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

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**Solution:**

1. Calculate the total sum of all 11 numbers using the average:
   - Total sum = Average × Number of terms = $71 	imes 11 = 781$

2. Calculate the sum of the first five numbers:
   - Sum of first five = $67 	imes 5 = 335$

3. Calculate the sum of the last four numbers:
   - Sum of last four = $91.5 	imes 4 = 366$

4. Find the sum of the sixth and seventh numbers:
   - Sum of sixth and seventh = Total sum - (Sum of first five + Sum of last four)
   - $= 781 - (335 + 366) = 781 - 701 = 80$

5. Given the ratio of sixth to seventh number is $9 : 7$.
   - Let sixth number = $9x$ and seventh number = $7x$

6. Using the sum of sixth and seventh numbers:
   - $9x + 7x = 80$
   - $16x = 80$
   - $x = 5$

7. Calculate the sixth number:
   - Sixth number = $9x = 9 	imes 5 = 45$

**Answer:** $45$

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