Average: Consecutive Numbers, Replacement, Runs, Wrong Entries and Ages
Average: Consecutive Numbers, Replacement, Runs, Wrong Entries and Ages
Average = sum ÷ number of items. Almost every NTPC average question is solved by working with SUMS, not averages: find the old total, change it, divide again. The shortcuts below let you skip even that for replacement, cricket and wrong-entry questions.
1. Core Facts
Average = Sum of observations / Number of observations Sum = Average × Number
| Set | Average |
|---|---|
| First n natural numbers | (n + 1)/2 |
| First n odd numbers | n |
| First n even numbers | n + 1 |
| Squares of first n natural numbers | (n + 1)(2n + 1)/6 |
| Any arithmetic progression (consecutive numbers, multiples) | (first term + last term)/2 |
| n consecutive numbers with average m | They run from m − (n − 1)/2 to m + (n − 1)/2 |
2. Shortcut Rules
| Situation | Rule |
|---|---|
| One member replaced, average rises by d over n members | New member = old member + n × d |
| One member joins, average rises by d (n members before) | New member = old average + (n + 1) × d |
| k new members join, average of all becomes A' | Sum of newcomers = (n + k)A' − nA |
| Batsman: kth innings score S raises average by d | Old average = S − k × d; new average = S − (k − 1) × d |
| Wrong entry corrected | New average = old average + (correct − wrong)/n |
| First k and last k of n = 2k − 1 numbers | Middle number = (sum of first k) + (sum of last k) − (total sum) |
| Ages after t years | Every member's age rises by t, so the average rises by t (if no member joins or leaves) |
✗ A new man replaces a 65 kg man and the average of 8 rises by 2.5 kg, so the new man weighs 67.5 kg | ✓ The total rises by 8 × 2.5 = 20 kg, so he weighs 65 + 20 = 85 kg
✗ Average age of a family now is 30, so after 5 years it is 35 even after a baby is born | ✓ The baby joins with a small age, so recompute the total and divide by the new count
हिंदी नोट: औसत के प्रश्नों में औसत नहीं, कुल योग के साथ काम कीजिए: योग = औसत × संख्या। किसी सदस्य के बदलने पर नया सदस्य = पुराना सदस्य + संख्या × औसत में वृद्धि।
Exam Pointer: Verified NTPC patterns: average of the first n natural or odd numbers (January 2021, June 2025 Graduate CBT-1), and average age with a teacher or a newborn included (January 2021, June 2022 CBT-2). Replacement, cricket averages, wrong entries and overlapping groups had no verified NTPC shift in our check and are tagged syllabus-based; they test the same "work with totals" idea.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Average of consecutive numbers, odd and even numbers, and multiples
[PYQ: NTPC CBT-1 20-Jan-2021 Shift-1 | NTPC Graduate CBT-1 10-Jun-2025 Shift-1]
EXAM LEVEL
Q. Find the average of all odd numbers from 1 to 49.
These are the first 25 odd numbers, whose average is 25. Check: (1 + 49)/2 = 25.
Answer: 25
EXAMATLAS LEVEL
Q. Find the average of all multiples of 7 between 100 and 300.
The first multiple above 100 is 105 and the last below 300 is 294. Multiples form an AP, so the average = (105 + 294)/2 = 199.5. No need to count the terms (there are 28).
Answer: 199.5
Pattern 2: Average changes when a member joins, leaves or is replaced
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The average weight of 8 persons increases by 2.5 kg when a new person replaces one weighing 65 kg. Find the new person's weight.
Total increase = 8 × 2.5 = 20 kg, so the newcomer weighs 65 + 20 = 85 kg.
Answer: 85 kg
EXAMATLAS LEVEL
Q. The average age of 30 students is 14 years. When 5 new students join, the average age becomes 14.5 years. Find the average age of the new students.
Old total = 420. New total = 35 × 14.5 = 507.5. The newcomers' total = 87.5, so their average = 87.5/5 = 17.5 years. Shortcut: each newcomer brings 14 plus enough to lift 35 people by 0.5: 14 + 35 × 0.5/5 = 17.5.
Answer: 17.5 years
Pattern 3: Cricket averages: runs in an innings change the average
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A batsman's average after 15 innings is 32. How many runs must he score in the 16th innings to raise his average by 3?
New average 35 over 16 innings means a total of 560; he has 480, so he needs 80. Shortcut: 32 + 16 × 3 = 80.
Answer: 80
EXAMATLAS LEVEL
Q. A batsman scores 92 runs in his 17th innings and thereby increases his average by 3. Find his new average, and the runs he needs in the 18th innings to make his average 45.
Old average = 92 − 17 × 3 = 41, so the new average = 44 (total 748). For an average of 45 over 18 innings he needs 810 in all, so 810 − 748 = 62 runs.
Answer: New average 44; 62 runs
Pattern 4: Wrong entries corrected
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The average of 20 numbers is 45. Later it was found that 64 was misread as 46. Find the correct average.
The total was short by 18, so the average rises by 18/20 = 0.9. Correct average = 45.9.
Answer: 45.9
EXAMATLAS LEVEL
Q. The average marks of 40 students was 62. Two marks, 48 and 75, were wrongly recorded as 84 and 57. Find the correct average.
Correct pair sum = 123, recorded pair sum = 141, so the total was 18 too high. Correct average = 62 − 18/40 = 61.55. The swapped digits look harmless but the sums differ.
Answer: 61.55
Pattern 5: Average age with a teacher or new family member, over time
[PYQ: NTPC CBT-1 8-Jan-2021 Shift-1 | NTPC CBT-2 17-Jun-2022 Shift-1 | NTPC CBT-1 16-Jan-2021 Shift-2]
EXAM LEVEL
Q. The average age of 30 boys is 13 years. When the teacher's age is included, the average becomes 14 years. Find the teacher's age.
New total = 31 × 14 = 434, old total = 390, so the teacher is 44. Shortcut: 13 + 31 × 1 = 44.
Answer: 44 years
EXAMATLAS LEVEL
Q. The average age of a family of 4 is 30 years today. A baby is born 5 years from now. What will be the average age of the family 9 years from now?
Today's total = 120. In 9 years the four members add 36, making 156. The baby, born at year 5, is 4 years old at year 9. Total = 160 for 5 members, average = 32 years. Answering 39 (30 + 9) ignores the baby.
Answer: 32 years
Pattern 6: Overlapping groups: find the middle number
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The average of 11 numbers is 50. The average of the first 6 is 49 and of the last 6 is 52. Find the 6th number.
The 6th number is counted in both groups: 6 × 49 + 6 × 52 − 11 × 50 = 294 + 312 − 550 = 56.
Answer: 56
EXAMATLAS LEVEL
Q. The average of 13 results is 60. The average of the first 7 is 57 and of the last 7 is 64. Find the 7th result.
7 × 57 + 7 × 64 − 13 × 60 = 399 + 448 − 780 = 67.
Answer: 67
Pattern 7: Consecutive numbers with a given average
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The average of 5 consecutive even numbers is 36. Find the largest.
The middle number equals the average, 36. The numbers are 32, 34, 36, 38, 40, so the largest is 40.
Answer: 40
EXAMATLAS LEVEL
Q. The average of 7 consecutive numbers is n. If the next 3 consecutive numbers are also included, what is the new average?
The numbers are n − 3 to n + 3; adding n + 4, n + 5, n + 6 gives 10 consecutive numbers from n − 3 to n + 6. Their average = ((n − 3) + (n + 6))/2 = n + 1.5.
Answer: n + 1.5
60-Second Revision
- Sum = average × count; always change the sum, then divide.
- First n natural (n + 1)/2, odd n, even n + 1; AP average = (first + last)/2.
- Replacement: new = old + n × rise; cricket: old average = score − k × rise.
- Wrong entry: average changes by (correct − wrong)/n.
- Overlap: middle = sum of first k + sum of last k − total.
- Ages: add t to every member, then add newcomers with their own ages.