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 Option 45 (or 45).

Let us solve the problem step-by-step to find the sixth number.

Step 1: Calculate the total sum of all 11 numbers.
We are given that the average of 11 numbers is 71.
$\text{Sum of 11 numbers}=11\times 71=781$

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

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

Step 4: Find the sum of the remaining two numbers (the 6^th and 7^th numbers).
The total sum of 11 numbers includes the sum of the first 5 numbers, the sum of the 6^th and 7^th numbers, and the sum of the last 4 numbers (since 5 + 2 + 4 = 11).
$\text{Sum of (6}{\text{th}}^{}\text{+ 7}{\text{th}}^{}\text{numbers)}=781-(335+366)$
$\text{Sum of (6}{\text{th}}^{}\text{+ 7}{\text{th}}^{}\text{numbers)}=781-701=80$

Step 5: Determine the 6^th number using the given ratio.
The ratio of the 6^th number to the 7^th number is given as 9 : 7.
Let the 6^th number be 9x and the 7^th number be 7x.
$9x+7x=80$
$16x=80$
$x=\frac{80}{16}=5$

Now, calculate the 6^th number:
$\text{6}{\text{th}}^{}\text{number}=9\times 5=45$

Thus, the sixth number is 45.