The average of thirteen numbers is 80. The average of the first five numbers is 74.5 and that of the next five numbers is 82.5. The 11^th number is 6 more than the 12^th number is 6 less than the 13^th number. What is the average of the 11^th and the 13^th numbers?

Solution (By Examveda Team)

Average of 13 numbers = 80
⇒ Sum of 13 numbers = 80 × 13 = 1040
Average of the first 5 numbers = 74.5
⇒ Sum of the first 5 numbers = 74.5 × 5 = 372.5
Average of next 5 numbers (6^th to 10^th) = 82.5
⇒ Sum of the next 5 numbers (6^th to 10^th) = 82.5 × 5 = 412.5
⇒ Sum of the last 3 numbers = 1040 - 372.5 - 412.5 = 255
Let the 12^th number be x
11^th number = x + 6 and 13^th number = x + 6
According to the question,
⇒ x + 6 + x + x + 6 = 255
⇒ x = 81
⇒ 11^th number = x + 6 = 81 + 6 = 87
⇒ 13^th number = x + 6 = 81 + 6 = 87
∴ Average of 11^th and 13^th number = $$\frac{{87 + 87}}{2}$$  = 87