What is the average of all the numbers between 9 and 90 (inclusive) that are exactly divisible by 8?

Introduction / Context:
This aptitude question checks your understanding of arithmetic progressions and averages of evenly spaced numbers. When numbers are equally spaced, such as multiples of a fixed number within a range, there is a very simple shortcut for finding their average without adding all of them one by one.

Given Data / Assumptions:

We consider numbers between 9 and 90 that are divisible by 8.
These numbers are positive integers and must be multiples of 8.
We take the range as inclusive where applicable, meaning we check if endpoints are multiples of 8.
We are required to find the average of all such multiples of 8 within the range.

Concept / Approach:
Multiples of 8 within a continuous range form an arithmetic progression (AP) with a common difference of 8. For any arithmetic progression, the average (or mean) of all its terms is simply equal to the average of the first and last terms, that is, (first term + last term) / 2. So our task reduces to identifying the smallest and largest multiples of 8 between 9 and 90 and applying this property.

Step-by-Step Solution:
Step 1: Find the first multiple of 8 greater than 9. Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, ... The first multiple of 8 greater than 9 is 16. Step 2: Find the last multiple of 8 less than or equal to 90. Looking at the list, 88 is the last multiple of 8 below 90. Step 3: Use the AP average shortcut. Average of all these multiples = (first + last) / 2 = (16 + 88) / 2 = 104 / 2 = 52.

Verification / Alternative Check:
The full list of multiples of 8 between 9 and 90 is 16, 24, 32, 40, 48, 56, 64, 72, 80 and 88. These are 10 terms with constant difference 8. The middle pair is 48 and 56 and their average is (48 + 56) / 2 = 52. In a symmetric arithmetic progression, this must equal the overall average, which confirms our previous result.

Why Other Options Are Wrong:
50 and 51 are slightly lower than the correct average and would correspond to a range whose endpoints are closer together than 16 and 88. They do not fit the true arithmetic progression here.
53 is slightly higher than the correct average, again inconsistent with the symmetric placement of numbers around 52.

Common Pitfalls:
Many students try to add all the numbers and divide, which is time consuming and error prone. Others mistakenly average 9 and 90 or misidentify the first or last multiple of 8 in the interval. Remember to always first list or identify the exact multiples within the range and then use the (first + last) / 2 shortcut for arithmetic progressions.

Final Answer:
The required average of all numbers between 9 and 90 that are divisible by 8 is 52.