Deviation method • Adding or removing a member • Replacement • Weighted average of groups • Consecutive numbers • Batting averages • Correcting errors
The average of a set of numbers is their sum divided by how many there are. That definition is all you need, but most exam questions are faster if you think in terms of the total rather than the average itself:
Total = average × number of items
When a member joins, leaves or is replaced, the total changes by a known amount. Tracking that change is quicker than recalculating the whole average, and it avoids long additions.
To average a list of numbers that are all close together, assume a base near the middle, write how far each number is above or below it, and average those deviations. Add that small average to the base.
Find the average of 47, 52, 49, 55 and 51. With base 50, the deviations are −3, +2, −1, +5 and +1, which add to +4. Their average is 4 ÷ 5 = 0.8, so the average is 50 + 0.8 = 50.8.
This works because subtracting the same base from every number lowers the average by exactly that base. The arithmetic shrinks from adding five two-digit numbers to adding five single digits.
If a new item raises the average of n items by d, it must supply its own share at the new average and also lift each of the n old items by d. So new item = new average + n × d.
The average of 10 numbers is 24. An 11th number raises the average to 25. The new number covers its own 25 and adds 1 to each of the 10 others: 25 + 10 × 1 = 35.
For a replacement, the group size stays the same, so the change in the total is simply n × (change in average).
The average age of 8 men rises by 2 years when a man aged 30 is replaced. The total rose by 8 × 2 = 16, so the new man is 30 + 16 = 46.
The average of 20 numbers was found to be 45, but 64 had been read as 46. The total is 18 too small, so the true average is 45 + 18 ÷ 20 = 45.9.
When groups of different sizes are combined, weight each group's average by its size.
Section A has 30 students averaging 60 and section B has 20 averaging 70. The combined average is (30 × 60 + 20 × 70) ÷ 50 = 64, closer to 60 because section A is larger.
For numbers in arithmetic progression, such as consecutive numbers, the average equals the middle term, which is also the average of the first and last terms.
| Series | Average |
|---|---|
| First n natural numbers | (n + 1) ÷ 2 |
| First n even numbers | n + 1 |
| First n odd numbers | n |
Five consecutive odd numbers average 27, so 27 is the middle one. The largest is two steps of 2 above it: 27 + 4 = 31.
Likewise the first 50 natural numbers average (50 + 1) ÷ 2 = 25.5.
Cricket questions are adding-a-member questions in disguise. A batting average is total runs divided by innings. One new innings changes the total by the runs scored and the count by one.
A batsman averages 40 after 15 innings and scores 88 in the 16th. His new average is (15 × 40 + 88) ÷ 16 = 688 ÷ 16 = 43.
When the question gives the rise in average instead, use the replacement idea: the extra runs above the old average are shared across all the innings, including the new one.
Average speed has its own rule; see the average speed formula. Weighted averages also underpin alligation and mixture, and the formula list is in the Quant cheat code.
Q1. Find the average of 72, 68, 75, 71 and 69. easy
✅ Answer: (c) 71
💡 Explanation: Take 70 as the base. The deviations are +2, −2, +5, +1 and −1, which add to +5. Their average is +1, so the average is 70 + 1 = 71.
Q2. The average of 12 numbers is 30. When one number is removed, the average of the rest is 29. What number was removed? easy
✅ Answer: (a) 41
💡 Explanation: The total was 12 × 30 = 360 and is now 11 × 29 = 319. The removed number is 360 − 319 = 41.
Q3. The average weight of 10 people rises by 1.5 kg when one person weighing 60 kg is replaced by a new person. What does the new person weigh? medium
✅ Answer: (d) 75 kg
💡 Explanation: The total rises by 10 × 1.5 = 15 kg, so the newcomer is 15 kg heavier than the person replaced: 60 + 15 = 75 kg.
Q4. A class has 40 boys with an average score of 50 and 60 girls with an average score of 55. What is the class average? medium
✅ Answer: (b) 53
💡 Explanation: Weight each average by its group size: (40 × 50 + 60 × 55) ÷ 100 = (2,000 + 3,300) ÷ 100 = 53. The simple average of 50 and 55 ignores the group sizes.
Q5. A batsman scores 96 in his 21st innings and raises his average by 2. What is his average after the 21st innings? hard
✅ Answer: (a) 56
💡 Explanation: The 21st score lifted all 21 innings by 2, which took 42 runs above the old average. So the old average was 96 − 42 = 54, and the new average is 56.
Q6. What is the average of the first 40 even numbers? easy
✅ Answer: (c) 41
💡 Explanation: The first n even numbers 2, 4, …, 2n have average n + 1. For n = 40 that is 41, which is also the average of the first and last terms, (2 + 80) ÷ 2.
Yes. Any base gives the same answer, because the deviations simply shift by the same amount. Choose a round number near the middle of the data so the deviations are small and easy to add.
A simple average treats both groups as the same size. If one group is larger, its average should count for more. Multiply each average by its group size, add, and divide by the total number of members.
All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.
How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 14 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-23 against the sources listed above.
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