AMC 8 · 2008 · #10

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

mean-median-mode-rangefraction-arithmeticmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: mean-median-mode-rangemulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Room A has 6 people whose ages average 40. Room B has 4 people whose ages average 25. If everyone gathers in one room, what is the average age of the combined group of 10 people?

Pick an answer.

(A)

32.5

(B)

(C)

33.5

(D)

(E)

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Transform the Structure

You cannot just average 40 and 25 — the two rooms have different sizes, so a plain $\frac{40+25}{2}$ throws away that fact. Tool #16 (Transform the Structure) says: turn each average back into a total. Once both rooms are described by their total age (sum), the combined room is just sum-of-sums divided by count-of-counts. Tool #7 (Identify Subproblems) breaks the work into two clean steps — find each room's total, then merge.

1STEP 1

Room A's total age is its average times its count: 40 across 6 people gives 240.

Room A total = 6 × 40 = 240

2STEP 2

Room B's total age the same way: 25 across 4 people gives 100.

Room B total = 4 × 25 = 100

3STEP 3

Pour both rooms together: the totals add to 340 across 10 people.

Combined total = 240 + 100 = 340, Combined count = 6 + 4 = 10

4STEP 4

Apply the average formula once more to the whole group: 340 over 10 is 34.

Average =

\frac{340}{10}

= 34 → (D)

Answer

The answer 34 sits between 25 (Room B's average) and 40 (Room A's average), which it has to — a combined average can never escape the range of the two parts. It also sits closer to 40 than to 25, which fits because Room A has more people (6 vs 4), so it pulls the average toward 40. A quick gut check: midpoint of 25 and 40 is 32.5; we should land above 32.5 because the bigger group is the older one, and indeed 34 ＞ 32.5.

💡Key takeaway

When two groups of different sizes are combined, you can't just average the averages. Turn each average back into a total, add the totals, and divide by the new headcount — the bigger group always pulls harder.

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