Order & Ranking
🔒 Log in to trackHow ranking questions work
🔒 Log in to trackPeople (or objects) stand in a line or carry ranks, and you must convert between viewpoints: top↔bottom, left↔right, front↔back. The entire topic runs on two ideas:
- Mirror conversion: if there are positions, the person at position from one end sits at position from the other. The '+1' exists because the person's own seat is counted on both sides.
- Two-viewpoint total: if the same person is -th from one end and -th from the other, then .
Everything else — 'how many between', overlaps, joined queues — is these two formulas plus careful reading. Draw a tiny line of boxes for any question that involves three or more people; the 5 seconds of drawing prevents every 'off by one' answer.
Detailed notes
Rank, position and order: the three conversions
Every question here involves a line of people (or a stack of objects) and a rank measured from one end. Three facts convert anything into anything.
Fact 1 — the two ends are mirrors. In a row of T people, the person r-th from the left is -th from the right. Example: 10th from the left in a row of 40 → = 31st from the right. The +1 exists because the person is counted once at each end.
Fact 2 — two ranks make the total. If one person is a-th from one end and b-th from the other end, the row holds
Again the −1: the person was counted in both numbers. Example: 9th from the left and 11th from the right → = 19 people.
Fact 3 — a swap copies positions. When two people interchange places, each simply takes over the other's seat. To find a new rank, convert the other person's rank into the same end first: Shyam 9th from the right in a row of 25 → = 17th from the left; after a swap, Ram sits 17th from the left.
Stacks and rows of objects. Books, boxes and shelves follow the same logic vertically: 'above' is like 'to the left'. The only new skill is assembling a scattered description into one ordered list — draw the order top to bottom as each sentence arrives, then read the asked slot.
Facing direction changes everything. A row facing north has its left and right in the usual sense; a row facing south has them swapped (their left is your right). Questions flag this deliberately — underline the facing word before using any left/right clue.
People ahead vs behind. In a queue of T, a person r-th from the front has people ahead and behind. The −1 on 'ahead' (the person is not ahead of themselves) is the most common slip in the whole topic.
Quick revision
- Mirror: r-th from one end = from the other.
- Total: for ranks a and b from opposite ends of the same person.
- Swap: convert both people to the same end; each takes the other's number.
- Ahead = r − 1; behind = T − r.
- Facing south flips left/right — read the facing word first.
Why the formulas work (and why that matters). Every shortcut in this topic is bookkeeping on one sentence: a rank is one more than the number of people before you. The mirror formula is that sentence counted from the other side (the people before you become the people after you, plus you); the total formula is the two before-counts overlapping on you, hence −1. When an exam twists the wording ('four people between them', 'equal number ahead and behind'), derive the equation from this sentence instead of memorising a new formula — it takes ten seconds and never breaks.
The twisted variants worth pre-solving. 'Equal ahead and behind' → → . 'k people between A and B, same end' → . 'A is exactly in the middle of an odd row' → . Practise these three derivations once and the variants stop costing time.
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 flips between the two endsvery common4 practice Q
One rank and the total are given; the same person's rank from the other end is asked.
- Other end = total − rank + 1.
- Keep the +1: the person sits at both counts.
- For queues: ahead = rank − 1, behind = total − rank.
Why it works: the two counts meet at the same person.
Example: In a row of 40 students, Ravi is 10th from the left. What is his position from the right?
= 31st from the right.
Type 2: Total from two opposite ranksvery common3 practice Q
One person's rank from both ends is given; the size of the row/class is asked.
- Total = a + b − 1.
- Sanity-check: total ≥ each rank.
- Works for rows, queues and class ranks alike.
Why it works: the person was counted once in each rank; subtract the double count.
Example: A student is 23rd from the left and 12th from the right. How many students are in the row?
= 34.
Type 3: Interchange of positionscommon2 practice Q
Two people swap places; a new rank after the swap is asked.
- Convert the partner's rank to the asked end (mirror if needed).
- The swapper now holds exactly that converted number.
- The other person takes the first person's old number.
Why it works: a swap is a copy of positions between two people.
Example: In a row of 25, Ram is 8th from the left and Shyam is 9th from the right. They swap places. What is Ram's new position from the left?
Shyam's seat: = 17th from the left → Ram now 17th.
Type 4: Assembling a scattered stack or rowcommon3 practice Q
Objects/friends described by above-below or left-right clues; a slot or extreme is asked.
- Convert each clue into order arrows (above = >).
- Merge into one list, drawing as you read.
- Read the asked slot off the finished list.
Why it works: scattered clues describe one total order; drawing recovers it.
Example: Red is above Blue; Green is below Blue but above White; Yellow is at the bottom. Which box is at the top?
Red > Blue > Green > White > Yellow → top = Red.
Formulas
Position from the other end.
Same person, ranked from both ends.
Two positions measured from the SAME end.
Shortcut tricks
⚡ Draw the end-zones
Sketch a short row of boxes. Fill in what you know at the ends: 'a-th from left' means a−1 people before her. Counting the unknowns at each end replaces every formula with a picture.
Example: Q. In a row of 40, A is 11th from the left. How many are to her right?
Sol. 11th from left → 10 before her; 40 − 11 = 29 to her right (she is 30th from the right).
a-th from an end = (a−1) people beyond that end. The rest is subtraction.
Where students lose marks
Forgetting the +1/−1 corrections — the examiner sets traps exactly one away from the answer.
Mixing viewpoints: subtracting a 'from the right' rank from a 'from the left' rank directly.
Assuming the answer must be one less or one more than a listed option — compute first.
Practice sets — 12 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 12 questions
Suggested time 9 min · wrong answers go to your mistake notebook automatically.