Averages
Averages are easy to compute and easy to misread. Most questions are about what happens to an average when a member joins, leaves or is replaced — and those are all solved by thinking in totals, not in averages.
Updated 19 September 2026 · Commonly 1-2 questions, often a replacement or new-member problem.
How to think about averages
Always convert back to the total
The reliable method for any average question is: total = average × count. Change the total and the count as the question describes, then divide again. Trying to reason directly about the average is where mistakes creep in.
Replacement shifts the total, not the count
When one person is replaced by another, the count stays the same and the total changes by the difference between them. If the average of n members shifts by d, the total shifted by n × d — that single line solves most replacement questions.
Weighted averages need the weights
Combining two groups is not averaging the two averages unless the groups are the same size. Multiply each average by its own count, add the totals, and divide by the combined count.
Formulas and shortcuts
- Average
Average = Sum of values / Number of values - Total from average
Sum = Average × Count - Replacement
New member = Old member + (n × change in average) - Combined average of two groups
(n₁A₁ + n₂A₂) / (n₁ + n₂) - Average of first n natural numbers
(n + 1) / 2
Solved examples
Q1. The average age of 10 students is 15 years. When the teacher joins, the average becomes 16. What is the teacher's age?
- Work in totals. Students total = 10 × 15 = 150
- With the teacher there are 11 people averaging 16, so the new total = 11 × 16 = 176
- The teacher's age is the difference: 176 − 150
Answer: 26 years
Q2. The average weight of 10 people increases by 2 kg when a new person replaces one weighing 50 kg. What is the new person's weight?
- The count stays at 10; only the total changed.
- Total increase = 10 × 2 = 20 kg
- New person = person who left + total increase = 50 + 20
Answer: 70 kg
Q3. The average of 11 numbers is 30. The average of the first 6 is 25 and of the last 6 is 35. Find the 6th number.
- The 6th number sits in both groups, so it gets counted twice when you add them.
- Sum of first 6 = 6 × 25 = 150
- Sum of last 6 = 6 × 35 = 210
- Their sum = 360, which is all 11 numbers plus the 6th counted once more.
- Sum of all 11 = 11 × 30 = 330
- 6th number = 360 − 330
Answer: 30
Practice questions with answers
1. The average of 10, 20, 30, 40 and 50 is:
Answer: B. 30 — Sum = 150, count = 5, average = 30.
2. The average of the first 5 even natural numbers (2, 4, 6, 8, 10) is:
Answer: B. 6 — Sum = 30, count = 5, average = 6.
3. The average of the first 10 natural numbers is:
Answer: B. 5.5 — (n + 1)/2 = 11/2 = 5.5.
4. The average of 5 numbers is 20. If one number is removed the average becomes 18. The removed number is:
Answer: B. 28 — Original total = 100. New total = 4 × 18 = 72. Removed = 100 − 72 = 28.
5. A class of 20 boys averages 60 marks and 10 girls average 75. The combined average is:
Answer: B. 65 — ((20 × 60) + (10 × 75)) / 30 = (1200 + 750) / 30 = 1950 / 30 = 65.
6. The average age of 8 people increases by 1 year when a 30-year-old is replaced. The new person is:
Answer: C. 38 — Total increase = 8 × 1 = 8. New person = 30 + 8 = 38.
Where marks get lost
- Averaging two group averages when the groups are different sizes.
- Changing the count on a replacement problem — a replacement keeps the count fixed.
- Forgetting that an overlapping element gets double-counted when two groups are added.
- Reasoning about the average directly instead of converting to totals.
Reading is not practising.
Run a timed set on averages and see which step you actually lose time on. Free, no card needed.
Start a timed set