AMC 8 · 2003 · #10

Grade 6 geometry-2d

Archetype Arithmetic Expression Decomposition

area-trianglesarea-rectanglesratio-proportionfraction-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: area-trianglesmulti-digit-arithmeticratio-proportion

📏 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:

How many cookies will be in one batch of Trisha's cookies?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Four friends bake cookies of the same thickness using equal amounts of dough per batch. Art's trapezoid cookies have parallel bases $5$ in and $3$ in with height $3$ in, and one batch contains $12$ of them. Trisha's cookies are right triangles with legs $3$ in and $4$ in. How many cookies are in one batch of Trisha's?

Givens: All cookies have the same thickness, so dough $\propto$ top area; Each batch uses the same total amount of dough; Art's cookie is a trapezoid with bases $5$ in and $3$ in, height $3$ in; Art's batch has $12$ cookies (from the shared problem hint); Trisha's cookie is a right triangle with legs $3$ in and $4$ in; Answer choices: (A) $10$, (B) $12$, (C) $16$, (D) $18$, (E) $24$

Unknowns: The number of cookies in one batch of Trisha's

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units

Same thickness plus same total dough means dough amount is just top area. So the whole question splits into three independent subproblems (Tool #7): (a) area of one Art cookie, (b) total dough area = $12$ Art cookies, (c) area of one Trisha cookie. Then Trisha's batch count is just (b) $\div$ (c). Tool #8 (Analyze the Units) keeps the bookkeeping honest: in $^2$ per cookie times cookies gives in $^2$ of dough, then in $^2$ of dough divided by in $^2$ per cookie returns plain cookies — the units cancel cleanly, which is the whole reason this approach works.

Execute — Answer: E

#7 Identify Subproblems 6.G.A.1 Step 1

Subproblem (a): area of one Art cookie.
The trapezoid has parallel bases $b_1 = 5$ in, $b_2 = 3$ in and height $h = 3$ in.

$$A_{\text{Art}} = \tfrac{1}{2}(5 + 3)\,(3) = \tfrac{1}{2}\,(8)(3) = 12 \text{ in}^2$$

💡 Grade 6 "area of special quadrilaterals" — average the two bases, multiply by the height.

#8 Analyze the Units 6.RP.A.3 Step 2

Subproblem (b): total dough area in one batch.
Art makes $12$ cookies, each $12$ in$^2$, so the batch uses $12 \times 12$ square inches of dough.
Because every friend uses the same amount of dough, this is also Trisha's batch total.

$$D = 12 \text{ cookies} \times 12 \tfrac{\text{in}^2}{\text{cookie}} = 144 \text{ in}^2$$

💡 Multiplying cookies by in$^2$ per cookie cancels "cookies" and leaves in$^2$ — exactly the dough total we need.

#7 Identify Subproblems 6.G.A.1 Step 3

Subproblem (c): area of one Trisha cookie.
The right triangle has legs $3$ in and $4$ in, which serve as base and height.

$$A_{\text{Trisha}} = \tfrac{1}{2}\,(3)(4) = 6 \text{ in}^2$$

💡 Grade 6 "area of right triangles" — half of base times height.

#8 Analyze the Units 6.RP.A.3 Step 4

Combine the subproblems.
Divide the total dough by the area of one Trisha cookie to count how many fit.

$$\dfrac{D}{A_{\text{Trisha}}} = \dfrac{144 \text{ in}^2}{6 \text{ in}^2/\text{cookie}} = 24 \text{ cookies} \;\Rightarrow\; \textbf{(E)}$$

💡 Dividing in$^2$ by in$^2$ per cookie cancels in$^2$ and returns "cookies" — the unit check confirms the answer is a count.

[1] #7 6.G.A.1 Subproblem (a): area of one Art cookie. The trapezoid has parallel bases $b_1 =

[2] #8 6.RP.A.3 Subproblem (b): total dough area in one batch. Art makes $12$ cookies, each $12$

[3] #7 6.G.A.1 Subproblem (c): area of one Trisha cookie. The right triangle has legs $3$ in an

[4] #8 6.RP.A.3 Combine the subproblems. Divide the total dough by the area of one Trisha cookie

Review

Reasonableness: Sanity check by comparing cookie sizes. Trisha's cookie ($6$ in$^2$) is exactly half the area of Art's ($12$ in$^2$), so with the same dough Trisha should bake twice as many cookies: $2 \times 12 = 24$. That matches (E). The trap answers track common slips: (B) $12$ copies Art's count without rescaling, (A) $10$ and (C) $16$ come from miscomputing one of the areas, and (D) $18$ matches forgetting the $\tfrac{1}{2}$ in the triangle area ($\tfrac{144}{3 \cdot 4} = 12$ then adjusted wrongly). Only (E) survives.

Alternative: Tool #5 (Look for a Pattern) via direct ratio: skip the dough total altogether. Number of cookies is inversely proportional to area per cookie, so $\dfrac{N_{\text{Trisha}}}{N_{\text{Art}}} = \dfrac{A_{\text{Art}}}{A_{\text{Trisha}}} = \dfrac{12}{6} = 2$. Then $N_{\text{Trisha}} = 2 \times 12 = 24$. Same answer (E) without ever computing the $144$ in$^2$ total.

CCSS standards used (min grade 6)

6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons (Computing the trapezoid area $\tfrac{1}{2}(5+3)(3) = 12$ for Art's cookie and the right-triangle area $\tfrac{1}{2}(3)(4) = 6$ for Trisha's cookie.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Using the rate "in$^2$ per cookie" to convert between total dough and cookie count: $12 \text{ cookies} \times 12 \tfrac{\text{in}^2}{\text{cookie}} = 144 \text{ in}^2$, then $144 \text{ in}^2 \div 6 \tfrac{\text{in}^2}{\text{cookie}} = 24$ cookies.)

⭐ Same dough, same thickness, so cookie count just depends on each cookie's area. Trisha's cookie is half the area of Art's, so she bakes twice as many: $2 \times 12 = 24$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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