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

Problems 8,9 and 10 use the data found in the accompanying paragraph and table:

Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and
France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)

The average price of his '70s stamps is closest to

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Prices per stamp: Brazil $6$¢, France $6$¢, Peru $4$¢, Spain $5$¢; '70s counts from the table: Brazil $12$, France $12$, Peru $6$, Spain $13$; Answer choices: (A) $3.5$ cents, (B) $4$ cents, (C) $4.5$ cents, (D) $5$ cents, (E) $5.5$ cents

Unknowns: The average price per stamp across all of Juan's '70s stamps

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

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.

Execute — Answer: E

#8 Analyze the Units 6.RP.A.3 Step 1

Subproblem 1a: cost of each country's '70s stamps.
Multiply price per stamp by the count in the '70s column.

$\text{Brazil: } 12 \times 6 = 72$¢, $\;\text{France: } 12 \times 6 = 72$¢, $\;\text{Peru: } 6 \times 4 = 24$¢, $\;\text{Spain: } 13 \times 5 = 65$¢

💡 Cents-per-stamp times stamps cancels "stamps" and gives cents — exactly the cost we need country by country.

#7 Identify Subproblems 6.SP.B.5 Step 2

Subproblem 1b: total cost.
Add the four country costs.

$$72 + 72 + 24 + 65 = 233 \text{ cents}$$

💡 All cents add up to one grand total — the numerator of the average.

#7 Identify Subproblems 6.SP.B.5 Step 3

Subproblem 2: total number of '70s stamps.
Add the four counts from the table.

$$12 + 12 + 6 + 13 = 43 \text{ stamps}$$

💡 All stamps count once — the denominator of the average.

#8 Analyze the Units 6.SP.B.5 Step 4

Combine the two subproblems.
Divide total cost by total stamps to get the average price per stamp, then pick the closest choice.

$$\dfrac{233 \text{ cents}}{43 \text{ stamps}} \approx 5.42 \text{ cents/stamp} \;\Rightarrow\; \text{closest to } 5.5 \;\Rightarrow\; \textbf{(E)}$$

💡 Cents divided by stamps cancels to cents per stamp. The value $5.42$ sits between $5$ and $5.5$, but $|5.42 - 5.5| = 0.08 < |5.42 - 5| = 0.42$, so $5.5$ wins.

[1] #8 6.RP.A.3 Subproblem 1a: cost of each country's '70s stamps. Multiply price per stamp by t

[2] #7 6.SP.B.5 Subproblem 1b: total cost. Add the four country costs.

[3] #7 6.SP.B.5 Subproblem 2: total number of '70s stamps. Add the four counts from the table.

[4] #8 6.SP.B.5 Combine the two subproblems. Divide total cost by total stamps to get the averag

Review

Reasonableness: 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.

Alternative: Tool #6 (Guess and Check) on the choices: at average $= 5.5$ cents over $43$ stamps, the total would be $5.5 \times 43 = 236.5$¢; at average $= 5$ cents the total would be $5 \times 43 = 215$¢. The actual total $233$¢ is much closer to $236.5$ than to $215$ (gap of $3.5$ vs. $18$), so $5.5$ wins, confirming (E).

CCSS standards used (min grade 6)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Multiplying the rate cents-per-stamp by the count of stamps for each country to get cents: $12 \times 6 = 72$, $6 \times 4 = 24$, $13 \times 5 = 65$.)
6.SP.B.5 Summarize numerical data sets, including reporting the number of observations and measures of center (Using the definition of mean (total $\div$ count) on the combined data: $233 \text{ cents} \div 43 \text{ stamps} \approx 5.42$ cents per stamp.)

⭐ 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).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

More like this