Average
Finding the number when average of numbers is given
Question (1) Find the average of all prime numbers between 10 and 30.
Solution
Prime numbers between 10 and 30 are 11, 13, 17, 19, 23, 29
Here, there are 6 prime numbers between 10 and 30
Thus, the average of prime numbers between 10 and 30
Thus, average of prime numbers between 10 and 30 = 18.66 Answer
Question (2) The average of 29 numbers is 45. If the average of the first 14 of them is 21 and the average of the last 14 of them is 16, then find the 15th number.
Solution
Given, average of 29 numbers = 45
And, average of first 14 numbers = 21
And, average of last 14 numbers = 16
Then, the 15th number = ?
As given, average of 29 numbers = 45
Thus, sum of those given 29 numbers
= 45 × 29 = 1305
Again, given, average of first 14 numbers = 21
Thus, sum of these 14 numbers
= 21 × 14 = 294
Again, given, average of last 14 numbers = 16
Thus, sum of given last 14 numbers
= 16 × 14 = 224
Thus, 15th number = sum of 29 numbers – (sum of first 14 numbers + sum of last 14 numbers)
= 1305 – (294 + 224)
= 1305 – 518 = 787
Thus, 15th number = 787 Answer
Question (3) The average of 11 numbers is 12. If the average of the first 5 of them is 8 and the average of the last 5 of them is 11, then find the 6th number.
Solution
Given, average of 11 numbers = 12
And, average of first 5 numbers = 8
And, average of last 5 numbers = 11
Then, the 6th number = ?
As given, average of 11 numbers = 12
Thus, sum of those given 11 numbers
= 11 × 12 = 132
Again, given, average of first 5 numbers = 8
Thus, sum of these 5 numbers
= 5 × 8 = 40
Again, given, average of last 5 numbers = 11
Thus, sum of given last 5 numbers
= 5 × 11 = 55
Thus, 6th number = sum of 11 numbers – (sum of first 5 numbers + sum of last 5 numbers)
= 132 – (40 + 55)
= 132 – 95 = 37
Thus, 6th number = 37 Answer
Question (4) The average of 17 numbers is 8. If the average of the first 8 of them is 9 and the average of the last 8 of them is 7, then find the 9th number.
Solution
Given, average of 17 numbers = 8
And, average of first 8 numbers = 9
And, average of last 8 numbers = 7
Then, the 9th number = ?
As given, average of 17 numbers = 8
Thus, sum of those given 17 numbers
= 17 × 8 = 136
Again, given, average of first 8 numbers = 9
Thus, sum of first 8 numbers
= 8 × 9 = 72
Again, given, average of last 8 numbers = 7
Thus, sum of given last 8 numbers
= 8 × 7 = 56
Thus, 9th number = sum of 17 numbers – (sum of first 8 numbers + sum of last 8 numbers)
= 136 – (72 + 56)
= 136 – 128 = 8
Thus, 9th number = 8 Answer
Question (5) There are 21 numbers. The average of these 21 numbers, the first 10 numbers, and the last 10 numbers are 12, 12, and 12 respectively. Then find the 11th number.
Solution
Given, average of 21 numbers = 12
And, average of first 10 numbers = 12
And, average of last 10 numbers = 12
Then, the 11th number = ?
As given, average of 21 numbers = 12
Thus, sum of those given 21 numbers
= 21 × 12 = 252
Again, given, average of first 10 numbers = 12
Thus, sum of first 12 numbers
= 12 × 10 = 120
Again, given, average of last 10 numbers = 12
Thus, sum of given last 12 numbers
= 10 × 12 = 120
Thus, 11th number = sum of 21 numbers – (sum of first 10 numbers + sum of last 10 numbers)
= 152 – (120 + 120)
= 152 – 240 = 12
Thus, 11th number = 12 Answer
Question (6) The average of 15 numbers is 9. If the average of the first 8 numbers is 8 and the last eight numbers is 10, then what is the 8th number?
Solution
Given, average of 15 numbers = 9
And, average of first 8 numbers = 8
And, average of last 8 numbers = 10
Then, the 8th number = ?
As given, average of 15 numbers = 9
Thus, sum of those given 15 numbers
= 15 × 9 = 135
Again, given, average of first 8 numbers = 8
Thus, sum of first 8 numbers
= 8 × 8 = 64
Again, given, average of last 8 numbers = 10
Thus, sum of given last 8 numbers
= 8 × 10 = 80
Thus, 8th number = (sum of first 8 numbers + sum of last 8 numbers) – sum of 15 numbers
= (64 + 80) – 135
= 144 – 135 = 9
Thus, 8th number = 9 Answer
Question (7) The average of 19 numbers is 12. If the average of the first 10 numbers is 12 and the last ten numbers is 14, then what is the 10th number?
Solution
Given, average of 19 numbers = 12
And, average of first 10 numbers = 12
And, average of last 10 numbers = 14
Then, the 8th number = ?
As given, average of 19 numbers = 12
Thus, sum of those given 19 numbers
= 19 × 12 = 128
Again, given, average of first 10 numbers = 12
Thus, sum of first 10 numbers
= 10 × 12 = 120
Again, given, average of last 10 numbers = 14
Thus, sum of given last 10 numbers
= 10 × 14 = 140
Thus, 10th number = (sum of first 10 numbers + sum of last 10 numbers) – sum of 19 numbers
= (120 + 140) – 128
= 260 – 128 = 32
Thus, 10th number = 32 Answer
Question (8) The average of 21 numbers is 10. If the average of the first 11 numbers is 12 and the last eleven numbers is 21, then what is the 11th number?
Solution
Given, average of 21 numbers = 10
And, average of first 11 numbers = 12
And, average of last 11 numbers = 21
Then, the 8th number = ?
As given, average of 21 numbers = 10
Thus, sum of those given 21 numbers
= 21 × 10 = 210
Again, given, average of first 11 numbers = 12
Thus, sum of first 11 numbers
= 11 × 12 = 132
Again, given, average of last 11 numbers = 21
Thus, sum of given last 11 numbers
= 11 × 21 = 231
Thus, 11th number = (sum of first 11 numbers + sum of last 11 numbers) – sum of 21 numbers
= (132 + 231) – 210
= 363 – 210 = 153
Thus, 11th number = 153 Answer
Question (9) The average of 9 numbers is 10. If the average of first 5 numbers is 8 and last five numbers is 9, then what is the 5th number?
Solution
Given, average of 9 numbers = 10
And, average of first 5 numbers = 8
And, average of last 5 numbers = 9
Then, the 5th number = ?
As given, average of 9 numbers = 10
Thus, sum of those given 9 numbers
= 9 × 10 = 90
Again, given, average of first 5 numbers = 8
Thus, sum of first 5 numbers
= 5 × 8 = 40
Again, given, average of last 5 numbers = 9
Thus, sum of given last 5 numbers
= 5 × 9 = 45
Thus, 5th number = (sum of first 5 numbers + sum of last 5 numbers) – sum of 9 numbers
= (40 + 45) – 90
= 45 – 90 = 5
Thus, 5th number = 5 Answer
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