Order and Ranking

In the exam this comes as one short sentence: "Vimal is 28th from the top and 65th from the bottom of his class. How many students are there?" One name, two ranks from opposite ends, and the total is asked. Let us see why a single line of arithmetic is enough.

Suppose Radhika is 14th from the left end and 19th from the right end of a row. How many persons stand on her left? Thirteen. How many on her right? Eighteen. Add Radhika herself and the whole row is 13 + 1 + 18 = 32 persons.

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Now see the shortcut hidden inside this. If we simply add her two ranks we get 14 + 19 = 33. That is one too many, because Radhika was counted once from the left and once from the right. Take away the extra count and you get the row. N = L + R − 1.

The method has three small checks, and each takes two seconds.

• Are both ranks of the same person? If two different people are named, this formula does not apply (see Type 8).
• Are the two ranks from opposite ends? If both are from the same end, there is nothing to add.
• Then write N = L + R − 1 and put the numbers.

Sometimes the question hides the numbers: "his rank from the right is 7 more than his rank from the left, and he is 13th from the left". Turn the words into numbers first (R = 13 + 7 = 20) and only then use the formula. The same formula works for classes ("rank from the top and from the bottom"), queues and any other line.

The second rank may also hide inside a count. "Only 12 persons are seated to the right of Sneha" means Sneha is 13th from the right, because Sneha herself must be counted. Turn such a count into a rank first, and only then add.

The exam gives you the size of the line and one position, and asks for the same person from the other end: "In a row of 20 children, Roshan is 10th from the right. What is his position from the left?" A tempting option repeats the number you were given ("10th from the left"). The right answer is 11th, and the reason is the whole of this type.

Here the total is given along with one rank, and you must find the rank from the other end. Think of a class of 45 students in which Aman is 17th from the top. Everyone above him, 16 students, is not counted when we start from the bottom. Everyone else is counted: Aman himself and the 28 students below him. That is 29.

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So the rank from the bottom is 45 − 16 = 29. Since 16 is simply 17 − 1, we can write the whole thing in one go: rank from the other end = N + 1 − rank from this end.

Why N + 1 and not N? Because the person himself is counted from both ends. That is the same fact as before: the two ranks add up to N + 1. This gives you a free self-check: after finding the answer add the two ranks. If the sum is not N + 1, something went wrong.

Method: (1) note which end the given rank is from; (2) write N + 1 − r; (3) check that both ranks add up to N + 1.

Many questions add a second person: "In a row of 54 students, Gagan is 13th from the right end. Nitin is 17 places to the left of Gagan. What is Nitin's position from the left end?" Moving to the left takes Nitin away from the right end, so his rank from the right grows: 13 + 17 = 30. Now convert once: 54 + 1 − 30 = 25th from the left. The number 30 is only the middle step and is usually kept among the options as a trap. A step towards an end lowers that end's rank; a step away from it raises that rank.

The question can be asked backwards, too: "Aman is 29th from the bottom in a class of 45; what is his rank from the top?" The same formula works in both directions, because swapping the two ends is a symmetric operation. Sometimes a second person enters the picture ("Sagar's rank is 5 less than Aman's rank from the bottom"). Find Aman's other-end rank first, make the small adjustment, and then convert once more if the question asks for the original end.

This is the most frequent direct question of the chapter: "All 193 persons are standing in a row facing north. Ccd is 14th from the right end and Vks is 124th from the left end. How many persons are between Ccd and Vks?" The numbers are big, so nobody draws the row. Everything depends on a single comparison and one subtraction.

To count the persons between two people, put both on one ruler and measure the gap between their positions. The persons between are that gap minus 1, because the two end persons are not counted.

Case 1: both ranks from the same end. Say Pooja is 12th and Sneha is 27th from the top. Gap = 27 − 12 = 15, so 14 persons are between them. Formula: |a − b| − 1.

Case 2: ranks from opposite ends, and N is known. Now the ranks are L (A from the left) and R (B from the right). First convert B to the left side: B's position from the left is N + 1 − R. The gap is |L − (N + 1 − R)| and the persons between are one less. Everything collapses into one safe formula:

Persons between = |N + 1 − L − R| − 1.

[[fig:0]]

Look at what the number L + R tells you. If L + R is less than N, A and B have not crossed (top picture) and the answer is N − L − R. If L + R is more than N + 1, they have crossed (bottom picture) and the answer is L + R − N − 2. If L + R = N, they stand side by side and nobody is between. If L + R = N + 1, they are one and the same person.

The picture is the proof: in the top row the red part is what is left after cutting L from the left and R from the right. In the bottom row the cuts overlap, so the overlap must be removed. Always compare L + R with N before choosing the formula; blind use of N − L − R gives a negative number when the persons have crossed.

One more thing to watch: a rank can hide in a count. "Only 12 people are seated to the right of Sneha" makes Sneha 13th from the right (Type 1). Turn the count into a rank first, then compare L + R with N.

In the latest Railway papers this type comes in the "exactly between" form: "Johnny is 24th from the left. Anuj is 28th from the left and sits exactly between Johnny and Rishabh. Rishabh is 16th from the right. How many people are in the row?" Nobody is told the total, yet two small steps give it. We take the "exactly between" form first, then the older "k persons between" form with its maximum and minimum.

Part A: exactly between. "Exactly between" means equal distance on both sides. Johnny is at 24 and Anuj at 28, so Anuj is 4 places from Johnny. Rishabh must be 4 places beyond Anuj, at 28 + 4 = 32 from the left. Written as a concept, with all three ranks counted from the same end: third = 2 × middle − first.

[[fig:0]]

Now Rishabh has two ranks: 32 from the left and 16 from the right. They belong to one person and come from opposite ends, so N = 32 + 16 − 1 = 47 (see the picture). The method is always the same: (1) find the third person's rank by the mirror concept, (2) add his two ranks and drop 1. Take care that the first two ranks are from the same end; the concept cannot be used if one of them is from the left and the other from the right.

Part B: k persons between, maximum and minimum. Sometimes the question says only that exactly k persons stand between two people, one given from the left and the other from the right, and asks for the largest or the smallest row.

Now the total is unknown. A is given from the left, B is given from the right, and exactly k persons stand between them. The question does not say who is on which side, so two rows are possible.

Case 1: B is to the right of A. A is at position L from the left. Then k persons, then B. So B's position from the left is L + k + 1. B's rank from the right is R, so N = (L + k + 1) + R − 1, that is N = L + R + k. This is the longest row (top picture).

Case 2: B is to the left of A. Now B is k + 1 places before A, so B's position from the left is L − k − 1. Then N = (L − k − 1) + R − 1, that is N = L + R − k − 2. This is the shortest row (bottom picture), and the two rank-zones overlap.

[[fig:1]]

Maximum N = L + R + k. Minimum N = L + R − k − 2.

Notice that the difference between the maximum and the minimum is always 2k + 2, a fact that can answer a "difference" question without any calculation. A note of care: the minimum row is valid only if L − k − 1 is at least 1. If A is too close to the left end, the second case is impossible and only one row exists.

Read the question for words that decide the case. "To the right of" or "after" fixes Case 1. "Maximum" or "minimum" tells you which case to compute. If the question gives no side and no "maximum/minimum", it is incomplete, so there will be a fixed side hidden in the words; read slowly.

No numbers here, only comparisons. A typical question from the latest papers: "The lengths of 6 rectangles A, B, C, D, E and F are compared. No two are equal. A is the second longest. F is longer than C and B but shorter than D and E. E is not the longest. Which rectangle is the longest?" Every clue is an arrow, and the answer is read off the chain.

This type has no numbers, only comparisons: taller, heavier, older, richer, faster. Each clue is an arrow from the bigger to the smaller. Your job is to join the arrows into one long chain at the persons they share.

[[fig:0]]

Take: Radhika is heavier than Sumit. Tanya is lighter than Sumit. Umesh is heavier than Radhika. Join at the shared names: Umesh > Radhika > Sumit > Tanya. Now every question can be read off the chain. The heaviest is Umesh, the lightest is Tanya, and the number of persons heavier than Sumit is 2.

The quick way to an extreme. To find the longest, list the names that never stand on the smaller side of any clue ("… than X", "less than X"). In the rectangle question, A, F, C and B all stand on a smaller side somewhere (A is second, so someone is above it). Only D and E are left, and "E is not the longest" removes E. So D is longest. For the shortest, use the names that never stand on the bigger side. If two candidates remain and no clue separates them, the answer is "cannot be said".

Three concepts keep you safe.

• Merge only at a common person. A > B and C > B meet at B but say nothing about A and C (bottom picture). Two chains with no common person cannot be joined.
• Certain only if everyone is linked. A person is certainly the tallest only if every other person has an arrow leading away from him. If one person floats outside the chain, the extreme may belong to him.
• Not tallest is not shortest. "X is not the tallest" tells you only that somebody is above X.

Symbol-coded statements. The question may define symbols, for example P @ Q means P is taller than Q, and P $ Q means P is shorter than Q. Translate each statement into an arrow and flip the "shorter" statements so that all arrows point the same way. Then test each conclusion on the chain.

Minimum and maximum rank. Count the persons who are certainly above X and the persons who are certainly below X. With N persons in all, X's best (smallest) rank from the top is 1 + certainly-above; the worst is N − certainly-below. In the same spirit, "Tarun is taller than exactly 3 of 8 persons" means 4 persons are taller than Tarun.

Floors and stacks [New Pattern]. The same chain idea stands upright when persons live on different floors of a building.

Floor puzzles are ranking puzzles standing upright. Persons live on different floors of a building, and the clues use "above", "below", "immediately above" and "does not live on". Unless the question says otherwise, floor 1 is the lowest floor and the highest number is the top.

Each kind of clue builds something different.

• Above / below builds a chain: Anil above Bhavna is one arrow. There may be any number of floors in between.
• Immediately above builds a glued block: two persons on consecutive floors, one on top of the other. Treat the pair as one tall tile.
• Exactly k floors between fixes the gap at k + 1 floors.
• Fixed floor ("lives on the second floor") puts the person straight in a floor. Do this first.
• Does not live on crosses out one cell. It only helps when the other cells are almost full.

[[fig:1]]

The method is: (1) place every fixed floor; (2) join the chain and the glued blocks; (3) slide the block into the floors that are still free, remembering that persons must follow the chain order; (4) use "does not live on" to remove the last doubt. In the picture, C is on floor 5. D and E form a block. B is above E, so the block cannot go above 3; and A is on floor 1, so the block D–E sits on floors 2–3 and B is forced to 4.

Fix the floors, glue the neighbours, then slide the blocks into the free floors.

Two-statement form [New Pattern]. A question such as "Who is the tallest among five persons?" is followed by Statement I and Statement II, and you must decide whether the data is enough. Draw Statement I alone, then Statement II alone on a fresh drawing, and only then both together. A person is fixed as the tallest only when every other person lies below him on some arrow path; a name missing from a statement is unplaced. Choose the option in this order: I alone, II alone, both together, even both together not enough. Never jump to "both together" before testing each statement alone. The full sufficiency format is also treated in the Data Sufficiency chapter.

[[fig:2]]

Here is how the Railway papers phrase it: "A, B, C, D, X and Y each scored different marks. Only two people scored more than A. Only B scored less than D. X scored more than Y but less than C. How many people scored more than X?" The word "only" is a number in disguise, and the whole puzzle opens once you read it that way.

The word "only" in these questions is a number in disguise. It fixes a rank exactly, so such a clue should be turned into a rank before anything else. After that the loose clues ("more than", "less than") are used to place the remaining persons.

Here is how the four common sentences convert, counting from the top (highest marks first) in a group of N persons.

Do not mix the second sentence with the first. "Only two score more than X" puts X third from the top. "X scores more than only two" puts X third from the bottom, because two persons are below him. Same number, opposite end.

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In the picture, "only one person scores more than T" fixes T at rank 2. "Only U scores less than R" fixes R at rank 4 and U at rank 5. Now only two ranks, 1 and 3, are empty, and the clue S scores more than V fills them: S first, V third.

Turn every "only" sentence into a fixed rank first. Then place the rest, and stop when the asked rank is filled.

Two small conversions finish the job. "How many scored more than X" is X's rank from the top minus 1. "Who is the shortest (lightest, lowest)" is the last box. So you rarely need to arrange everyone; fill the boxes only as far as the question demands.

Many of these questions need only part of the arrangement. If the question asks "who ranks fourth", do not waste time deciding the order of the persons in ranks 1 to 3. Also keep in mind that an "only" clue is a promise of nothing more: if a clue says "only Gagan scores less than Hari", nobody else can lie below Hari.

Two forms of the same idea appear in the papers. Interchange: "In a row, Amit is 12th from the left and Sunil is 15th from the right. After they interchange their positions, Amit becomes 17th from the left. How many persons are in the row?" Shift: "In a row of 21 girls, when Rita was shifted by five places towards the right, she became 12th from the left. What was her earlier position from the right?" Both rest on one thought: the places of the line do not change; only the people move between them.

When two persons interchange (swap) their places, the places do not change at all. Only the people sitting on them change. So after the swap, A is sitting in B's old place and B is sitting in A's old place.

[[fig:0]]

This one thought solves the whole type. A's new rank (counted from some end) is exactly B's old rank from that same end. And B's new rank is A's old rank. Two things never change: the total number of persons N, and the number of persons between the two.

A question gives you one fact before the swap and one after, and usually from different ends. The usual path is:

• Step 1: Write A's old rank and A's new rank from the same end.
• Step 2: A's new rank is B's old rank from that end. Now B's rank is known from both ends, so N = (B's rank from this end) + (B's rank from the other end) − 1.
• Step 3: With N known, convert anything else using N + 1 − r.

After a swap, A's new place = B's old place. N and the gap stay the same.

A useful side result: the shift in A's position is the gap between the two. If A moves from 14th to 23rd from the left, he has moved 9 places, and so has B in the opposite direction. In the picture the two persons move 10 places in opposite directions. This often saves the whole calculation when the question asks "how many persons are between them?".

Be careful about the ends. A "rank from the front" for one person and a "rank from the back" for the other are not directly comparable. Convert first, then use the swap fact.

Shift of k places. Here one person walks along the line by k places while everybody else stays where they are (or the question only cares about this one person). Think of the line as a ruler and the person as a bead that slides k ticks.

[[fig:1]]

If the bead moves towards the left end by k places, its distance from the left end becomes smaller by k, and its distance from the right end becomes larger by k. The reverse happens if it moves towards the right end. The total number of persons N does not change.

Moving towards an end: that end's rank falls by k, the other end's rank rises by k.

The questions ask for the old rank, so you work backwards. If the person is now r-th from the left end after moving k places towards the left, he was r + k from the left before. If he had moved away from the left end, he was r − k.

The method has three steps. (1) Decide which end the person moved towards. (2) Undo the shift on the rank of that end: old rank = new rank + k when he moved towards it, new rank − k when away from it. (3) Convert to the other end with N + 1 − r if the question wants the other end.

When a person moves twice, such as "8 places to the right and then 11 places to the left", do not do two separate conversions. Take the net shift: 8 right and 11 left is 3 left. Always check that the new position still lies between 1 and N; if a rank comes out zero or bigger than N, you have shifted in the wrong direction.

In this type the clues are about different people, and the exam hides many forms in it: a third person placed by "places to the right of"; two persons at the extreme ends with a stranger between; lamps or students whose two ranks are both given; and short line puzzles of five or six persons. All of them are solved on one line.

In this type the ranks belong to different people: Mohit from the left, Neha from the right, and a third person Kapil placed relative to them. The formula N = L + R − 1 is useless here because L and R are not of the same person. We need a picture.

Draw a line, not one box for every person. The line is a ruler measured from the left, and every person is a point on it.

Convert every clue to a position from the left. Use the right-end clue last, to get N.

Take the question: Mohit is 12th from the left, Neha is 14th from the right, Kapil is 5 places to the right of Mohit and 3 places to the left of Neha. Mohit is at 12. Kapil is at 12 + 5 = 17. Neha is 3 places further, at 20. Now Neha's right rank is known, so N = 20 + 14 − 1 = 33.

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Every answer is now read straight from the line. Persons between Mohit and Neha = 20 − 12 − 1 = 7. Neha's rank from the right is 14, which agrees with 33 + 1 − 20. Always make this last check; it catches a wrong clue direction at once.

Persons at the two ends. Two persons are named at the extreme ends, and a third stands inside: "Janhvi and Pranitha are at the two extreme ends of a row. Only 26 students are between Janhvi and Ravi, and only 14 between Ravi and Pranitha. How many students are in the line?" The line breaks into pieces: Janhvi, 26 persons, Ravi, 14 persons, Pranitha. Count each piece once: N = 26 + 14 + 3 = 43, where the 3 are Janhvi, Ravi and Pranitha themselves. Persons between + the named persons = N. This is valid only when the two outer persons really stand at the ends.

Check the data first. Sometimes both ranks of two different things are given: "50 lamps in a row; a golden lamp is 18th from the left and 33rd from the right; a silver lamp is 25th from the left and 26th from the right. How many lamps are between them?" Test each lamp with L + R = N + 1. Here 18 + 33 = 51 and 25 + 26 = 51, and N + 1 = 51, so the data is consistent. The left positions 18 and 25 then give 25 − 18 − 1 = 6 lamps between them. The option "this arrangement is impossible" is right only when a lamp fails this test.

Short line puzzles. With five or six named persons and clues like "N is immediately to the right of D, who stands at an end", draw empty boxes in a row and fill them in this order: (1) end and fixed places, (2) glued pairs ("immediately"), (3) loose clues ("to the right of", "not at an end"), (4) the last person takes the last gap. A clue like "L is not at an end" is used to throw away arrangements, not to place anyone. In a line facing north, right means the east side, so "to the right of D" means a box after D when you read from the left. The same method works for lines arranged by height: the shortest stands in front, so "taller than only one student" means second from the front.

Two clue words deserve special care.

• Exactly in the middle (the mirror concept of Type 4 gives the position). If C is exactly in the middle of A and B, the distance from A to C equals the distance from C to B. So the number of persons between A and B is odd and the gap between them is even.
• Only C between A and B. This fixes the gap at 2 places, but not the side. If A is 8th from the left and B is 9th from the right with only C between, B can be to the right of A (N = 18) or to the left of A (N = 14). A real question fixes the side in its words ("B is to the right of A"); read for it, and if it is not fixed, test both sides.

This type is about counting heads in parts of the line, and the latest papers add a class with boys and girls: "84 students; the number of girls is three times the number of boys. A boy is 18th from the top and 13 girls are ahead of him. How many boys are behind him?" The same idea also gives the old forms: ahead and behind counts, and passed, failed and absent students.

Some questions never mention ranks. They say: "Vikas has 7 persons ahead of him, and the number behind him is twice the number ahead." Here ahead means the persons nearer the start of the line, and behind means the persons nearer the end.

If X is at rank t from the start, then ahead of him there are t − 1 persons and behind him there are N − t persons. Put X in the middle and the whole line is split in three parts:

N = (persons ahead) + 1 + (persons behind).

[[fig:0]]

That "+ 1" is X himself, the person everybody forgets. Also remember that "ahead" never includes X, so "7 persons ahead" means X is 8th from the front.

When the question gives a relation, such as "twice as many behind as ahead", turn it into numbers: ahead = 7, behind = 2 × 7 = 14, N = 22. When two relations are given, name the unknowns: let ahead = a and behind = b, write the two relations as equations, and solve. For a person who moves from the front group to the back group, subtract 1 from "ahead" and add 1 to "behind"; the total remains the same.

Take care with the direction words. The queue may face north, or the line may face the teacher; if the question defines "ahead" by a direction, make sure which end is the front. In most of these questions, "ahead" is the same end as "front" or "first". Do not mix "ahead of" with "taller than": the former is about position, the latter is about comparison (Type 5).

In real classes not everyone is in the same list. Some pass, some fail, some stay absent, and boys and girls may be ranked separately. A rank is always counted inside one group, and the group is named in the question: "among those who passed", "among the girls".

So the first job is to find the group the ranks belong to. If Pooja is 7th from the top and 22nd from the bottom among the students who passed, then the group of passed students has 7 + 22 − 1 = 28 persons. The failed and the absent students are not inside this ranking at all.

[[fig:1]]

Only after that do you add the other groups, and the words of the question tell you which ones to add.

• Passed = the group in which the ranks are counted.
• Appeared (took the examination) = passed + failed. Absent students did not take it.
• Registered / total candidates / strength of the class = passed + failed + absent.

Appeared = passed + failed. Registered = passed + failed + absent.

For a gender split, the same counting holds: persons ahead of a girl = boys ahead + girls ahead. If a girl is 5th among the girls and 12th among all the students, 11 persons are ahead of her overall, 4 of them girls, so 7 are boys. When the question gives a relation between groups ("boys passed are 5 more than girls passed"), turn it into numbers group by group and add at the end.

Splitting by group. Work group by group. Boys = 84 ÷ 4 = 21 and girls = 63. Ahead of the boy are 17 persons; 13 of them are girls, so 4 are boys. Boys behind him = 21 − 4 − 1 = 16, where the 1 is the boy himself. Behind in a group = group total − ahead in that group − 1 (himself, if he belongs to that group). If the person is a girl, the 1 is taken from the girls' count, not from the boys'.

Finally, watch the word "total". Our book reads "total students in the class" as the whole class, absent students included, and "students who took the exam" as passed plus failed. If the question is unclear, read the exact words once more before you add the absentees.