1. Basics
Average = sum ÷ number. Sum = average × number. If every value changes by k, the average changes by k.
2. Shortcuts
| Case | Result |
|---|---|
| First n natural numbers | (n + 1)/2 |
| First n even / odd numbers | n + 1 / n |
| Consecutive numbers | Average = middle term |
| Replacement raises average of n by d | New − old = n × d |
| Combined average | (n₁a₁ + n₂a₂)/(n₁ + n₂) |
| Average speed, equal distances | 2xy/(x + y) |
| Batsman's new average after k runs in (n + 1)th innings raises average by d | New average = k − n × d |
3. Solved examples
Average of 5 consecutive odd numbers is 27 → largest = 27 + 4 = 31.
Average weight of 8 rises by 2.5 kg when one of 65 kg is replaced → new = 65 + 20 = 85 kg.
Batsman scores 87 in the 17th innings, average rises by 3 → new average = 87 − 16 × 3 = 39.
(ASO 2021) 30 numbers average 12; first 20 average 11, next 9 average 10 → last = 360 − 220 − 90 = 50.
Average of 11 results is 50; first 6 average 49, last 6 average 52 → 6th = 294 + 312 − 550 = 56.
Teacher joins 30 students (average 15) and average becomes 16 → teacher = 496 − 450 = 46.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. The average of 79, 34, 69, 80, 33 is:
(a) 73.75 (b) 56 (c) 59 (d) 61
Answer: (c) 59
Solution: Sum = 295. Average = 295/5 = 59.
Example 2. The average of the first 20 natural numbers is:
(a) 21 (b) 10 (c) 10.5 (d) 11.5
Answer: (c) 10.5
Solution: First n natural: (n+1)/2. First n odd: n. First n even: n+1. Here: 10.5.
ASO exam level
Example 3. The average weight of 30 boys in a class is 40 kg and that of the other 20 boys is 35 kg. The average weight of all the boys is:
(a) 38 kg (b) 37.5 kg (c) 39 kg (d) 43 kg
Answer: (a) 38 kg
Solution: Total = 30×40 + 20×35 = 1,900. Average = 1,900/50 = 38 kg.
Example 4. The average weight of 12 persons increases by 1.5 kg when a person weighing 50 kg is replaced by a new person. The new person weighs:
(a) 68 kg (b) 32 kg (c) 51.5 kg (d) 73 kg
Answer: (a) 68 kg
Solution: Total rises by 12 × 1.5 = 18 kg. New person = 50 + 18 = 68 kg.
Example 5. The average age of 40 students is 12 years. When the teacher's age is included, the average rises by 2 years. The teacher's age is:
(a) 92 years (b) 95 years (c) 14 years (d) 94 years
Answer: (d) 94 years
Solution: New total = 41 × 14 = 574. Old total = 480. Teacher = 94 years.
Example 6. The average age of 30 students is 12 years. When the teacher's age is included, the average rises by 1 year. The teacher's age is:
(a) 13 years (b) 43 years (c) 44 years (d) 42 years
Answer: (b) 43 years
Solution: New total = 31 × 13 = 403. Old total = 360. Teacher = 43 years.
Example 7. The average of 7 numbers is 19. The average of the first 4 is 18 and of the last 4 is 21. The middle (4th) number is:
(a) 20 (b) 25 (c) 19.5 (d) 23
Answer: (d) 23
Solution: First 4 + last 4 = 72 + 84 = 156. This counts the 4th number twice. 4th = 156 − 7×19 = 23.
Example 8. The average age of A, B and C is 40 years and that of B, C and D is 44 years. If A is 36 years old, D's age is:
(a) 51 years (b) 40 years (c) 48 years (d) 44 years
Answer: (c) 48 years
Solution: (B + C + D) − (A + B + C) = 3×44 − 3×40 = 12. D = 36 + 12 = 48 years.
Example 9. The average age of A, B and C is 40 years and that of B, C and D is 44 years. If A is 30 years old, D's age is:
(a) 44 years (b) 42 years (c) 34 years (d) 45 years
Answer: (b) 42 years
Solution: (B + C + D) − (A + B + C) = 3×44 − 3×40 = 12. D = 30 + 12 = 42 years.
Example 10. A batsman scores 120 runs in his 17th innings and raises his average by 4. His average after the 17th innings is:
(a) 56 (b) 52 (c) 60 (d) 7.06
Answer: (a) 56
Solution: Let old average = x. 16x + 120 = 17(x + 4) → x = 52. New average = 56.
Example 11. A batsman scores 120 runs in his 17th innings and raises his average by 4. His average after the 17th innings is:
(a) 7.06 (b) 52 (c) 56 (d) 60
Answer: (c) 56
Solution: Let old average = x. 16x + 120 = 17(x + 4) → x = 52. New average = 56.
Example 12. The average of 30 numbers is 12. The average of the first 20 is 11 and of the next 9 is 10. The last number is:
(a) 12 (b) 50 (c) 40 (d) 62
Answer: (b) 50
Solution: Last = 30×12 − 20×11 − 9×10 = 360 − 220 − 90 = 50.
Advanced level
Example 13. The average of 50 observations was 60. Later it was found that 52 was wrongly taken as 25. The correct average is:
(a) 59.46 (b) 62.7 (c) 61.54 (d) 60.54
Answer: (d) 60.54
Solution: Total increases by 52 − 25 = 27. New average = 60 + 27/50 = 60.54.
Example 14. The average of 50 observations was 41. Later it was found that 52 was wrongly taken as 25. The correct average is:
(a) 42.54 (b) 41.54 (c) 43.7 (d) 40.46
Answer: (b) 41.54
Solution: Total increases by 52 − 25 = 27. New average = 41 + 27/50 = 41.54.
Example 15. A man covers three equal distances at 20, 30 and 60 km/h. His average speed for the whole journey is:
(a) 29 km/h (b) 36.67 km/h (c) 32 km/h (d) 30 km/h
Answer: (d) 30 km/h
Solution: Average speed = 3/(1/20 + 1/30 + 1/60) = 30 km/h (harmonic mean, not simple average).
Topic-wise practice for Averages (Quant): Consecutive numbers, wrong entry corrected, new member, cricket runs. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Average of given numbers
- Average of consecutive odd, even and natural numbers
- A new member added or removed
- Wrong entry found and corrected
- Average age of a group, teacher included
- Cricket batting average and runs needed
- Combined average of two groups
- Average of first and last terms
- Monthly income and expenditure average