Order & Ranking

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

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Ranks and positions

⏱ 4 min read🧩 5 question types🎯 16 practice Q

The idea in one minute

These questions are about one person's rank in a class, a queue or a row. Use the two shortcuts: total minus rank plus 1, and rank plus rank minus 1. When people join or leave the row, count the people before him first. Then add 1.

The other end, again

The rank from the other end of a class or a row is a simple sum.

\text{rank from the bottom} = \text{class size} - \text{rank from the top} + 1

Example: in a class of 60, Neha is 23rd from the top. From the bottom she is $60 - 23 + 1 = 38$ th.

The total from two ranks

If a person's two ranks are given, the total is their sum minus 1.

Example: Sonu is 14th from the left and 20th from the right. The row has $14 + 20 - 1 = 33$ people.

Three quick facts to remember:

12th from the top in a class of 45 is 34th from the bottom.
8th from the left and 21st from the right gives $8 + 21 - 1 = 28$ people.
11th from either end gives $11 + 11 - 1 = 21$ people.

Tip: The total is always bigger than each of the two ranks. If not, you misread.

People above and below

The rank counts the person too. The number of people below him leaves him out.

People below him = class size $-$ his rank from the top.
People above him = his rank from the top $- 1$ .

Example: Reena is 12th from the top in a class of 40. So $40 - 12 = 28$ students are below her.

People join or leave the front

Count the people before him first. Then add 1.

$k$ people join ahead of him: his rank becomes old rank $+ k$ .
$k$ people who were ahead of him leave: his rank becomes old rank $- k$ .
People who join or leave behind him do not change his rank from the front.

Example: Ishaan is 18th in a queue. 6 people join in front of him. Then 3 people who were ahead of him leave. His new rank is $18 + 6 - 3 = 21$ st.

People leave from the middle of a row

Only the people to the right of the leavers move. They slide left by the number who left.

Example: A is 16th from the left. 5 people leave from the left end. A slides to $16 - 5 = 11$ th.

Watch: Look at where the leavers stood. People on the same side as the person change his rank. People on the other side change only the total.

Rank from the other end after people leave

When people to his left leave, his rank from the right does not change. Only the total falls.

Example: in a row of 45, Gopal is 20th from the left. His rank from the right is $45 + 1 - 20 = 26$ . Eight people to his left leave. He is now 12th from the left in a row of 37. His rank from the right is $37 + 1 - 12 = 26$ . It did not change.

Class ranks with a twist

If some students fail, remove them from the total before you use the formula. The formula stays the same. Only the total changes.

Example: a class of 50 has 4 failed students, and Anil is 10th from the top among the rest. Now the total is $50 - 4 = 46$ . His rank from the bottom is $46 + 1 - 10 = 37$ .

Question types you will see

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

Type 1very common2 practice Q

Rank from the other end

How to spot it:

The size of the class or row and a rank from the top or left are given. The rank from the other end is asked.

p' = n + 1 - p

Method

Write the class size and the rank.
Subtract the rank from the size.
Add 1.

Why it works:

The person is counted from both ends, so add back the shared seat.

Try this

In a class of 75, Varun ranks 28th from the top. What is his rank from the bottom?

Show solution

$75 - 28 + 1$ .
$= 48$ .

Answer

48th from the bottom

Type 2very common2 practice Q

Total from two ranks

How to spot it:

One person's rank from both ends is given. The total is asked.

n = a + b - 1

Method

Add the two ranks.
Subtract 1.
Check that the total is bigger than each rank.

Why it works:

The person is counted twice, once from each end.

Try this

Divya is 16th from the left and 24th from the right in a row. How many people are in the row?

Show solution

$16 + 24 - 1$ .
$= 39$ .

Answer

Type 3common2 practice Q

People join or leave ahead

How to spot it:

People join or leave a queue. The person's new position is asked.

r_{\text{new}} = r_{\text{old}} + \text{joiners} - \text{leavers}

Method

Count only the people who joined or left ahead of him.
Add joiners and subtract leavers from his rank.
Ignore people behind him.

Why it works:

Only the people before him change his rank.

Try this

Ishaan is 18th in a queue. 6 people join in front of him. Then 3 people who were ahead of him leave. What is his new position?

Show solution

After the joiners: $18 + 6 = 24$ .
After the leavers: $24 - 3 = 21$ .

Answer

21st

Type 4common2 practice Q

Rank after people leave the row

How to spot it:

Some people stand up and leave. A remaining person's new rank is asked.

Method

Note whether the leavers stood on his left or right.
Recount his rank from the side where they left. Subtract the leavers.
Reduce the total by the number who left. Then find the rank asked.

Why it works:

Only leavers on his own side change his rank. All leavers change the total.

Try this

In a row of 45, Gopal is 20th from the left. 8 people to his left leave the row. What is Gopal's position from the right now?

Show solution

Row now: $45 - 8 = 37$ people. Gopal: $20 - 8 = 12$ th from the left.
From the right: $37 + 1 - 12 = 26$ .

Answer

26th from the right

Type 5occasional

The same rank from both ends

How to spot it:

The person is said to be the same rank from both ends. The total is asked.

n = 2r - 1

Method

Use $a = b = r$ .
Total = $r + r - 1$ .
He is the middle person.

Why it works:

The person is in the exact middle, with the same number of people on each side.

Try this

Deepak is 14th from either end of a line. How many people are in the line?

Show solution

$14 + 14 - 1$ .
$= 27$ .

Answer

Formula sheet

Rank from the other end

p' = n + 1 - p

n = class size or row size.

Total from two ranks

n = a + b - 1

The same person, ranked from both ends.

People joining or leaving ahead

r_{\text{new}} = r_{\text{old}} \pm k

Plus for k joiners ahead. Minus for k leavers who were ahead.

Shortcuts that save time

⚡ Check that the sum of the two ranks is total plus 1

For one person, the two ranks add up to the total plus 1. Test it before you mark the answer.

Example

Sonu is 14th from the left and 20th from the right. How many are in the row?

Show solution

$14 + 20 - 1 = 33$ .
Check: $14 + 20 = 34 = 33 + 1$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using $n = a + b$ with no minus 1.

The person is counted twice, so use $n = a + b - 1$ .

Mistake 02

Reading '11th from either end' as two different people.

It is one person in the exact middle. Total $= 11 + 11 - 1 = 21$ .

Mistake 03

Changing his rank when people join or leave behind him.

Only people ahead of him change his rank from the front.

Mistake 04

Calling 'people below him' the same as his rank from the bottom.

People below = his rank from the bottom $- 1$ .

Mistake 05

Forgetting to reduce the total when people leave.

Every leaver reduces the total by 1, even if his rank does not change.

Quick revision

Read this the night before the exam.

Other end: size $-$ rank $+ 1$ .
Two ranks: total $= a + b - 1$ .
Joiners ahead: rank $+ k$ . Leavers ahead: rank $- k$ .
People behind him never change his rank from the front.
Below him = rank from the bottom $- 1$ .
If some students fail, remove them from the total first.

Practice: 16 questions

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

Topic test · 10 questions

Suggested time 5 min · wrong answers go to your mistake notebook automatically.

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