In this guide
The average of a set of numbers is their total divided by how many there are. It tells you the typical value of the group. If five candidates score 12, 15, 18, 21 and 24, the total is 90 and the average is 90 ÷ 5 = 18.
Average questions look varied: marks, ages, weights, runs, salaries. Underneath, nearly all of them use one rule. Once you train yourself to turn every average into a total, most of these questions take under a minute.
Total = Average × Number of items
Evenly spaced numbers
When numbers go up by the same step each time, the average is simply the middle value, or (first + last) ÷ 2. For 12, 15, 18, 21, 24: (12 + 24) ÷ 2 = 18. No adding needed.
| Set | Average |
|---|---|
| First n natural numbers (1, 2, … n) | (n + 1) ÷ 2 |
| First n even numbers (2, 4, … 2n) | n + 1 |
| First n odd numbers (1, 3, … 2n − 1) | n |
| Any evenly spaced list | (first + last) ÷ 2 |
Changing every number
- Add (or subtract) the same number to every item, and the average rises (or falls) by that number.
- Multiply (or divide) every item by the same number, and the average is multiplied (or divided) by it.
If the average of five numbers is 30 and each is increased by 4, the new average is 34. No totals needed.
When the group changes
| Situation | Method |
|---|---|
| Someone joins | New total − old total = the newcomer's value |
| Someone leaves | Old total − new total = the leaver's value |
| Someone is replaced | New person = old person + (number in group × change in average) |
| Two groups combined | Combined average = (total of group 1 + total of group 2) ÷ (n1 + n2) |
| A value was copied wrongly | New average = old average + (correct − wrong) ÷ number |
A quick shortcut for someone joining: the newcomer's value = new average + (old number of people × rise in average). If 30 students average 14 years and a teacher joining lifts it to 15, the teacher is 15 + 30 × 1 = 45 years old.
Solved examples
Example 1. Find the average of the first 10 even numbers.
- The numbers are 2, 4, … 20.
- Average = (2 + 20) ÷ 2 = 11, which matches the rule n + 1.
Example 2. Section A has 30 students with average marks of 60. Section B has 20 students with average marks of 70. Find the average of all 50.
- Totals: 30 × 60 = 1,800 and 20 × 70 = 1,400.
- Combined total = 3,200.
- Average = 3,200 ÷ 50 = 64.
- Note it is not 65, the simple middle of 60 and 70, because section A is bigger.
Example 3. The average age of 30 students is 14 years. When the teacher's age is included, the average becomes 15 years. Find the teacher's age.
- Old total = 30 × 14 = 420.
- New total = 31 × 15 = 465.
- Teacher = 465 − 420 = 45 years.
Example 4. The average weight of 8 people rises by 2.5 kg when one person weighing 65 kg is replaced by a new person. Find the new person's weight.
- The total rises by 8 × 2.5 = 20 kg.
- The new person weighs 65 + 20 = 85 kg.
Example 5. The average of 20 numbers was found to be 45. Later it was found that 64 had been copied as 46. Find the correct average.
- Old total = 20 × 45 = 900.
- Correct total = 900 − 46 + 64 = 918.
- Correct average = 918 ÷ 20 = 45.9.
Example 6. A batter's average after 10 innings is 40. They score 62 in the 11th innings. Find the new average.
- Total after 10 innings = 400.
- Total after 11 innings = 462.
- New average = 462 ÷ 11 = 42.
- Shortcut: 40 + (62 − 40) ÷ 11 = 40 + 2 = 42.
Common mistakes
| Mistake | Correct way |
|---|---|
| Averaging two averages directly | Use totals when the groups differ in size |
| Forgetting that the group size changes | A joiner makes it n + 1; a leaver makes it n − 1 |
| In replacement, using n + 1 | The group size stays n |
| Correcting the average by (correct − wrong) alone | Divide the difference by the number of items |
| Adding long lists one by one | For evenly spaced lists, use (first + last) ÷ 2 |
Practice set
- Find the average of 7, 14, 21, 28 and 35.
- Find the average of the first 20 natural numbers.
- The average of 5 numbers is 30. Each number is increased by 4. Find the new average.
- The average age of 25 students is 12 years. When the teacher is included, it becomes 13 years. Find the teacher's age.
- The average weight of 10 people rises by 1.5 kg when a person weighing 58 kg is replaced. Find the new person's weight.
- Group A has 20 members with an average of 50; group B has 30 members with an average of 60. Find the combined average.
- The average of three consecutive even numbers is 24. Find the largest.
- A batter averages 32 runs after 8 innings. How many runs are needed in the 9th innings to raise the average to 35?
Answers:
- Evenly spaced: (7 + 35) ÷ 2 = 21.
- (20 + 1) ÷ 2 = 10.5.
- 30 + 4 = 34.
- 26 × 13 − 25 × 12 = 338 − 300 = 38 years.
- 58 + 10 × 1.5 = 73 kg.
- (1,000 + 1,800) ÷ 50 = 56.
- The middle number is 24, so the numbers are 22, 24, 26. Largest: 26.
- Needed total 9 × 35 = 315; current total 8 × 32 = 256; runs needed = 59.
What to do next
- Rewrite the five "group changes" rules from memory.
- Solve 15 average questions using totals only, then try the shortcuts.
- Use averages with ratio and ages, where the same total method appears.
- Revise percentage for questions that mix averages and percentages.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
Get the next SSC GD guide by email
New guides every week. No spam, unsubscribe any time.