Ranking Puzzles

When someone's rank from one end is known:

Total = Rank from top + Rank from bottom − 1

Rearranging:

• Rank from bottom = Total − Rank from top + 1
• Rank from top = Total − Rank from bottom + 1
• Total = Rank from top + Rank from bottom − 1

Example:

In a class of 40, Riya is 15th from top.

Rank from bottom = 40 − 15 + 1 = 26

Example 1 (find total):

A is 8th from the left and 5th from the right in a row. How many students are in the row? Total = 8 + 5 − 1 = 12

Example 2 (find rank):

In a row of 25 students, B is 10th from the right. Find his position from the left. Position from left = 25 − 10 + 1 = 16

Example 3 (inter-rank):

A is 12th from the left, B is 15th from the right. They interchange places — now A is 18th from the left. Find total students.

After interchange: A is now where B was (15th from right → now 18th from left) Total = 18 + 15 − 1 = 32

Check: Originally A was 12th from left, B was 15th from right = 32−15+1 = 18th from left. ✓

Example 4 (no overlap check):

In a row, P is 7th from the left and Q is 9th from the right. Together they account for 7 + 9 = 16 positions. If total > 16, they are different people occupying different spots.

If total = 14: 7 + 9 − 1 = 15 > 14 → they overlap → only 14 − 7 = 7 people between and including P and Q from both ends, some positions counted twice.

Rank in a row: physical position (seat numbers, queue positions) Rank by marks: performance ranking

For marks-based ranking:

If Rahul scored 85 and there are 3 students above him, his rank = 4. If there are 5 below him and total = 30: Rank from bottom = 30 − 4 + 1 = 27.

Type 1: Single row/column — find total or a rank

(Covered above — use Total = top rank + bottom rank − 1)

Type 2: Two-person position swap

Given positions of two people, they swap — find original/new positions or total.

Type 3: Minimum/Maximum total

"A is 7th from left, B is 8th from right. They are different persons. What is the MINIMUM number of students?" Min total = 7 + 8 = 15 (they are in adjacent seats, no overlap)

Type 4: Marks ranking (tie adjustment)

3 students scored the same → each gets rank 5. Next rank is 8 (not 6).

1. In a row of 30 students, Ram is 12th from the right. What is his position from the left?
2. Priya is 8th from the top in a class. There are 11 students below her. How many students are in the class?
3. In a queue, Sam interchanges with Tom. Sam was 5th from the front and is now 8th from the front. Tom was 12th from the rear. Find the total.
4. A is ranked 4th from the top and 6th from the bottom in a test. How many students appeared in the test?
5. Stretch: In a row of 20, A is 8th from left. B is 5 places to the right of A. What is B's position from the right?