Introduction
Questions on rank interchange ask how ranks change when two people swap places in a line. This pattern is common in seating and ranking problems and tests your ability to convert positions between ends and update ranks quickly.
Learning the conversion and swap method prevents mistakes when ranks are given from different ends or when totals are involved.
Pattern: Rank after Interchange
Pattern
Key concept: Convert ranks to the same reference (people ahead or position from one end), then swap the positions and recompute ranks from the required end.
Useful formulas:
- Position from right = Total - Position from left + 1.
- Total = Left position + Right position - 1 (when both positions of the same person are given).
- After swap, each person's new position = the other's old position (convert as needed to express in requested reference).
Step-by-Step Example
Question
In a row of 20 students, A is 5th from the left and B is 8th from the right. They interchange their places. What is A's new rank from the right and B's new rank from the left?
Solution
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Step 1: Write the given positions and total
Total students, n = 20. A = 5th from left. B = 8th from right. -
Step 2: Convert B's position to left reference (so we know B's left-position)
Position from left = n - (position from right) + 1 = 20 - 8 + 1 = 13th from left. -
Step 3: Note the positions being swapped
A's old position (left) = 5. B's old position (left) = 13. After interchange:- A will occupy left position 13 (B's old place).
- B will occupy left position 5 (A's old place).
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Step 4: Convert A's new left-position to right-position
A's new rank from right = n - (A's new left-position) + 1 = 20 - 13 + 1 = 8th from right. -
Step 5: Convert B's new left-position to the requested left-rank (already left)
B's new rank from left = 5th from left. -
Final Answer:
A is 8th from the right; B is 5th from the left. -
Quick Check:
A now at left 13 → right = 20 - 13 + 1 = 8 ✅. B now at left 5 → left rank = 5 ✅.
Quick Variations
1. If both ranks are given from left and right for the same two people, convert both to one side first then swap.
2. If total is not given, derive it when one person's left and right ranks are provided (Total = left + right - 1).
3. When multiple swaps occur, update positions sequentially - treat each swap as a new state.
4. If asked for positions "between" after interchange, compute new positions then apply difference formulas.
Trick to Always Use
- Step 1 → Convert all ranks to the same reference (prefer left or 'position from left').
- Step 2 → Swap the numeric positions directly.
- Step 3 → Convert back to the requested reference (right/top) using n - pos + 1.
Summary
Summary
- Always convert ranks to a common side (left/top) before swapping positions.
- Use the formula Right = Total - Left + 1 to switch ends.
- After interchange, assign each person the other's numeric position, then convert if needed.
- Quick check by recomputing the opposite-side rank to confirm consistency.
Example to remember:
If A = 4th from left and B = 3rd from right in n people → convert B to left, swap left positions, then convert to requested references.
Hint: Convert the given right-position to left, swap left positions, then convert back if needed.
Common Mistakes: Not converting ranks to the same reference before swapping.