The average of thirteen numbers is 57. The average of first three numbers is 48, and that of the next seven numbers is 58. The 11^th number is two times the 12^th number and 12^th number is 3 less than the 13^th number. What is the average of the 11^th and 13^th numbers?

Given:

The average of 13 numbers = 57

The average of the first 3 numbers = 48

The average of the next 7 numbers = 58

11th number = 2 × 12th number

12th number = (13th number - 3)

Formula used:

Average = Total sum / Number of items

Total sum of numbers = Average × Number of items

Calculation:

Total sum of 13 numbers = 57 × 13 = 741

Total sum of the first 3 numbers = 48 × 3 = 144

Total sum of the next 7 numbers = 58 × 7 = 406

Sum of the last 3 numbers (11th, 12th, and 13th) = 741 - (144 + 406)

⇒ Sum of the last 3 numbers = 741 - 550 = 191

Let the 13th number be x.

⇒ 12th number = x - 3

⇒ 11th number = 2 × (x - 3) = 2x - 6

Sum of the last 3 numbers = (2x - 6) + (x - 3) + x = 191

⇒ 2x - 6 + x - 3 + x = 191

⇒ 4x - 9 = 191

⇒ 4x = 191 + 9

⇒ x = 50

13th number = x = 50

12th number = x - 3 = 50 - 3 = 47

11th number = 2 × (x - 3) = 2 × 47 = 94

Average of the 11th and 13th numbers = (11th number + 13th number) / 2

⇒ Average = (94 + 50) / 2 = 144 / 2 = 72

∴ The correct answer is option (4).