Rank from Top & Bottom

🎯 Learning Objective

Use Total = top + bottom − 1 for one person and convert a rank from one end to the other.

💡 Concept

• The same person counted from both ends is counted twice → Total = (rank from top) + (rank from bottom) − 1
• The −1 is exactly that one person, shared by both counts
• Rank from bottom = Total − rank from top + 1 (and the reverse)
• The same rule works for left/right in a row and for top/bottom in a merit list
• For small groups, draw dots — 10 seconds of dots beats a wrong formula

🧮 Key Formulas

Total = (rank from top) + (rank from bottom) − 1

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Rank from bottom = Total − rank from top + 1

✏️ Easy Example

Q. In a class of 40 students, Ravi is 12th from the top. What is his rank from the bottom?

1. Rank from bottom = Total − rank from top + 1
2. = 40 − 12 + 1

Answer: 29th from the bottom

🇮🇳 Real-Life Example

Ticket queue at the station: 11 people ahead of you and 28 behind, plus you — the line has 11 + 1 + 28 = 40. Ranking maths is just queue maths.

📝 Exam-Level Example

Q. In a row, Sita is 15th from the left end and 8th from the right end. How many people are in the row?

1. Sita is counted in both the 15 and the 8
2. Total = 15 + 8 − 1

Answer: 22 people

📝 Exam-Level Example

Q. In a class of 45 students, Amit is 20th from the top. Sonia's rank is 5 places below Amit's. What is Sonia's rank from the bottom?

1. Sonia is 20 + 5 = 25th from the top
2. Rank from bottom = 45 − 25 + 1

Answer: 21st from the bottom

🪄 Memory Trick

Ask first: is the question about the SAME person from both ends, or two different people? Same person → Total = top + bottom − 1. That single check avoids most errors.

⚠️ Common Mistakes

• ❌ Forgetting the −1 — the person is counted twice
• ❌ Dropping the +1 while converting a rank to the other end
• ❌ Mixing 'from top' and 'from bottom' directions in one step

🏆 Exam Tips

• ✅ Verify with a small dot diagram whenever the numbers are tiny
• ✅ Underline the direction words — 'from top', 'from left' — before you calculate

📌 Summary

• Total = top + bottom − 1 for the same person
• Other end: Total − rank + 1
• Same rule for rows and merit lists
• Dots never lie — draw when unsure