The average of five consecutive odd numbers is 61. What is the difference between the largest and the smallest of these five odd numbers?

Introduction / Context:
This problem uses the idea of consecutive odd numbers and their average. For any set of consecutive numbers (or consecutive odd or even numbers), the average is simply the middle term. Once we identify the sequence, finding the smallest and largest numbers and their difference becomes straightforward.

Given Data / Assumptions:

• There are 5 consecutive odd numbers.
• The average (arithmetic mean) of these 5 numbers is 61.
• We are asked to find: largest number - smallest number.

Concept / Approach:
For an odd count of consecutive numbers, the average equals the middle term. So, in a sequence of 5 consecutive odd numbers: n1, n2, n3, n4, n5 where n1 < n2 < n3 < n4 < n5 and the numbers differ by 2 each. Then: average = n3. So, we can directly find the middle odd number, then move two steps backward and two steps forward to find the smallest and largest numbers respectively.

Step-by-Step Solution:
Step 1: Since there are 5 consecutive odd numbers, the middle number is the average. Step 2: Therefore, the middle odd number = 61. Step 3: Consecutive odd numbers differ by 2. Step 4: The five numbers are: n1 = 61 - 4 = 57, n2 = 61 - 2 = 59, n3 = 61, n4 = 61 + 2 = 63, n5 = 61 + 4 = 65. Step 5: Smallest number = 57, largest number = 65. Step 6: Difference = largest - smallest = 65 - 57 = 8.
Verification / Alternative check:
Check average: (57 + 59 + 61 + 63 + 65) / 5. Sum = 57 + 59 = 116, +61 = 177, +63 = 240, +65 = 305. Average = 305 / 5 = 61, matching the given average.
Why Other Options Are Wrong:
Option A (4), C (12), D (16), and E (20) correspond to either too small or too large gaps for five consecutive odd numbers centered at 61. With five consecutive odd numbers, the difference between smallest and largest must be 4 steps of 2, i.e. 8, and cannot be any other value.
Common Pitfalls:
Some students mistakenly treat the average as the first or last term instead of the middle term. Others may incorrectly use common difference 1 instead of 2 for odd numbers.
Final Answer:
The difference between the highest and lowest numbers is 8.