How to Solve Averages Problems
Solve averages aptitude problems using the sum-and-count method — additions, removals, weighted averages — with a worked example and practice questions.
Expected Interview Answer
The average of a set is the sum of all values divided by the count, and most averages problems are solved by tracking the total sum rather than the average itself.
Average = Sum ÷ Count, so Sum = Average × Count — converting back to totals is the key move whenever a value is added, removed, or replaced. When a new value joins a group, the change in average times the new count tells you the deviation the new value contributes. A weighted average, unlike a simple average, requires each group’s count as a weight, not just the raw values. Always work in totals when values change, then divide once at the end.
- Working in totals avoids fraction errors
- Handles additions, removals and replacements uniformly
- Extends cleanly to weighted averages
AI Mentor Explanation
A batter’s batting average is total runs divided by dismissals — not an average of averages. If they’ve scored 400 runs in 8 innings (average 50) and then score 60 in the 9th, the new total is 460 over 9 innings, average 51.1 — you always recompute from the sum, never average the old average with the new score directly. Averages problems work the same way: convert average × count back to a total sum whenever a value changes.
Worked example (new value joins the group)
Original group
- 8 numbers, avg 20
- Total = 160
New group
- 9 numbers, avg 21
- Total = 189
New value
- 189 − 160 = 29
Step-by-Step Explanation
Step 1
Convert average to sum
Sum = Average × Count for the original group.
Step 2
Apply the change
Add, remove, or replace values directly on the sum.
Step 3
Recompute the new sum
New Sum = New Average × New Count if the new average is given.
Step 4
Solve for the unknown
Isolate the missing value from Old Sum and New Sum.
What Interviewer Expects
- Sum = Average × Count as the core conversion
- Correct handling of additions, removals and replacements
- Distinguishing weighted average from simple average
- Recognizing when true average (total/total) is needed instead of averaging rates
Common Mistakes
- Averaging two averages directly without weighting by count
- Forgetting to update the count when a value is added or removed
- Averaging speeds instead of using total distance over total time
- Sign errors when a value is removed rather than added
Best Answer (HR Friendly)
“Always convert the average back into a total by multiplying by the count — averages problems are almost always about tracking that total sum. When a value is added, removed, or replaced, update the sum directly, then divide once at the end by the new count to get the new average.”
Follow-up Questions
- How do you find a missing value when the average of n numbers is given?
- How does replacing one value in a group change the average?
- When must you use a weighted average instead of a simple average?
- Why is average speed a special case that isn’t a simple average?
MCQ Practice
1. The average of 5 numbers is 18. If one number is removed, the average of the remaining 4 is 20. The removed number is?
Original sum = 90; new sum = 80; removed number = 90 − 80 = 10.
2. A class of 30 students has an average score of 60. A new student joins, raising the average to 61. The new student scored?
Old total = 1800; new total = 61×31 = 1891; new student = 1891 − 1800 = 91.
3. A car travels 60km at 30km/h and 60km at 60km/h. Its average speed for the trip is?
Total distance = 120km; total time = 2 + 1 = 3h; average speed = 120/3 = 40 km/h.
Flash Cards
Core average formula? — Average = Sum ÷ Count, so Sum = Average × Count.
How to find a new joining value? — New Sum − Old Sum, using Sum = Average × Count for each.
Why not average two averages directly? — Unequal group sizes need a weighted average, not a simple mean of means.
Average speed over unequal times? — Total distance ÷ total time — not the average of the speeds.