Average - All Formulas & Shortcuts

📘 Basic Formulas

1. Simple Average

Average = Sum of all values / Number of values Average = (a₁ + a₂ + … + aₙ) / n

2. Finding Sum

Sum = Average × Number of values Sum = Average × n

3. Adding/Removing Elements

New number added:

New Average = (Old Sum + New Number) / (n + 1)

Number removed:

New Average = (Old Sum - Removed Number) / (n - 1)

⚡ Quick Shortcuts

4. Weighted Average

Weighted Avg = (n₁a₁ + n₂a₂) / (n₁ + n₂)

5. Average of Consecutive Numbers

Average = (First + Last) / 2 Average = Middle number (if odd count)

Examples:

• 1,2,3,4,5 → Avg = (1+5)/2 = 3 OR middle = 3
• 10,11,12,13 → Avg = (10+13)/2 = 11.5

6. Average of First n Natural Numbers

Average = (n + 1) / 2

Example: First 100 natural numbers → (100+1)/2 = 50.5

7. Average of First n Even Numbers

Average = n + 1

Example: First 50 even numbers → 50 + 1 = 51

8. Average of First n Odd Numbers

Average = n

Example: First 50 odd numbers → 50

💡 Special Patterns

9. Age Problems

Average age increases by x years after y years:

Total age increase = n × y (where n = number of people)

Replacement:

Avg increase × n = New person’s age - Old person’s age

10. Speed/Distance Average

Equal distances at different speeds:

Average Speed = 2ab/(a+b) (Harmonic Mean)

🔗 Related Resources

• Average Practice Questions
• Average Theory
• All Formulas