Problem 3 · 2002 AMC 8
Easy
Arithmetic & Operations
total-then-divide
What is the smallest possible average of four distinct positive even integers?
Show answer
Answer: C — 5.
Show hints
Hint 1 of 2
A smaller average comes from smaller numbers — and "distinct" forbids reusing one, so grab the four tiniest even numbers there are.
Still stuck? Show hint 2 →
Hint 2 of 2
Those four are evenly spaced. For an evenly-spaced list, the average is just the middle — you can skip the adding.
Show solution
Approach: use the four smallest distinct positive even integers
- To shrink an average, feed it the smallest allowed values. "Distinct positive even" means the four smallest are 2, 4, 6, 8.
- These are evenly spaced, so their average sits dead center — halfway between 4 and 6, which is 5. (Check: (2 + 4 + 6 + 8) ÷ 4 = 20 ÷ 4 = 5.)
- *Worth keeping:* the mean of any evenly-spaced list equals its middle value (the average of the first and last) — no summing required.
Mark:
· log in to save