AMC 8 · 2002 · #3

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

mean-median-mode-rangemultiplesmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 1 insight

Problem

Pick four different positive even integers. What is the smallest average their four values can have?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

Since the average is the sum divided by 4, the smallest average comes from the smallest sum. To get the smallest sum from four distinct positive even integers, list the positive even integers in order (2, 4, 6, 8, 10, …) and take the first four. Tool #2 (Make a Systematic List) gives us a strict ordering so we cannot miss a smaller candidate.

1STEP 1

List the positive even integers in order — the first four are the smallest distinct choices.

2, 4, 6, 8, 10, 12, …

2STEP 2

The first four are 2, 4, 6, 8 — all distinct positive even numbers — and their sum is 20.

2 + 4 + 6 + 8 = 20

3STEP 3

Dividing the sum by four gives the average 5 — no other valid set has a smaller sum, so this is the minimum.

\frac{20}{4}

= 5 → (C)

Answer

Check the answer is achievable and minimal. The set {2, 4, 6, 8} gives average 5 — a real example, so 5 is reachable. Now try to beat it: any other set of four distinct positive even integers must drop one of these four and pick a different even number that is at least 10 (the next even integer not in the set). Swapping, say, 8 for 10 raises the sum from 20 to 22, raising the average to 5.5. No swap can lower the sum, so 5 is indeed the smallest possible average.

💡Key takeaway

Smallest average comes from the smallest sum — so reach for the smallest four positive even integers and take their mean.

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