Average, Problems on Ages
1. Average
- Average of first n natural numbers = (n + 1)/2
- Average of first n even numbers = (n + 1)
- Average of first n odd numbers = n
- Average of squares of first n natural numbers = (n+1)(2n+1)/6
- Average of cubes of first n natural numbers = n(n+1)²/4
- Average of first n multiples of m = m(n+1)/2
- If a is added to each observation, new average = (x + a)
- If a is subtracted from each observation, new average = (x - a)
- If each observation is multiplied by a, new average = ax
- If each observation is divided by a, new average = x/a
- Average of (n₁ + n₂) = (n₁A₁ + n₂A₂)/(n₁ + n₂)
- Average of (n₁ - n₂) = (n₁A₁ - n₂A₂)/(n₁ - n₂)
When a person is included or excluded from a group: Age/Weight of that person = Number of persons × (Increase/Decrease in average) ± New Average
2. Exercise (Questions 1-10)
Q1. The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?
- (a) 0
- (b) 1
- (c) 10
- (d) 19
Q2. Nine persons went to a hotel for taking their meals. Eight of them spent ₹12 each and the ninth spent ₹8 more than the average expenditure of all the nine. What was the total money spent?
- (a) 115
- (b) 117
- (c) 119
- (d) 122
Q3. The average age of a group of person going for picnic is 16 years. Twenty new persons with average age 15 years join, due to which average becomes 15.5 years. Find the number of persons initially.
- (a) 20
- (b) 18
- (c) 22
- (d) None of these
Q4. A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half. The number of pupils in the class is:
- (a) 10
- (b) 20
- (c) 40
- (d) 73
Q5. In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach a target of 282 runs?
- (a) 6.25
- (b) 6.50
- (c) 6.75
- (d) 7.00
Q6. The average expenditure of a labourer for 6 months was ₹85 and he fell into debt. In the next 4 months by reducing his monthly expenses to ₹60 he not only cleared off his debt but also saved ₹30. His monthly income is
- (a) 70
- (b) 72
- (c) 75
- (d) 78
Q7. A batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3. Find his average after 17th inning.
- (a) 36
- (b) 39
- (c) 42
- (d) 45
Q8. A car owner buys petrol at Rs 7.50, Rs 8.00 and Rs 8.50 per litre for three successive years. What is his average cost per litre if he spends Rs 4000 each year?
- (a) 8
- (b) 9
- (c) 7.98
- (d) 8.50
Q9. The average of two numbers is XY. If one number is X, then the other number is
- (a) Y
- (b) Y
- (c) 2XY - X
- (d) X(Y-1)
Q10. In Arun's opinion, his weight is greater than 65 kg but less than 72 kg. His brother thinks it's greater than 60 kg but less than 70 kg. His mother thinks it cannot be greater than 68 kg. If all are correct, what is the average of different probable weights?
- (a) 67 kg
- (b) 68 kg
- (c) 69 kg
- (d) None of these
3. Exercise (Questions 11-20)
Q11. The average age of a board of 8 functional directors is the same as it was 3 years ago, a younger man having been substituted for one of the directors. How much younger was the new man than the director whose place he took?
- (a) 24 years
- (b) 26 years
- (c) 28 years
- (d) None of these
Q12. The average monthly sales for the first eleven months was Rs 12000, but due to illness during last month, the average for the year came down to Rs 11375. What was the value of sales during the last month?
- (a) 2100
- (b) 875
- (c) 700
- (d) 4500
Q13. The average age of a group of person going for picnic is 16 years. Twenty new persons with average age 15 years join, due to which average becomes 15.5 years. Find the number of persons initially.
- (a) 20
- (b) 18
- (c) 22
- (d) None of these
Q14. The average age of 36 students in a group is 14 years. When teacher's age is included, the average increases by one. What is the teacher's age?
- (a) 31
- (b) 36
- (c) 51
- (d) Cannot be determined
Q15. A batsman in his 12th innings makes a score of 65 and thereby increases his average by 2 runs. What is his average after the 12th innings if he had never been 'not out'?
- (a) 42
- (b) 43
- (c) 44
- (d) 45
Q16. The average number of printing error per page in a book of 512 pages is 4. If total errors in first 302 pages is 1208, the average errors per page in remaining pages is
- (a) 0
- (b) 4
- (c) 840
- (d) 90
Q17. The average of a batsman for 40 innings is 50 runs. His highest score exceeds his lowest score by 172 runs. If these two innings are excluded, his average drops by 2 runs. Find his highest score.
- (a) 172
- (b) 173
- (c) 174
- (d) 175
Q18. A batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3. Find his average after 17th inning.
- (a) 36
- (b) 39
- (c) 42
- (d) 45
Q19. Nine men went to a hotel. 8 of them spent ₹3 each over their meals and the ninth spent ₹2 more than the average expenditure of all the nine. The total money spent was
- (a) 26
- (b) 40
- (c) 29.25
- (d) 27
Q20. The mean of 30 values was 150. It was detected that one value 165 was wrongly copied as 135. Find the correct mean.
- (a) 151
- (b) 149
- (c) 152
- (d) None of these
4. Exercise (Questions 21-30)
Q21. A car owner buys petrol at Rs 7.50, Rs 8.00 and Rs 8.50 per litre for three successive years. What is his average cost per litre if he spends Rs 4000 each year?
- (a) 8
- (b) 9
- (c) 7.98
- (d) 8.50
Q22. A family consists of grandparents, parents and three grandchildren. The average age of grandparents is 67 years, parents is 35 years and grandchildren is 6 years. What is the average age of the family?
- (a) 28 years
- (b) 31 years
- (c) 32 years
- (d) None of these
Q23. The average of two numbers is XY. If one number is X, then the other number is
- (a) Y
- (b) Y
- (c) 2XY - X
- (d) X(Y-1)
Q24. The average of 10 numbers is 40.2. Later it is found that two numbers have been wrongly copied. The first is 18 greater than actual and second added is 13 instead of 31. Find the correct average.
- (a) 40.2
- (b) 40.4
- (c) 40.6
- (d) 40.8
Q25. If a, b, c, d, e are five consecutive odd numbers, then the average in terms of 'a' will be:
- (a) a+2
- (b) a+3
- (c) a+4
- (d) a
Q26. If the mean score in mathematics of a class of 50 students is 45, then the total marks obtained by them are:
- (a) 45
- (b) 95
- (c) 2250
- (d) 50
Q27. The average age of 80 boys in a class is 15. The average age of a group of 15 boys is 16 and another 25 boys is 14. What is the average age of the remaining boys?
- (a) 15.25
- (b) 14
- (c) 14.75
- (d) Cannot be determined
Q28. The average of four consecutive odd numbers is 36. What is the smallest of these numbers?
- (a) 31
- (b) 35
- (c) 43
- (d) None of these
Q29. The total of ages of a class of 75 girls is 1050. The average age of 25 of them is 12 yrs and another 25 is 16 yrs. Find the average age of the remaining girls.
- (a) 12 yrs
- (b) 13 yrs
- (c) 14 yrs
- (d) 15 yrs
Q30. The average age of a group of 16 persons is 28 years and 3 months. Two persons each 58 years old left the group. The average age of the remaining persons is
- (a) 26
- (b) 24
- (c) 22
- (d) 20
Complete Answer Key
| 1. (d) | 2. (b) | 3. (a) | 4. (c) | 5. (a) |
| 6. (d) | 7. (b) | 8. (c) | 9. (c) | 10. (d) |
| 11. (a) | 12. (d) | 13. (a) | 14. (c) | 15. (b) |
| 16. (b) | 17. (c) | 18. (b) | 19. (c) | 20. (a) |
| 21. (c) | 22. (b) | 23. (c) | 24. (a) | 25. (c) |
| 26. (c) | 27. (a) | 28. (d) | 29. (c) | 30. (b) |