Replacement & New Entry Problems

🎯 Learning Objective

Handle 'a person joins/leaves/is replaced' questions with the deviation shortcut.

💡 Concept

• When average of N people changes by d after a replacement → new − old member weight difference = N × d
• New person's value = old person's value ± (N × change)
• When someone JOINS and average changes: new value = old average + (new count × change)

🧮 Key Formulas

New value = Replaced value + N × (rise in average)

✏️ Easy Example

Q. Average weight of 8 people increases by 2 kg when a 60 kg person is replaced. Find the new person's weight.

1. Extra weight added = 8 × 2 = 16 kg
2. New person = 60 + 16

Answer: 76 kg

🇮🇳 Real-Life Example

A new player joins your 11-member team and the average age rises by 1 — that one player brought 11 extra 'years' into the pool.

📝 Exam-Level Example

Q. The average age of a class of 30 is 15 years. When the teacher is included, the average becomes 15.5. Find the teacher's age.

1. Teacher = new average + old count × rise
2. = 15.5 + 30 × 0.5 = 15.5 + 15
3. (check: 31×15.5 − 30×15 = 480.5 − 450 = 30.5 ✓)

Answer: 30.5 years

🪄 Memory Trick

Newcomer's value = new average + (old count × rise). One line, done.

⚠️ Common Mistakes

• ❌ Using new count instead of old count in the shortcut
• ❌ Sign errors when average decreases

🏆 Exam Tips

• ✅ Decreasing average → subtract instead of add
• ✅ Verify with totals if time permits

📌 Summary

• Replacement: difference = N × d
• Join: newcomer = new avg + old count × rise
• Watch the direction of change