How to Solve Average Problems in Seconds

Averages look like the easiest topic in the paper. Add the numbers, divide by how many there are. Done.

Then the questions get harder — "the average of 40 students is 62; when the teacher's marks are included it becomes 63" — and suddenly students are adding long columns of numbers and losing two minutes each time.

There is a faster way to think about averages, and it turns most of these questions into one line of mental arithmetic.

The idea: think in deviations, not totals

The average is a balance point. Every value above it is balanced by values below it.

So instead of summing everything, pick a convenient reference number and work with the differences from it.

Find the average of 62, 65, 61, 64, 63.

The long way sums to 315 and divides by 5. The fast way: take 63 as a reference.

• Deviations: −1, +2, −2, +1, 0
• Sum of deviations = 0
• Average = 63 + (0 ÷ 5) = 63

Find the average of 246, 250, 244, 252.

Reference 248:

• Deviations: −2, +2, −4, +4 → sum = 0
• Average = 248

For numbers clustered together — which is most exam data — this is dramatically faster than adding.

The consecutive-numbers shortcut

For any evenly spaced set of numbers, the average is simply the middle value, or the average of the two middle values.

• Average of 1 to 99 = 50
• Average of 2, 4, 6, …, 40 = (2 + 40) ÷ 2 = 21
• Average of first 20 odd numbers = 20

No addition at all. This alone answers a surprising number of exam questions.

Adding or removing a member

This is the classic pattern and where deviations really pay off.

The average of 40 students is 62. Including the teacher, the average becomes 63. What are the teacher's marks?

Adding one person raised the average by 1 for all 41 people. So the teacher supplied:

• the new average for themselves: 63
• plus 1 mark for each of the other 40: 40

Teacher's marks = 63 + 40 = 103

The general rule:

> New member's value = new average + (number of old members × change in average)

A cricketer's average after 10 innings is 45. After the 11th innings it rises to 47. What did he score?

• 47 + (10 × 2) = 67

If the average falls, the change is negative and the same formula works.

Replacing a member

The average weight of 8 people is 60 kg. One person weighing 55 kg is replaced and the average becomes 61 kg.

Total change in average = +1 across 8 people = 8 kg extra in total. So the new person weighs 55 + 8 = 63 kg.

> Change in one member = number of members × change in average

Weighted averages

When two groups of different sizes combine, you cannot simply average the averages.

Class A: 30 students, average 70. Class B: 20 students, average 60. Combined average?

• Total = (30 × 70) + (20 × 60) = 2100 + 1200 = 3300
• Combined average = 3300 ÷ 50 = 66

Note it is 66, not 65 — the answer leans towards the bigger group. That leaning is the entire idea of a weighted average, and it is the most commonly tested trap in this topic.

The allegation shortcut

For two groups, the ratio of group sizes is the inverse of the ratio of their distances from the combined average.

In the example above: A is 4 above 66, B is 6 below 66. Distances 4 : 6 = 2 : 3, so the sizes are 3 : 2 — which matches 30 : 20 ✓

This lets you find a missing group size in one step.

Quick reference

Practising it

Averages reward mental arithmetic more than almost any other topic, because the deviations method only feels fast when small additions are automatic.

A sensible order:

1. Deviations method on clustered sets
2. Evenly spaced shortcuts
3. Add / remove / replace questions
4. Weighted averages and allegation

Our mental maths daily drill is designed for exactly the quick addition and subtraction this topic leans on.

Frequently asked questions

Which class introduces averages? Usually Class 7, though the idea of a mean appears earlier in data handling.

Is the deviations method allowed in exams? Yes — it is standard mathematics. If working must be shown, state the reference value and the deviations; it is often shorter to write than a long sum.

Why is the combined average not the average of the two averages? Because the groups are different sizes. The larger group has more influence. Averaging the averages only works when both groups are exactly equal in size.

What is the most common mistake? Averaging averages, and forgetting that removing a member changes the count as well as the total.

How does this connect to data interpretation? Closely. Most data interpretation sets ask for an average of a row or column, so the speed you build here pays off there too.

Speed comes from the arithmetic underneath

Every method above assumes small sums and divisions are instant. Where they are not, the shortcut still feels slow.

Our Quantitative Aptitude course builds that speed alongside the techniques, for Class 6 and above.

Book a free demo class to see where your child stands.

Cover image: Shixart1985 — CC BY 2.0.