The average of 11 numbers is 71 and average of first five numbers is 67. If average of last four numbers is 91.5 and the ratio of sixth and seventh number is 9 : 7, then find the sixth number.

The correct option/answer is 45.

Let's break down the problem step-by-step to find the sixth number.

Step 1: Find the sum of all 11 numbers
We are given that the average of 11 numbers is 71.
Sum of all 11 numbers = 11 * 71 = 781

Step 2: Find the sum of the first five numbers
The average of the first five numbers is 67.
Sum of first 5 numbers = 5 * 67 = 335

Step 3: Find the sum of the last four numbers
The average of the last four numbers is 91.5.
Sum of last 4 numbers = 4 * 91.5 = 366

Step 4: Find the sum of the remaining (sixth and seventh) numbers
The total numbers are 11. The first 5 numbers, the last 4 numbers, and the sixth and seventh numbers make up the entire set.
Sum of 11 numbers = (Sum of first 5) + (Sixth number + Seventh number) + (Sum of last 4)
781 = 335 + (Sixth number + Seventh number) + 366
781 = 701 + (Sixth number + Seventh number)
Sixth number + Seventh number = 781 - 701 = 80

Step 5: Use the given ratio to find the sixth number
We are given that the ratio of the sixth and seventh numbers is 9 : 7.
Let the sixth number be 9x and the seventh number be 7x.
Their sum is:
9x + 7x = 80
16x = 80
x = 80 / 16 = 5

Therefore, the sixth number is:
9x = 9 * 5 = 45