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 answer is 45.

Let's solve the problem step-by-step:

Step 1: Calculate the total sum of all 11 numbers.
The average of 11 numbers is given as 71.
Sum of 11 numbers = 11 * 71 = 781

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

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

Step 4: Find the sum of the remaining two numbers (the 6th and 7th numbers).
The sum of all 11 numbers is equal to the sum of the first 5 numbers, the last 4 numbers, and the 6th and 7th numbers.
Sum of (6th + 7th numbers) = Total Sum - (Sum of first 5 numbers + Sum of last 4 numbers)
Sum of (6th + 7th numbers) = 781 - (335 + 366)
Sum of (6th + 7th numbers) = 781 - 701 = 80

Step 5: Determine the sixth number using the given ratio.
The ratio of the 6th number to the 7th number is 9 : 7.
Let the 6th number be:
$9x$
and the 7th number be:
$7x$
The sum of these two numbers is:
$9x+7x=80$
$16x=80$
$x=5$
Now, find the sixth number by multiplying 9 by x:
Sixth number = $9\times 5=45$