Order and ranking for LIC AAO

Order and ranking questions look like free marks, and most of the time they are. The trouble is that they are easy to get wrong by exactly one. Add where you should subtract, forget to count the person themselves, and you land on an option the setter placed there for you. The fix is to know where the "+ 1" comes from, so you never have to remember it.

Where the formula comes from

Put a person in a row. Count from the left and you count everyone up to and including them. Count from the right and you do the same. Add the two counts and you have counted every person once, except the person themselves, who has been counted twice.

Position from left + position from right = total + 1.

Everything else follows from rearranging it.

Ranks in a class work the same way. "From the top" plays the part of "from the left", and "from the bottom" plays the part of "from the right".

People between two persons

Suppose A is p-th from the left and B is q-th from the right in a row of N.

• If they do not overlap (A is to the left of B), the people between them = N − (p + q).
• If they overlap (A is to the right of B, so p + q > N + 1), the people between them = p + q − N − 2.

You do not need to memorise the second formula. The safer method is to convert both positions to the same end, then subtract and take away one. If A is 11th from the left and B is 26th from the left, the people between them are 26 − 11 − 1 = 14.

Comparison chains

"P is taller than Q, R is shorter than S…" questions are ranking without numbers. Write every clue with the same sign, always >, and join the pieces into one chain. If a person cannot be placed relative to everyone else, the chain splits into branches. Any question that depends on the unplaced link has no definite answer.

Worked examples

Example 1. In a row of 45 people, A is 17th from the left. What is A's position from the right?

• 45 − 17 + 1 = 29th.

Example 2. In a class, B is 13th from the top and 22nd from the bottom. How many students are in the class?

• 13 + 22 − 1 = 34.

Example 3. In a row of 40, P is 11th from the left and Q is 15th from the right. How many people sit between them?

• Q from the left = 40 − 15 + 1 = 26th.
• Between positions 11 and 26: 26 − 11 − 1 = 14. The formula agrees: 40 − (11 + 15) = 14.

Example 4 (minimum number). R is 18th from the left and S is 16th from the right. Exactly four people sit between them. What is the minimum number of people in the row?

• If R is to the left of S, the row holds 18 + 4 + 16 = 38 people.
• If S is to the left of R, S sits five places to the left of R: 18 − 5 = 13th from the left. The row then holds 13 + 16 − 1 = 28 people.
• Check: S at 13 and R at 18 leave positions 14 to 17 between them, which is four people. Minimum 28.

Example 5 (interchange). In a row, C is 9th from the left and D is 14th from the right. They swap places, and C is now 16th from the left. How many people are in the row?

• After the swap, C sits in D's old seat. That seat is 16th from the left and 14th from the right.
• Total = 16 + 14 − 1 = 29.
• Check: D's old seat was 29 − 14 + 1 = 16th from the left, which matches.

Example 6 (passed and failed). A student ranks 12th from the top and 28th from the bottom among those who passed. Six students failed and four were absent. How many students are in the class?

• Students who passed = 12 + 28 − 1 = 39.
• Class = 39 + 6 + 4 = 49.

Example 7 (chain). P is taller than Q. R is shorter than S. Q is taller than S. T is shorter than R. Who is the tallest, and who stands in the middle?

• Rewrite: P > Q, S > R, Q > S, R > T.
• Join: P > Q > S > R > T.
• P is the tallest. With five people, the middle is third: S.

Example 8 (chain with a gap). M is heavier than N but lighter than O. P is heavier than O. Q is lighter than O. Who is the heaviest, and who is the lightest?

• Rewrite: M > N, O > M, P > O, O > Q.
• Chain: P > O > M > N, with Q somewhere below O.
• P is the heaviest. The lightest is N or Q, and nothing compares them, so the lightest cannot be determined.

Common mistakes

• Forgetting the − 1 when finding the total. Left + right counts the person twice.
• Mixing ends. Converting one position to "from the left" and forgetting to convert the other.
• Missing the overlap case in "minimum number" questions.
• Adding failed or absent students twice. Ranks among passed students only count passed students. Add the others once, at the end.
• Filling a gap in a chain with a guess. If nothing links Q to N, the answer is "cannot be determined".

Practice set

1. In a row of 50, E is 21st from the right. What is E's position from the left?
2. F is 8th from both ends of a row. How many people are in the row?
3. G scored more than H but less than J. K scored more than J. Who scored the least?
4. In a row of 35, X is 10th from the left and Y is 12th from the right. How many people sit between them?
5. In a class of 42, Z ranks 7th from the top. What is Z's rank from the bottom?
6. In a row, A is 10th from the left and B is 7th from the right. They swap places, and A is now 18th from the left. How many people are in the row?
7. M is 14th from the left and N is 9th from the right. Exactly three people sit between them. What is the minimum number of people in the row?
8. Among five friends, U scored more than V and W scored less than V. X scored more than U, and Y scored less than U but more than V. Who scored the third highest?

Answers

1. 30th. 50 − 21 + 1 = 30.
2. 15. 8 + 8 − 1 = 15.
3. H. The chain is K > J > G > H.
4. 13. Y from the left = 35 − 12 + 1 = 24th. Between 10 and 24: 24 − 10 − 1 = 13.
5. 36th. 42 − 7 + 1 = 36.
6. 24. A now sits in B's old seat: 18th from the left and 7th from the right. 18 + 7 − 1 = 24.
7. 18. Without overlap the row would hold 14 + 3 + 9 = 26. With N to the left of M, N is 14 − 4 = 10th from the left, and the total is 10 + 9 − 1 = 18. Check: positions 11, 12 and 13 lie between them.
8. Y. The chain is X > U > Y > V > W.

What to do next

• Do ten ranking questions with no formula sheet. For each, write the "counted twice" reason in one line before you calculate.
• Practise the same counting in alphanumeric series, where positions from the right cause the same errors.
• Move on to puzzles, where comparison chains become grids.
• Add ranking to your week 2 topics in the reasoning plan.