Order & Ranking

medium importance~1-2 Q in Tier 111 formulas⚡ 6 shortcuts4 subtopics

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How ranking questions work

⏱ 5 min read🧩 6 question types🎯 12 practice Q

The idea in one minute

In a row or a queue, a person's rank tells how many people stand before them, plus one. Every ranking question comes from this one sentence. Learn two shortcuts: the rank from the other end, and the total from two ranks. Draw a small row of boxes whenever three or more people are involved.

The one idea

A rank counts the person too. If 4 people stand before you, your rank is 5.

Rule: Rank = number of people before you + 1. Every formula in this topic comes from this line.

Example: Sita is 5th in a queue of 12. Four people are ahead of her ( $5 - 1 = 4$ ). Seven are behind her ( $12 - 5 = 7$ ).

The other end

The same person has a different rank from the other end. Find it in one step.

\text{rank from the other end} = \text{total} - \text{rank} + 1

Example: in a row of 40, Ravi is 10th from the left. From the right he is $40 - 10 + 1 = 31$ st.

Why add 1? Ravi is counted once from each side. Take him out and count the people to his right: $40 - 10 = 30$ . Then add Ravi himself.

Total from two ranks

Suppose one person is $a$ -th from one end and $b$ -th from the other end. Both counts include that person. So he is counted twice.

\text{total} = a + b - 1

Example: 9th from the left and 11th from the right gives $9 + 11 - 1 = 19$ people.

Tip: The two ranks always add up to the total plus 1. Use this to check yourself: $9 + 11 = 20 = 19 + 1$ .

Ahead and behind

"Ahead" does not include the person. So it is one less than his rank.

Ahead of him = rank $- 1$ .
Behind him = total $-$ rank.
Ahead + behind + 1 = total.

Example: 14 people are ahead of a girl and 21 are behind her. The total is $14 + 21 + 1 = 36$ .

Two people swap places

When two people exchange seats, each person takes the other's seat. So a new rank is just the old rank of the other seat.

Example: a row has 25 people. Ram is 8th from the left. Shyam is 9th from the right, which is $25 + 1 - 9 = 17$ th from the left. After the swap, Ram is 17th from the left.

Watch: Convert both ranks to the same end before you compare seats.

Stacks and rows of objects

Books, boxes and shelves work the same way. "Above" is like "to the left". Write the order from top to bottom as each sentence comes. Then read the seat that is asked.

A full walk-through

Read this one slowly. A queue has 52 people. Kavya is 17th from the front.

People ahead of her: $17 - 1 = 16$ .
People behind her: $52 - 17 = 35$ .
Check: $16 + 35 + 1 = 52$ . The total matches.
Her rank from the back: $35 + 1 = 36$ . The formula gives the same: $52 + 1 - 17 = 36$ .

Example: Two ways gave 36. When both routes agree, you can trust the answer. Use this check in the exam whenever time allows.

Facing South

A row facing North has its usual left and right. A row facing South has them swapped. Their left is your right. Underline the facing word before you use any left or right clue.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common4 practice Q

Rank from the other end

How to spot it:

The total and one rank are given. The rank from the other end is asked.

p' = n + 1 - p

Method

Write the total and the rank you know.
Subtract the rank from the total.
Add 1.

Why it works:

The person is counted from both ends, so one seat is counted twice and you add it back.

Try this

In a queue of 52 people, Kavya is 17th from the front. What is her position from the back?

Show solution

Position from the back = $52 - 17 + 1$ .
$= 36$ .

Answer

36th from the back

Type 2very common3 practice Q

Total from two ranks

How to spot it:

One person's rank is given from both ends. The number of people is asked.

n = a + b - 1

Method

Add the two ranks.
Subtract 1 for the person counted twice.
Check: the total must be bigger than each rank.

Why it works:

The person is counted in both ranks, so one count must be removed.

Try this

Manoj is 19th from the left and 26th from the right in a row. How many people are in the row?

Show solution

Total = $19 + 26 - 1$ .
$= 44$ .

Answer

Type 3common2 practice Q

Two people swap places

How to spot it:

Two people exchange seats. A rank after the swap is asked.

Method

Convert the second person's rank to the same end as the question.
That seat number is the first person's new rank.
Read the answer from that seat.

Why it works:

Each person takes over the other's seat, so a new rank is an old seat number.

Try this

In a row of 30, Asha is 12th from the left and Bina is 10th from the right. They exchange places. What is Asha's new position from the left?

Show solution

Bina's seat from the left = $30 + 1 - 10 = 21$ .
Asha moves to that seat.

Answer

21st from the left

Type 4common3 practice Q

Building a stack or row from clues

How to spot it:

Boxes, books or people are described by 'above', 'below', 'left' or 'right'. A seat or an end is asked.

Method

Draw a column or a row of empty boxes.
Fill in the sure fact first, such as 'at the bottom'.
Place each clue as it comes. Then read the asked seat.

Why it works:

Each clue fixes one link. The links together make one ordered line.

Try this

Five boxes are in a stack. P is above Q. R is below Q but above S. T is at the bottom. Which box is in the middle?

Show solution

T is at the bottom.
Order from the top: P, Q, R, S, T.
The middle (third) box is R.

Answer

Type 5common

People ahead and behind

How to spot it:

The question gives the people ahead of someone and the people behind, and asks the total or the rank.

n = \text{ahead} + \text{behind} + 1

Method

Write the number ahead and the number behind.
Add the person himself: +1.
His rank from the front is ahead + 1.

Why it works:

Ahead and behind leave out the person, so he must be added back.

Try this

In a queue, 14 people stand ahead of Sneha and 21 stand behind her. How many people are in the queue?

Show solution

Total = $14 + 21 + 1$ .
$= 36$ .

Answer

Type 6occasional

Exactly in the middle

How to spot it:

The person is exactly in the middle, or there are as many people ahead as behind.

n = 2r - 1

Method

Write ahead = behind, so $r - 1 = n - r$ .
Solve for the total: $n = 2r - 1$ .
The middle person is in seat $\dfrac{n+1}{2}$ .

Why it works:

The middle person has the same number of people on each side.

Try this

Meena is exactly in the middle of a row. She is 13th from the left. How many people are in the row?

Show solution

Ahead of her: $13 - 1 = 12$ .
Behind her: also 12.
Total = $12 + 12 + 1 = 25$ .

Answer

Formula sheet

Rank from the other end

p' = n + 1 - p

n = total people, p = rank from one end.

Total from two ranks

n = a + b - 1

The same person is a-th from one end and b-th from the other end.

Between count (same end)

\text{between} = |a - b| - 1

Both ranks are from the same end.

Ahead and behind

\text{ahead} = r - 1,\quad \text{behind} = n - r

r = the person's rank from the front.

Shortcuts that save time

⚡ Check that the two ranks add to total plus 1

For one person, the two ranks from opposite ends always add up to the total plus 1. If they do not, you have misread a number.

Example

A boy is 5th from the left and 8th from the right. How many people are in the row?

Show solution

$5 + 8 - 1 = 12$ .
Check: $5 + 8 = 13 = 12 + 1$ .

Answer

⚡ Draw the end zones

Draw a short row of boxes. 'a-th from the left' means $a - 1$ people stand before the person. Counting people at each end is often faster than any formula.

Example

In a class of 45, Priya is 12th from the top. What is her rank from the bottom?

Show solution

People below her: $45 - 12 = 33$ .
Her rank from the bottom: $33 + 1 = 34$ .

Answer

34th

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Forgetting the +1 or the -1.

The person counts from both ends. Use $n + 1 - p$ and $a + b - 1$ .

Mistake 02

Subtracting a rank from the left directly from a rank from the right.

Convert both to the same end first.

Mistake 03

Calling 'ahead of him' the same as his rank.

Ahead = rank $- 1$ . Rank = ahead $+ 1$ .

Mistake 04

Guessing one more or one less than an option.

Work it out. Wrong options are set exactly one away.

Mistake 05

Using left and right the usual way for a row facing South.

Underline the facing word. South-facing left is your right.

Quick revision

Read this the night before the exam.

Rank = people before you + 1.
Other end: total $-$ rank $+ 1$ .
Total from two ranks: $a + b - 1$ .
Ahead = rank $- 1$ . Behind = total $-$ rank.
Swap: each person takes the other's seat.
A South-facing row has left and right swapped.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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