What is the average of all integers between 200 and 350 that are exactly divisible by 11?

The correct answer is 275.

To find the average of all integers between 200 and 350 that are exactly divisible by 11, we first need to determine the arithmetic progression (AP) formed by these numbers.

Step 1: Find the first integer greater than 200 divisible by 11.
Dividing 200 by 11 gives:

$200÷11=18.18$

The next integer is 19. So, the first multiple of 11 after 200 is:

$11\times 19=209$

Step 2: Find the largest integer less than 350 divisible by 11.
Dividing 350 by 11 gives:

$350÷11=31.81$

Taking the integer part 31, the last multiple of 11 before 350 is:

$11\times 31=341$

Step 3: Calculate the average of the sequence.
The numbers form an arithmetic progression where:
First term, a = 209
Last term, l = 341
Common difference, d = 11

For any arithmetic progression, the average of all the terms is simply the average of the first and last terms:

$\text{Average}=\frac{a+l}{2}$

Substituting the values:

$\text{Average}=\frac{209+341}{2}=\frac{550}{2}=275$

Thus, the average of all integers between 200 and 350 that are divisible by 11 is 275.