Problem 10 · 2002 AMC 8
Medium
Arithmetic & Operations
read-tabletotal-then-divideestimate-and-pick
Juan organizes the stamps in his collection by country and by the decade in which they were issued. He paid these prices at the stamp shop: Brazil and France, 6¢ each; Peru, 4¢ each; and Spain, 5¢ each. (Brazil and Peru are South American countries; France and Spain are European.) The table shows how many stamps he has from each country and decade.
The average price of his 1970s stamps is closest to which value?
| Country | '50s | '60s | '70s | '80s |
|---|---|---|---|---|
| Brazil | 4 | 7 | 12 | 8 |
| France | 8 | 4 | 12 | 15 |
| Peru | 6 | 4 | 6 | 10 |
| Spain | 3 | 9 | 13 | 9 |
Show answer
Answer: E — About 5.4 cents.
Show hints
Hint 1 of 2
Tempting trap: averaging the four prices (6, 6, 4, 5) gives 5.25 — but that pretends each country sent the same number of stamps. They didn't.
Still stuck? Show hint 2 →
Hint 2 of 2
Real average price = (total money spent on '70s stamps) ÷ (total '70s stamps). The counts *weight* the average, so the prices with more stamps pull harder.
Show solution
Approach: weighted average over the 1970s column
- The average price is total cost over total count — *not* the average of the listed prices, because the four countries contribute different numbers of stamps (that's a weighted average).
- '70s column: Brazil 12 & France 12 at 6¢, Peru 6 at 4¢, Spain 13 at 5¢. Cost = 12×6 + 12×6 + 6×4 + 13×5 = 72 + 72 + 24 + 65 = 233¢, over 12 + 12 + 6 + 13 = 43 stamps.
- 233 ÷ 43 ≈ 5.4 cents.
- *Sanity check:* most of these stamps cost 5¢ or 6¢ and only a handful cost 4¢, so the average should sit high in that 4-to-6 range — 5.4 fits; a naive 5.25 would have under-counted the many 6¢ stamps.
Mark:
· log in to save