AMC 8 · 2003 · #8

Grade 6 geometry-2d

Archetype Small-Domain Constrained Search

area-rectanglesarea-trianglesratio-proportionsystematic-enumeration systematic-enumerationidentify-subproblems ↑ Prerequisites: area-rectanglesarea-trianglesmulti-digit-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

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

Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.

$\circ$ Art's cookies are trapezoids.

$\circ$ Roger's cookies are rectangles.

$\circ$ Paul's cookies are parallelograms.

$\circ$ Trisha's cookies are triangles.

Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough?

Pick an answer.

(A)

Art

(B)

Roger

(C)

Paul

(D)

Trisha

(E)

There is a tie for fewest.

View mode:

Toolkit + CCSS Solution

Understand

Restated: Four friends use the same amount of cookie dough and the same thickness for their cookies. Art's cookies are trapezoids with parallel sides $3$ in and $5$ in and height $3$ in. Roger's are $4 \times 2$ rectangles. Paul's are parallelograms with base $3$ in and height $2$ in. Trisha's are right triangles with legs $3$ in and $4$ in. Art makes exactly $12$ cookies. Who makes the fewest cookies?

Givens: Art's cookie: trapezoid with parallel sides $3$ in and $5$ in, height $3$ in; Roger's cookie: rectangle, $4$ in $\times\ 2$ in; Paul's cookie: parallelogram, base $3$ in, height $2$ in; Trisha's cookie: right triangle, legs $3$ in and $4$ in; All cookies have the same thickness, and each friend uses the same amount of dough; Art makes $12$ cookies; Answer choices: (A) Art, (B) Roger, (C) Paul, (D) Trisha, (E) tie for fewest

Unknowns: Which friend bakes the fewest cookies from one batch of dough

Understand

Plan

Primary tool: #1 Draw a Diagram

Secondary: #11 Find an Invariant

The figures are already drawn for us, so Tool #1 (Draw a Diagram) just means reading each shape and computing its top area with the right formula. Tool #11 (Find an Invariant) is the key idea: because every friend uses the same amount of dough and the same thickness, the total top area per friend is the same constant. That means the number of cookies is inversely proportional to one cookie's area — the friend with the biggest cookie ends up with the fewest cookies. So the question "who makes the fewest" becomes "who has the largest single-cookie area."

Execute — Answer: A

#1 Draw a Diagram 6.G.A.1 Step 1

Compute Art's cookie area.
The trapezoid has parallel sides $3$ in and $5$ in with height $3$ in.
Use the trapezoid area formula: half the sum of the parallel sides times the height.

$$A_{\text{Art}} = \dfrac{3 + 5}{2} \times 3 = 4 \times 3 = 12 \text{ in}^2$$

💡 Grade 6 area of a trapezoid: average the two parallel sides, then multiply by the height.

#1 Draw a Diagram 6.G.A.1 Step 2

Compute Roger's cookie area.
The rectangle is $4$ in by $2$ in.

$$A_{\text{Roger}} = 4 \times 2 = 8 \text{ in}^2$$

💡 Grade 6 area of a rectangle: length times width.

#1 Draw a Diagram 6.G.A.1 Step 3

Compute Paul's cookie area.
The parallelogram has base $3$ in and height $2$ in (the dashed segment in the figure marks the perpendicular height).

$$A_{\text{Paul}} = 3 \times 2 = 6 \text{ in}^2$$

💡 Grade 6 area of a parallelogram: base times perpendicular height.

#1 Draw a Diagram 6.G.A.1 Step 4

Compute Trisha's cookie area.
The triangle is right-angled with legs $3$ in and $4$ in, so the two legs serve as base and height.

$$A_{\text{Trisha}} = \dfrac{1}{2} \times 3 \times 4 = 6 \text{ in}^2$$

💡 Grade 6 area of a right triangle: half of base times height.

#11 Find an Invariant 6.RP.A.3 Step 5

Use the invariant.
Same dough and same thickness means the total cookie area per friend is the same.
The number of cookies is inversely proportional to one cookie's area, so the friend with the largest single-cookie area makes the fewest cookies.
Compare: $12 > 8 > 6 = 6$.

$$12 > 8 > 6 = 6 \;\Rightarrow\; \text{Art's cookie is largest} \;\Rightarrow\; \textbf{(A)}$$

💡 Constant total area split into cookies of size $a$ gives count $=$ (total)$/a$. Bigger $a$ means smaller count.

[1] #1 6.G.A.1 Compute Art's cookie area. The trapezoid has parallel sides $3$ in and $5$ in wi

[2] #1 6.G.A.1 Compute Roger's cookie area. The rectangle is $4$ in by $2$ in.

[3] #1 6.G.A.1 Compute Paul's cookie area. The parallelogram has base $3$ in and height $2$ in

[4] #1 6.G.A.1 Compute Trisha's cookie area. The triangle is right-angled with legs $3$ in and

[5] #11 6.RP.A.3 Use the invariant. Same dough and same thickness means the total cookie area per

Review

Reasonableness: Pin down the constant. Art makes $12$ cookies of area $12$ in$^2$ each, so the total top area per batch is $12 \times 12 = 144$ in$^2$. Then Roger bakes $144 / 8 = 18$ cookies, Paul bakes $144 / 6 = 24$, and Trisha bakes $144 / 6 = 24$. The counts are $12, 18, 24, 24$ — Art has the fewest, matching (A). The tie answer (E) is a trap: Paul and Trisha do tie, but for the most cookies, not the fewest.

Alternative: Tool #6 (Guess and Check) on the answer choices: the question asks who bakes fewest, so test "biggest cookie wins." Without any algebra, eyeball the four shapes — Art's trapezoid clearly covers more area than the others, while Paul's parallelogram and Trisha's triangle look about the same size. Confirm by computing only Art's area ($12$) and one other (say Roger's $8$): Art's is biggest, so Art bakes the fewest. Answer (A).

CCSS standards used (min grade 6)

6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles (Computing the four single-cookie areas: trapezoid $12$, rectangle $8$, parallelogram $6$, right triangle $6$.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Reading "same total dough, same thickness" as a constant total area, so the cookie count is inversely proportional to one cookie's area — largest area means fewest cookies.)

⭐ When everyone uses the same amount of dough at the same thickness, fewer cookies means each one is bigger. Compute each shape's area with the Grade 6 polygon formulas, and the friend with the largest cookie is the one who bakes the fewest.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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