Bottom - 1 = 10 + 16 - 1 = 25.

  2. Step 2: Cross-check

    From total 25, end rank = 25 - 10 + 1 = 16 ✅

  3. Final Answer:

    25 runners → Option C

  4. Quick Check:

    Top + Bottom - 1 = 25 ✅
Hint: Top + Bottom - 1 = Total.
Common Mistakes: Missing the ‘-1’ correction factor.
5. In a group of players, John ranks 6th from the top and 9th from the bottom. Find total players in the group.
medium
A. 13
B. 15
C. 16
D. 14

Solution

  1. Step 1: Apply formula

    Total = Top + Bottom - 1 = 6 + 9 - 1 = 14.

  2. Step 2: Verify

    14 - 6 + 1 = 9 → matches given bottom rank.

  3. Final Answer:

    14 players → Option D

  4. Quick Check:

    Sum - 1 = 14 ✅
Hint: Add both ranks and subtract 1.
Common Mistakes: Adding both ranks directly without subtracting 1.

Mock Test

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