The average of 15 numbers is 72. The average of the first 6 numbers is 65. The average of the next 5 numbers is 20% higher than the average of the first 6. The 12th number is 8 less than the 15th, and the 13th is 5 greater than the 15th and if 14th number is 78. What is the average of the 12th and 13th numbers?

The correct option is 73.5.

Step 1: Calculate the total sum of all 15 numbers
The average of 15 numbers is given as 72.
$\text{Total sum of 15 numbers}=15\times 72=1080$

Step 2: Calculate the sum of the first 6 numbers
The average of the first 6 numbers is 65.
$\text{Sum of the first 6 numbers}=6\times 65=390$

Step 3: Calculate the average and sum of the next 5 numbers (7th to 11th numbers)
The average of the next 5 numbers is 20% higher than the average of the first 6 numbers.
$\text{Average of next 5 numbers}=65+(20\%\text{of}65)=65+13=78$
Now, find the sum of these 5 numbers:
$\text{Sum of next 5 numbers}=5\times 78=390$

Step 4: Calculate the sum of the remaining 4 numbers (12th, 13th, 14th, and 15th)
Subtract the sum of the first 11 numbers (first 6 + next 5) from the total sum of 15 numbers:
$\text{Sum of first 11 numbers}=390+390=780$
$\text{Sum of remaining 4 numbers (12th + 13th + 14th + 15th)}=1080-780=300$

Step 5: Express the numbers in terms of a variable and find the 15th number
We are given that the 14th number is 78.
Let the 15th number be $x$.
- The 12th number is 8 less than the 15th number: $\text{12th number}=x-8$
- The 13th number is 5 greater than the 15th number: $\text{13th number}=x+5$
- The 14th number = 78
- The 15th number = $x$

Now, write the equation for the sum of these 4 numbers:
$(x-8)+(x+5)+78+x=300$
Combine like terms:
$3x+75=300$
$3x=300-75=225$
$x=75$

Step 6: Find the 12th and 13th numbers and their average
- 12th number = $75-8=67$
- 13th number = $75+5=80$

Now, compute their average:
$\text{Average}=\frac{67+80}{2}=\frac{147}{2}=73.5$