Rank from two ends
What it looks like
Same person's rank from both ends; find the total.
How to solve
total = left + right - 1.
Mini example. 12th left + 18th right → 29 total.
Trap: Forgetting the -1.
Reasoning Ability - Lesson 8
Put everyone on one clean line.
Ranking is not one memorized formula. It is knowing exactly who is counted and from which side - then reading the position straight off the line.
← Left / TopRight / Bottom →
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2
3
R
5
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8
9
Ravi: 4th from left · 6th from right → total 9
Topic map
Draw the line, place the slots, count exactly.
Every ranking question is the same line: anchor each end, convert wording into slots, and count precisely what is asked.
The mental modelOrder and Ranking is a line-position problem. Every clue places an item from the left/top/front or right/bottom/back; the answer comes from converting relative language into exact slots.
← Left / TopRight / Bottom →
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2
A
4
5
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B
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9
Rule 01
Position from one end
Rank from left/top/front gives a direct slot.
Rule 02
Opposite-end rank
Convert right/bottom rank when total is known.
Rule 03
Total count formula
Same person both ends: total = r1 + r2 - 1.
Rule 04
Between count
People between = difference - 1.
Rule 05
Overlap & uncertainty
When total is unknown, ranges replace exact counts.
Rule 06
Rank interchange
Two people swap; update both ranks.
Rule 07
Comparison
Compare positions of two different people.
Rule 08
Height / score order
Same line logic for tallest, highest, lowest.
Checkpoint 09
Mastery check
Mini test + cheat sheet to lock it in.
Foundation - 01
The line logic behind every rank.
Once the line is visible, the formulas are just shortcuts you can re-derive.
Position from one end
A rank from the left, top, or front is a direct slot counted from that side. "5th from left" means the 5th seat from the left edge.
Use it
A single-end rank places the person exactly - no formula needed.
Watch out
Mixing which side you are counting from. Label the ends first.
← Left / TopRight / Bottom →
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2
3
4
X
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9
Try it yourself
Slide the rank - see both ends update.
A position from the left and a position from the right always add up to total + 1. Drag the sliders and watch the same person's two ranks stay locked to that rule.
People in the row9
Your rank from the left4th
You are 4 from the left in a row of 9, so you must be 6 from the right.
← Left / TopRight / Bottom →
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2
3
You
5
6
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9
Check: 4 + 6 = 10 = total + 1.
Method - 02
Anchor the sides, place the slots, count.
A tiny labeled line beats any formula recalled from memory under pressure.
STEP 1
Set orientation
Identify the line: left-right, top-bottom, front-back, or merit order. Label both ends.
STEP 2
Mark anchors
Place every known rank as a tick from its correct side.
STEP 3
Apply the rule
Use total = left + right - 1 only when the same person has both ranks.
STEP 4
Count what is asked
Rank, total, or number between? Count exactly that - then re-read the question.
Pro tip. For interchange questions, always draw the before and after line states. Trying to track a swap in your head is where ranks drift off by one.
Question types - 03
The 7 formats every exam recycles.
Spot the format and you instantly know whether to add, subtract, or just count.
Rank with total
What it looks like
Total known, rank from one end given.
How to solve
Opposite rank = total - rank + 1.
Mini example. 40 total, 9th top → 32nd bottom.
Trap: Dropping the +1.
People between
What it looks like
Two same-side ranks; count those between.
How to solve
between = difference - 1.
Mini example. 7th and 15th left → 7 between.
Trap: Counting the endpoints.
Rank comparison
What it looks like
Compare two different people's positions.
How to solve
Draw both on the line; read who is ahead.
Mini example. A 5th, B 9th → A is ahead.
Trap: Using the total formula on two people.
Interchange ranking
What it looks like
Two people swap; ranks change.
How to solve
Draw before/after; update both ranks.
Mini example. After swap, infer the total.
Trap: Updating only one rank.
Uncertain total
What it looks like
Total not given; positions overlap or not.
How to solve
Give a minimum / maximum range.
Mini example. At least / at most N people.
Trap: Assuming a fixed total.
Height / score order
What it looks like
Tallest, highest, lowest framed as a line.
How to solve
Apply the same line logic to the ordering.
Mini example. Tallest = rank 1 from top.
Trap: Reversing high/low.
Worked examples - 04
Three rankings solved on the line.
Total count, between count, and an opposite-rank conversion.
Problem
QuestionRavi is 12th from the left and 18th from the right in a row. How many students are there?
Answer
? ? ?
Place it on the line
← Left / TopRight / Bottom →
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4
Ravi
6
7
8
9
Step 1. Same person, ranks from both ends. Draw the line with Ravi marked.
1 / 3
Common traps - 05
The 5 traps that cost a clean mark.
Tap any card to flip from the wrong instinct to the correction.
Trap 01 - Forgetting to subtract one
total = left rank + right rank.
Tap for the fix
Trap 02 - Counting endpoints
Include A and B in the between count.
Tap for the fix
Trap 03 - Direction reversal
Mixing left with right, or highest with lowest.
Tap for the fix
Trap 04 - Formula on two people
Using total = r1 + r2 - 1 for two different people.
Tap for the fix
Trap 05 - Interchange confusion
Updating only one rank after a swap.
Tap for the fix
Best practices - 06
Six habits for error-free ranking.
Tiny drawing disciplines that kill off-by-one mistakes.
TIP 01
Draw a tiny line first
A short line with side labels beats any recalled formula.
TIP 02
Label every end
Write L/R or Top/Bottom next to each rank you place.
TIP 03
Circle the target
Mark the person the question actually asks about.
TIP 04
Subtract after the difference
For between questions, take the difference, then subtract one.
TIP 05
Write old and new
For interchange, note old position and new position explicitly.
TIP 06
Never assume the total
Use a total only when given or derivable from the same person.
Mini practice - 07
Test it on 4 lines.
Mixed difficulty, no timer. Every answer comes with a worked explanation and a diagram.
Question 1 of 4Easy
Score - 0/4
Sara is 8th from the left and 5th from the right. How many people are in the row?
Streak target: 3 in a row.
Cheat sheet - 08
One card to revise ranking.
The four formulas and when each applies.
Core formulas
| Situation | Formula |
|---|---|
| Same person, both ends | total = L + R - 1 |
| Opposite rank (total known) | total - rank + 1 |
| Between two same-side ranks | difference - 1 |
| Adjacent positions | between = 0 |
Watch for
| Different people | do not use total formula |
| Unknown total | give a min/max range |
| Interchange | draw before & after |
The exam-time process
- 01Fix the orientation and label both ends first.
- 02Place every rank as a tick from its correct side.
- 03Same person both ends → total = L + R - 1.
- 04Between two same-side ranks → difference - 1.
- 05Different people → draw both, never use the total formula.
- 06No total given → answer with a range.
Anchor the sides, place the slots, count exactly what is asked.
What's next - 09
Lesson done. Time to lock it in.
Practice the hard stuff, queue the next topic, or revisit what tripped you up.