Ranking Exit
Logical reasoning

Ranking

Position from left + position from right = total + 1. Everything in this topic is that one line.

Start with this

Ravi is 7th from the left and 12th from the right in a row. How many people are there? 7 + 12 - 1 = 18: Ravi was counted twice.

Ranking questions all come from one identity: position from left + position from right = total + 1.

Left + right - 1 = total

A person counted from both ends is counted twice, so subtract 1.

Between two people: if A is p-th from the left and B is q-th from the left, there are |p - q| - 1 people between them.

123456Ravi789101112131415161718leftright
7th from the left and 12th from the right: 18 people

Left + right - 1 = total: the rules

Total = left rank + right rank - 1
Right rank = total - left rank + 1

Worked example

In a class of 40, Priya is 15th from the top. What is her rank from the bottom?

  1. Bottom rank = 40 - 15 + 1.
    Why: Rearranged identity.
  2. = 26.

Answer: 26

Worked example

In a row, A is 10th from the left and B is 15th from the right. If they swap places, A becomes 18th from the left. How many people are in the row?

  1. After the swap A is where B was: 18th from the left and 15th from the right.
    Why: The swap tells us B's left position.
  2. 18 + 15 - 1 = 32.

Answer: 32

Shortcut: Total from both ends

Use it when: '15th from the top and 20th from the bottom'.

  1. 15 + 20 - 1 = 34.

Total from both ends: try it

Total students?

  1. 34.

Answer: 34

Your turn medium

A is 14th from the left, B is 17th from the right, and there are 5 people between them (A to the left of B). Total?

  1. 14 + 5 + 17 = 36.

Answer: 36

Your turn medium

A and B swap places. A was 8th from the left and moves to 15th from the left; B was 12th from the right. B's new rank from the right?

  1. A moved 7 places right, so B moved 7 places left.
  2. 12 + 7 = 19.

Answer: 19

Your turn hard

Rita is 12th from the left, Mona 4th from the right. They swap, and Rita is now 15th from the left. How many are in the row?

  1. Rita now sits where Mona was: 15th from left and 4th from right.
  2. 15 + 4 - 1.

Answer: 18

Recap

Now practise

6 questions with full solutions.

Open practice Back to the lesson