Introduction
Average questions are quick, formula-driven, and frequently combined with Age and Replacement scenarios —
a chapter where recognising the pattern instantly is more valuable than complex calculation.
Basic Formula
| Formula |
|---|
| Average = Sum of all observations ÷ Number of observations |
| Sum of observations = Average × Number of observations |
Average of Consecutive Numbers
| Series | Average |
|---|
| Consecutive natural numbers 1 to n | (n+1)/2 |
| First n odd numbers | n |
| First n even numbers | n+1 |
Q. Find the average of the first 20 odd numbers.
Average of first n odd numbers = n → Average = 20
Combined Average
Flowchart — Combining Two Groups' Averages
Combined Average = (n₁×A₁ + n₂×A₂) / (n₁ + n₂)
n = number of items in each group, A = average of each group.
Q. The average marks of 30 students in a class is 60. The average marks of another 20 students is 70. Find the combined average.
Combined average = (30×60 + 20×70) / (30+20) = (1800+1400)/50 = 3200/50 = 64
"New Person Replaces Old" Pattern
Flowchart — Effect of Replacement on Average
Change in Total = (Change in Average) × (Number of people)
↓
New person's value = Old person's value ± Change in Total
Q. The average weight of 10 students increases by 2 kg when a new student replaces one weighing 45 kg. Find the new student's weight.
Change in total = 2 × 10 = 20 kg increase
New student's weight = 45 + 20 = 65 kg
Average Age Problems
Q. The average age of a family of 5 members is 30 years. If the age of the youngest member is 10 years, find the average age of the family at the time of the youngest member's birth.
Total present age = 5 × 30 = 150 years
10 years ago, total age of the other 4 members = 150 − (5×10) = 100 years (subtract 10 years from each of the 5, but the youngest wasn't born, so subtract only from the 4 others: 150 − 10×4 = 110... use the direct method: Total age of remaining 4 members today = 150 − 10 = 140; 10 years ago their total = 140 − 40 = 100
Average age of family 10 years ago = 100 ÷ 4 = 25 years
💡 Exam Tip: Whenever you see "average increases/decreases by X when a value is replaced," immediately compute Change in Total = X × count first — this single step unlocks almost every replacement-pattern question.
✅ Practice Focus: Consecutive number average shortcuts · Combined average (weighted formula) · Replacement pattern (change in total = change in average × count) · Age-based average problems.