AMC 8 · 2002 · #9

Easy mode Grade 4

Problem

Juan keeps a stamp collection. He sorts his stamps by the country they came from and by the decade (the '50s, '60s, '70s, or '80s) when they were made.

He bought each stamp at the same shop. The price per stamp depends on the country:

Brazil: 6 cents each
France: 6 cents each
Peru: 4 cents each
Spain: 5 cents each

Brazil and Peru are in South America. France and Spain are in Europe.

The table below shows how many stamps Juan has from each country in each decade.

Look at just his South American stamps — that means Brazil and Peru. And look at just the ones made before the '70s — that means the '50s and '60s. How much did those stamps cost him in total? Give your answer in dollars and cents.

Pick an answer.

(A)

$0.40

(B)

$1.06

(C)

$1.80

(D)

$2.38

(E)

$2.64

View mode:

Toolkit + CCSS Solution

Understand

Restated: Juan's stamps are grouped by country (Brazil, France, Peru, Spain) and by decade (50s, 60s, 70s, 80s). Brazilian and French stamps cost $6$ cents each, Peruvian stamps $4$ cents each, and Spanish stamps $5$ cents each. Brazil and Peru are the South American countries. Find the total cost, in dollars and cents, of his South American stamps that were issued before the $70$s.

Givens: Per-stamp prices: Brazil $6$ cents, France $6$ cents, Peru $4$ cents, Spain $5$ cents; South American countries in the table: Brazil and Peru; Decades before the $70$s in the table: $50$s and $60$s; From the table: Brazil $50$s $= 4$, Brazil $60$s $= 7$, Peru $50$s $= 6$, Peru $60$s $= 4$; Answer choices: (A) $\$0.40$, (B) $\$1.06$, (C) $\$1.80$, (D) $\$2.38$, (E) $\$2.64$

Unknowns: The total cost in dollars and cents of the matching stamps

Understand

Plan

Primary tool: #7 Break into Subproblems

Secondary: #2 Make an Organized List

The table has $16$ cells but the question only cares about a small rectangle of them: two rows (Brazil, Peru) by two columns ($50$s, $60$s). Tool #2 (Make an Organized List) pulls out exactly those four cells with the right per-stamp price attached. Tool #7 (Break into Subproblems) then handles them one country at a time: total Brazil stamps from those decades times $6$ cents, total Peru stamps times $4$ cents, then add. Splitting by country is the right cut because price changes with country, not with decade.

Execute — Answer: B

#2 Make an Organized List 3.MD.B.3 Step 1

List the four cells the question selects.
"South American" keeps Brazil and Peru; "before the $70$s" keeps the $50$s and $60$s columns.
From the table, Brazil has $4$ stamps in the $50$s and $7$ in the $60$s; Peru has $6$ in the $50$s and $4$ in the $60$s.

$$\text{Brazil: } 4 \text{ (50s)}, 7 \text{ (60s)} \qquad \text{Peru: } 6 \text{ (50s)}, 4 \text{ (60s)}$$

💡 Grade 3 scaled data tables: read off only the cells the question asks for, ignore the rest.

#7 Break into Subproblems 4.OA.A.3 Step 2

Count Brazil's stamps from the two decades, then multiply by the $6$-cent Brazil price.

$$\text{Brazil cost} = (4 + 7) \times 6 = 11 \times 6 = 66 \text{ cents}$$

💡 Add first, then multiply: one country, one price, so a single multiplication finishes the Brazil subtotal.

#7 Break into Subproblems 4.OA.A.3 Step 3

Do the same for Peru, this time with the $4$-cent Peru price.

$$\text{Peru cost} = (6 + 4) \times 4 = 10 \times 4 = 40 \text{ cents}$$

💡 Peru is a separate subproblem because its per-stamp price differs from Brazil's.

#7 Break into Subproblems 4.MD.A.2 Step 4

Add the two country subtotals and convert from cents to dollars by dividing by $100$.

$66 + 40 = 106 \text{ cents} = \$1.06 \;\Rightarrow\; \textbf{(B)}$

💡 Grade 4 money: $100$ cents make $\$1$, so $106$ cents is one dollar and six cents.

[1] #2 3.MD.B.3 List the four cells the question selects. "South American" keeps Brazil and Peru

[2] #7 4.OA.A.3 Count Brazil's stamps from the two decades, then multiply by the $6$-cent Brazil

[3] #7 4.OA.A.3 Do the same for Peru, this time with the $4$-cent Peru price.

[4] #7 4.MD.A.2 Add the two country subtotals and convert from cents to dollars by dividing by $

Review

Reasonableness: Check the filter held: France ($6$ cents each) and Spain ($5$ cents each) were dropped (not South American), and the $70$s and $80$s columns were dropped (not before the $70$s). Eleven Brazil stamps at $6$ cents each is $\$0.66$; ten Peru stamps at $4$ cents each is $\$0.40$; together $\$1.06$, matching (B). The trap (A) $\$0.40$ is Peru only, (C) $\$1.80$ includes the $70$s, and (E) $\$2.64$ includes all four decades — easy to spot once you know the right filter.

Alternative: Tool #2 (Make an Organized List) on its own: write the four rows as (country, decade, count, price): (Brazil, 50s, $4$, $6$ cents), (Brazil, 60s, $7$, $6$ cents), (Peru, 50s, $6$, $4$ cents), (Peru, 60s, $4$, $4$ cents). Compute each row's cost ($24, 42, 24, 16$ cents) and add: $24 + 42 + 24 + 16 = 106$ cents $= \$1.06$. Answer (B).

CCSS standards used (min grade 4)

3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs (Reading the four relevant cells (Brazil/Peru in the $50$s/$60$s columns) out of the stamp count table.)
4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations (Adding stamp counts within each country and then multiplying by that country's per-stamp price ($11 \times 6 = 66$, $10 \times 4 = 40$).)
4.MD.A.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals (Combining the two country subtotals as $66 + 40 = 106$ cents and converting $106$ cents to $\$1.06$.)

⭐ Big tables hide small questions. Use the wording to throw out everything you don't need (France, Spain, the $70$s and $80$s), then handle each country at its own price. Two short multiplications and an add — $11 \times 6 + 10 \times 4 = 106$ cents $= \$1.06$ — finish the problem.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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