Order & Ranking
🔒 Log in to trackRanks and positions
🔒 Log in to trackSingle-person rank conversions and totals:
- Convert an end: 12th from top in a class of 45 → 45 + 1 − 12 = 34th from bottom.
- Find the total: 8th from the left and 21st from the right → n = 8 + 21 − 1 = 28.
- 'Same from both ends': 11th from either end → n = 11 + 11 − 1 = 21.
Watch the exam's phrasing variants: 'how many students scored less than him' = his rank from the bottom, 'precedes him' = people before his position, 'queue joined from the front' shifts every old position forward by the number of joiners.
Detailed notes
Single-person rank work
This subtopic drills the conversions one person at a time: one end to the other, two ends to the total, and the moving-queue variants where people join or leave.
The mirror conversion. From-rank r in a row of T: other end = . Practical route: rank from bottom = class size − rank from top + 1. A 23rd-from-the-top rank in a class of 60 is 38th from the bottom: .
The total from two ends. When the question gives one person's rank from both ends, the class size pops out: . Sonu 14th from the left and 20th from the right → people. Sanity-check the answer: it must be at least as big as each rank.
Queues that change size. People joining the front shift everyone back; people leaving the front pull everyone forward. Rule: adjust the number of people before the person first, then add one.
- k people join ahead of you → new rank = old + k.
- k people leave from ahead of you → new rank = old − k.
- If the joiners/leavers are behind you, your rank does not change.
People leaving mid-queue. '5 people left the row from the left side': anyone to the right of them slides left by 5. A person 16th from the left becomes 11th. Anyone to the left of the departed block keeps their rank but the total shrinks.
Ranks from both ends together. Some questions hand you rank from the top and from the bottom of a class with a twist ('A is 12th from the top and 41st from the bottom; how many students are there?'). Same formula: .
Watch the units. 'Rank' counts the person; 'people ahead/behind' does not. Convert explicitly: ahead = rank − 1; rank = ahead + 1. Questions swap these words between the question and the options on purpose.
Quick revision
- Other end: . Two ends: .
- Joiners ahead: rank + k. Leavers ahead: rank − k. Changes behind you: no change.
- Mid-row exits: slide only people to the right of the exit block.
- ahead = rank − 1, behind = T − rank; convert words to numbers before computing.
Two-sided puzzles. Harder variants give a rank from one end plus a second fact ('he is exactly midway', 'there are 3 people between him and the front'). Convert the fact into a before-count first: 'midway' means ahead = behind; '3 between him and the front' means ahead = 4. Once the before-count is known, rank = before + 1 and the total follows if the other side is described.
Class ranks vs rows. The formulas are identical; only the words move (top/bottom instead of left/right, pass/fail groups at the ends). One warning: 'X failed' questions where 'rank from the bottom among failures' is asked — strip the failed students out of the total before mirroring. The formula never changes, but the total does.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Rank from the other endvery common2 practice Q
Class/queue size known; rank from top/left/front given; the mirror rank is asked.
- Apply total − rank + 1.
- Read the end words carefully (top↔bottom, left↔right, front↔back).
Why it works: the two counts meet at the same person.
Example: In a class of 60, Neha ranks 23rd from the top. What is her rank from the bottom?
= 38th.
Type 2: Total from two ranksvery common2 practice Q
One person's rank from both ends is given; the total is asked.
- Add the ranks, subtract 1.
- Check the total is not smaller than either rank.
Why it works: the shared person is counted twice.
Example: Sonu is 14th from the left and 20th from the right in a row. How many people are in the row?
= 33.
Type 3: Queue joining and leavingcommon2 practice Q
People join or leave a queue; the person's new position is asked.
- Count only people ahead: rank = (people ahead) + 1.
- k join ahead → +k; k leave from ahead → −k.
- Changes behind the person never move their rank.
Why it works: a rank is one more than the number of people ahead.
Example: In a queue of 35, Moni is 12th. If 8 people ahead of her leave, what is her new position?
= 4th.
Type 4: Rank after people leave mid-rowcommon2 practice Q
Some people exit from one side of a standing row; a survivor's new rank is asked.
- If the exits are to the person's left, subtract the count.
- If they are to the right, the rank from the left is unchanged.
- Recompute the total if it is asked.
Why it works: survivors slide toward the emptied side.
Example: A is 16th from the left in a row of 40. If 5 people leave from the left end, what is A's new rank from the left?
The 5 departed were all ahead of A → = 11th.
Shortcut tricks
⚡ Sanity-span check
The two ranks of one person must satisfy a + b = n + 1. Before marking a total, add the given ranks: if the sum is not total-plus-one, one of the numbers was misread.
Example: Q. K is 12th from the left and 19th from the right. Total?
Sol. 12 + 19 = 31, so n = 31 − 1 = 30. Check: 12th from left has 11 left of her; 19th from right has 18 right of her; 11 + 1 + 18 = 30 ✓.
a + b = n + 1 is a two-second self-check built into every question.
Where students lose marks
Using n = a + b (no −1).
Treating '11th from either end' as two different people's ranks.
Missing queue shifts when people join or leave ahead of the person.
Practice sets — 16 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 5 min · wrong answers go to your mistake notebook automatically.