AMC 8 · 2002 · #10

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

mean-median-mode-rangegraph-readingmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

Juan paid Brazil and France 6 cents per stamp, Peru 4 cents per stamp, and Spain 5 cents per stamp. In the '70s column of his table he has 12 Brazil, 12 France, 6 Peru, and 13 Spain stamps. Find the average price per stamp for his '70s stamps, rounded to the closest choice.

Pick an answer.

(A)

$3.5 \text{ cents}$

(B)

$4 \text{ cents}$

(C)

$4.5 \text{ cents}$

(D)

$5 \text{ cents}$

(E)

$5.5 \text{ cents}$

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

How to solve

Strategy Analyze the Units

The unit of the answer is "cents per stamp," which Tool #8 (Analyze the Units) turns into a recipe: multiply (cents/stamp) by (stamps) to get cents for each country, add those to get total cents, then divide by total stamps to cancel "stamps" and leave "cents per stamp." Tool #7 (Identify Subproblems) breaks the bookkeeping into two clean subproblems — total cost and total count — that are computed independently and then divided. Together they keep a four-country weighted average from turning into a tangle.

1STEP 1

Multiply each country's price per stamp by its '70s count to get its cost.

Brazil: 12 × 6 = 72¢, France: 12 × 6 = 72¢, Peru: 6 × 4 = 24¢, Spain: 13 × 5 = 65¢

2STEP 2

Add the four country costs to get the total: 233 cents.

72 + 72 + 24 + 65 = 233 cents

3STEP 3

Add the four counts from the table to get the total: 43 stamps.

12 + 12 + 6 + 13 = 43 stamps

4STEP 4

Divide 233 by 43: about 5.42 cents per stamp, so the closest choice is (E).

(233 cents)/(43 stamps) ≈ 5.42 cents/stamp → closest to 5.5 → (E)

Answer

5.5 cents

Sanity check via bounds. Every '70s stamp costs between 4¢ (Peru) and 6¢ (Brazil/France), so the average must lie strictly between 4 and 6. That immediately rules out (A) 3.5 and any value outside this band. Most '70s stamps cost 5¢ or 6¢ (12 + 12 + 13 = 37 of 43), with only 6 Peru stamps at 4¢, so the average should sit above 5 and below 6 — closer to the 6 end than the 4 end. The computed 5.42 matches that picture, and 5.5 is the closest choice.

💡Key takeaway

A weighted average is just total cost divided by total count. Add up what Juan paid for his '70s stamps (233¢) and divide by how many he has (43) to get about 5.42¢ — the closest choice is 5.5¢, answer (E).

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