The average of 12 numbers is 75. The average of the first 5 numbers is 68. The average of the next 5 numbers is 20% more than the average of the first 5 numbers. The 11th number is 12 more than the 12th number. What is the average of the 11th and 12th numbers?

The correct answer is Option 76.

Let's solve the problem step-by-step by finding the sum of each group of numbers.

Step 1: Find the total sum of all 12 numbers.
The formula for the total sum is:
Sum = Average × Count

Given that the average of 12 numbers is 75:

$\text{Total Sum}=12\times 75=900$

Step 2: Find the sum of the first 5 numbers.
The average of the first 5 numbers is 68.

$\text{Sum of first 5 numbers}=5\times 68=340$

Step 3: Find the average and sum of the next 5 numbers (6th to 10th).
The average of the next 5 numbers is 20% more than the average of the first 5 numbers (68):

$\text{Average of next 5 numbers}=68+(0.20\times 68)=68+13.6=81.6$

Now, compute the sum of these 5 numbers:

$\text{Sum of next 5 numbers}=5\times 81.6=408$

Step 4: Find the sum of the remaining two numbers (11th and 12th numbers).
Subtract the sum of the first 10 numbers (first 5 + next 5) from the total sum of 12 numbers:

$\text{Sum of first 10 numbers}=340+408=748$

$\text{Sum of 11th and 12th numbers}=900-748=152$

Step 5: Find the average of the 11th and 12th numbers.
The average of any two numbers is their sum divided by 2:

$\text{Average of 11th and 12th numbers}=\frac{152}{2}=76$

Thus, the average of the 11th and 12th numbers is 76.