The average of 12 numbers is 48. The average of the first 5 numbers is 45 and the average of the next 4 numbers is 52. If the 10th number is 10 less than the 11th number and is 5 more than the 12th number, then the average of the 11th and 12th numbers is:
Show answer & explanation
Correct answer: D
Explanation
The correct answer is Option 4 i.e. 48.5.
We know that,
Average = Sum of observations/Number of observations
The average of 12 numbers is 48
Sum of 12 numbers = 12 × 48 = 576
The average of the first 5 numbers is 45
Sum of 5 numbers = 5 × 45 = 225
The average of the next 4 numbers is 42
Sum of 4 numbers = 4 × 52 = 208
The sum of the last 3 numbers = Total sum - (Sum of first 5 numbers + Sum of next 4 numbers)
⇒ 576 - (225 + 208)
⇒ 576 - 433 = 143
Let the 10th, 11th, and 12th numbers be x, y, and z respectively
⇒ x + y + z = 143.....(1)
The 10th number is 10 less than the 11th number
⇒ x = y - 10
⇒ y = x + 10.....(2)
The 10th number is 5 more than the 12th number
⇒ x = z + 5
⇒ z = x - 5.....(3)
Putting these values in eq 1
⇒ x + y + z = 143
⇒ x + (x + 10) + (x - 5) = 143
⇒ 3x + 5 = 143
⇒ 3x = 138
⇒ x = 46
So,
⇒ z = x - 5 = 46 - 5 = 41
⇒ y = x + 10 = 46 + 10 = 56
So,
The average of the 11th and 12th numbers = (y + z)/2
⇒ (56 + 41)/2 = 97/2 = 48.5