Order & Ranking

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

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People between two positions

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

The idea in one minute

These questions ask how many people stand between two people. First find out which end each rank comes from. If both ranks are from the same end, subtract and take away 1. If they are from opposite ends, subtract both from the total.

The same end

"Between" leaves out both people. So subtract the two ranks and drop 1.

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

Example: R is 9th from the top and S is 26th from the top. Between them: $26 - 9 - 1 = 16$ students.

Opposite ends

Suppose P is $a$ -th from the left and Q is $b$ -th from the right. Take both ranks off the total. What is left is the people in between.

\text{between} = \text{total} - a - b

Example: a row has 45 people. X is 14th from the left and Y is 18th from the right. Between them: $45 - 14 - 18 = 13$ .

Tip: No total given? Find it from any person who has ranks from both ends: $a + b - 1$ .

The middle person

Take two ranks from the same end. The person exactly in the middle has this rank from that end:

\dfrac{a + b}{2}

Example: 9th and 21st from the left. The middle person is $\dfrac{9 + 21}{2} = 15$ th from the left. There are 5 people on each side of him, between him and each of the two.

If $a + b$ is odd, there is no single middle person. Two people stand in the middle.

The overlap puzzle

A swap can reveal a rank that the row never stated. Build the total first. Then count between.

Example: Ravi is 15th from the left. Sohan is 11th from the right. After they swap, Ravi is 20th from the left.

Ravi took Sohan's seat. So Sohan's seat is 20th from the left and 11th from the right.
Total $= 20 + 11 - 1 = 30$ .
Originally, Ravi was 15th and Sohan's seat was 20th. Between them: $20 - 15 - 1 = 4$ .

Watch: The rank after the swap belongs to the other person's seat. It is not his old seat.

Given the gap, find the rank

If $k$ people stand between two people and they are counted from the same end, the second rank is:

b = a + k + 1

Example: Tina is 11th from the top. 7 students are between Tina and Uma, and Uma is below her. Uma is $11 + 7 + 1 = 19$ th from the top.

People outside the pair

"How many people are not between them?" Find between first. Then remove the two people themselves.

\text{outside} = \text{total} - \text{between} - 2

Example: total 52, between 16. Outside: $52 - 16 - 2 = 34$ .

Two ways to check

Check every answer once by drawing the row.

Example: the row has 45 people. X is 14th from the left. Y is 18th from the right, so Y is $45 + 1 - 18 = 28$ th from the left. Now use the same-end formula: $28 - 14 - 1 = 13$ . The direct formula $45 - 14 - 18 = 13$ agrees.

Two routes, one answer. If they differ, one rank was read from the wrong end.

Note: Between can never be negative, and between plus 2 can never be more than the total. If it is, you misread an end.

Question types you will see

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

Type 1very common3 practice Q

Between, same end

How to spot it:

Both ranks are counted from the same end (left, top or front). The people between them are asked.

|a - b| - 1

Method

Check that both ranks are from the same end.
Subtract the smaller from the larger.
Subtract 1.

Why it works:

The two people themselves are not between, so one seat is dropped.

Try this

P is 7th from the left and Q is 31st from the left in a row. How many people are between them?

Show solution

$31 - 7 = 24$ .
$24 - 1 = 23$ .

Answer

Type 2very common2 practice Q

Between, opposite ends

How to spot it:

One rank is from the left or top. The other is from the right or bottom. The total is given.

n - a - b

Method

Write the total, then both ranks.
Subtract both ranks from the total.
No total? Build it from a person with both ranks.

Why it works:

Taking both ranks off the total leaves exactly the people in the middle.

Try this

In a row of 52, X is 15th from the left and Y is 21st from the right. How many people are between X and Y?

Show solution

$52 - 15 - 21$ .
$= 16$ .

Answer

Type 3common2 practice Q

The middle person

How to spot it:

The question asks who or which seat is exactly in the middle of two people.

\dfrac{a + b}{2}

Method

Get both ranks from the same end.
Add them and halve the sum.
Check the answer is a whole number.

Why it works:

The middle is the average of the two seat numbers.

Try this

A is 12th from the left and B is 30th from the left. Which seat is exactly in the middle of them?

Show solution

$\dfrac{12 + 30}{2} = 21$ .
There are 8 people between A and the middle person.

Answer

21st from the left

Type 4occasional2 practice Q

Swap and count between

How to spot it:

A swap gives a new rank. The number of people between the two before the swap is asked.

Method

The person who moved took the other's seat. That seat has the new rank.
Find the total: new rank $+$ other's rank from the other end $- 1$ .
Count between with the same-end formula.

Why it works:

The swap reveals one rank the row never stated. Once the total is known, it is a plain count.

Try this

Ravi is 15th from the left and Sohan is 11th from the right. After they exchange places, Ravi is 20th from the left. How many people were between them before?

Show solution

Sohan's seat is 20th from the left and 11th from the right. Total $= 20 + 11 - 1 = 30$ .
Between: $20 - 15 - 1 = 4$ .

Answer

Type 5occasional

People outside the pair

How to spot it:

The question asks how many people are not between two people, or ahead of and behind them.

n - \text{between} - 2

Method

Find the people between first.
Subtract that from the total.
Subtract 2 for the two people themselves.

Why it works:

Total = between + the two people + everyone outside.

Try this

In a row of 52, X is 15th from the left and Y is 21st from the right. How many people are outside the pair (not between them and not X or Y)?

Show solution

Between $= 52 - 15 - 21 = 16$ .
Outside $= 52 - 16 - 2 = 34$ .

Answer

Type 6occasional

Find the rank from the gap

How to spot it:

The number of people between two people is given. One person's rank is known and the other's is asked.

b = a + k + 1

Method

Note which person is nearer to the start.
Add the gap $k$ and 1 to the nearer rank.
Subtract instead if the second person is closer to the start.

Why it works:

The second person sits after the $k$ people between and one more seat.

Try this

Tina is 11th from the top. 7 students are between Tina and Uma, and Uma is below Tina. What is Uma's rank from the top?

Show solution

$11 + 7 + 1$ .
$= 19$ .

Answer

19th from the top

Formula sheet

Between, same end

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

Both ranks are counted from the same end.

Between, opposite ends

\text{between} = n - a - b

n = total people, a and b = ranks from opposite ends.

Middle person

m = \dfrac{a + b}{2}

a and b from the same end. The result must be a whole number.

Rank from the gap

b = a + k + 1

k people are between a and b, counted from the same end.

Shortcuts that save time

⚡ Convert first, then count

Never subtract ranks from different ends. Convert one rank with $n + 1 - p$ . Then use the same-end formula.

Example

In a row of 40, A is 5th from the left and B is 10th from the right. How many people are between them?

Show solution

B from the left: $40 + 1 - 10 = 31$ .
Between: $31 - 5 - 1 = 25$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

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

Convert one rank to the other end first, or use $n - a - b$ .

Mistake 02

Forgetting the last -1 and counting the two people as well.

'Between' leaves out both people.

Mistake 03

Converting both ranks with $n + 1 - p$ .

Convert only one. Both ranks must end up on the same side.

Mistake 04

Using the person's old seat after a swap.

After a swap, the person sits in the other's seat.

Mistake 05

Continuing when the answer is negative.

A negative count means a misread end. Stop and convert again.

Quick revision

Read this the night before the exam.

Same end: $|a - b| - 1$ .
Opposite ends: total $- a - b$ .
Middle person: $\dfrac{a + b}{2}$ from the same end.
Swap puzzle: find the total first, then count between.
Given a gap $k$ : $b = a + k + 1$ .
Outside the pair = total $-$ between $- 2$ .

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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