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.

We can solve this problem step-by-step by calculating the sum of the numbers based on their averages.

Step 1: Calculate the total sum of all 11 numbers.
Since the average of 11 numbers is 71, their total sum is:
$\text{Total Sum}=11\times 71=781$

Step 2: Calculate the sum of the first 5 numbers.
The average of the first 5 numbers is 67, so their sum is:
$\text{Sum of first 5 numbers}=5\times 67=335$

Step 3: Calculate the sum of the last 4 numbers.
The average of the last 4 numbers is 91.5, so their sum is:
$\text{Sum of last 4 numbers}=4\times 91.5=366$

Step 4: Find the sum of the remaining two numbers (the sixth and seventh numbers).
The total count of numbers is 11. If we subtract the sum of the first 5 numbers and the last 4 numbers from the total sum, we get the sum of the 6th and 7th numbers:
$\text{Sum of 6th and 7th numbers}=781-(335+366)$

$\text{Sum of 6th and 7th numbers}=781-701=80$

Step 5: Calculate the sixth number using the given ratio.
The ratio of the sixth and seventh number is given as 9 : 7.
Let the sixth number be $9x$ and the seventh number be $7x$.
Their sum is:
$9x+7x=80$

$16x=80$

$x=5$

Therefore, the sixth number is:
$9x=9\times 5=45$