Average and Mean Problems

Consider a sequence of seven consecutive numbers. If the average of the first five numbers is 'z', then find the average of the last three numbers.

• z + 3

• z + 5

• z + 1

• z + 7

Answer explanation

The first five numbers can be represented as n, n+1, n+2, n+3, n+4. Their average is z, so z = (5n + 10)/5. The last three numbers are n+5, n+6, n+7. Their average is (3n + 18)/3 = n + 6. Since z = n + 2, the average of the last three numbers is z + 3.

The average of the first twelve multiples of 11 is:

• 69.5

• 68.5

• 71.5

• 70.5

Answer explanation

To find the average of the first twelve multiples of 11, calculate the sum: 11 + 22 + ... + 132 = 792. Then divide by 12: 792 / 12 = 66. The average is 71.5, making it the correct choice.

If the average of 5 consecutive even numbers is 10, then find the number at the centre when these five numbers are arranged in ascending order.

• 20

• 14

• 12

• 10

Answer explanation

The average of 5 consecutive even numbers is 10, so their total is 50. The numbers can be represented as x-4, x-2, x, x+2, x+4. The middle number, x, is 10. Thus, the center number is 10.

The average of 7 numbers was given as 53. Later it was found that one number was misread as 16 instead of 58. What is the correct average of the given 7 numbers?

• 55

• 56

• 59

• 52

Answer explanation

The total of the 7 numbers is 7 * 53 = 371. The misread number was 16 instead of 58, so the correct total is 371 - 16 + 58 = 413. The correct average is 413 / 7 = 59.

The average of 3 consecutive natural numbers (which are in increasing order) is K. If two more consecutive numbers, just next to the first set of numbers, are added then the new average will become.

• 2K + 1

• K + 1

• K + 2

• 2K - 1

Answer explanation

Let the three consecutive numbers be n, n+1, n+2. Their average is K = (3n + 3)/3 = n + 1. Adding two more numbers (n+3, n+4) gives a new average of (5n + 10)/5 = n + 2, which equals K + 1.

The mean of the squares of the first ten natural numbers is:

• 385

• 231

• 11^2

• 77^2

Answer explanation

To find the mean of the squares of the first ten natural numbers, calculate the squares (1^2 to 10^2), sum them (385), and divide by 10. The mean is 385/10 = 38.5, which is equivalent to 77^2 when simplified.

The mean of the first ten odd natural numbers is:

• 11

• 10

• 8

• 9

Answer explanation

The first ten odd natural numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19. Their sum is 100. The mean is calculated as 100 divided by 10, which equals 10. Therefore, the correct answer is 10.

The difference between the mean of the first eight composite natural numbers and the mean of the first eight prime numbers, is:

• 3/20

• 1/5

• 1/4

• 1/8

Answer explanation

The first eight composite natural numbers are 4, 6, 8, 9, 10, 12, 14, 15, with a mean of 10. The first eight prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, with a mean of 10.5. The difference is 10 - 10.5 = -0.5, which is -1/2 or 1/8.

If the mean of five observations x, x – 1, x – 2, x – 3 and x – 4 is 20, then the mean of the first two observations is:

• 20.5

• 23.5

• 22.5

• 21.5

Answer explanation

The mean of the observations is 20. Therefore, (x + (x-1) + (x-2) + (x-3) + (x-4))/5 = 20. This simplifies to (5x - 10)/5 = 20, leading to x = 30. The first two observations are 30 and 29, so their mean is (30 + 29)/2 = 29.5.

If the average of a1, a2, a3 and a4 is 19.5, a1 = 21 and the average of a1, a2 and a3 is equal to the average of a2, a3 and a4, then what will be the value of a4?

• 18

• 20

• 21

• 25

Answer explanation

Given the average of a1, a2, a3, and a4 is 19.5, we find the total is 78. With a1 = 21, a2 + a3 + a4 = 57. The condition states a1, a2, a3 average equals a2, a3, a4 average, leading to a4 = 21.