AMC 8 · 2003 · #10

Grade 6 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

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

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

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?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

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) ÷ (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.

1STEP 1

Subproblem (a): average Art's trapezoid bases (5 and 3 in), then multiply by the height for one cookie's area, 12 in².

A_Art =

\frac{1}{2}

(5 + 3) (3) =

\frac{1}{2}

(8)(3) = 12 in²

2STEP 2

Subproblem (b): Art bakes 12 such cookies, so a batch uses 144 in² of dough — the same total every friend gets.

D = 12 cookies ×

\frac{12\text{ in}^2}{\text{cookie}}

= 144 in²

3STEP 3

Subproblem (c): Trisha's right triangle has legs 3 and 4 in as base and height, giving one cookie's area 6 in².

A_Trisha =

\frac{1}{2}

(3)(4) = 6 in²

4STEP 4

Combine: divide the batch's 144 in² of dough by one Trisha cookie's 6 in² to get 24 cookies.

\frac{D}{A_{Trisha}}

\frac{144\text{ in}^2}{6\text{ in}^2/\text{cookie}}

= 24 cookies → (E)

Answer

Sanity check by comparing cookie sizes. Trisha's cookie (6 in²) is exactly half the area of Art's (12 in²), so with the same dough Trisha should bake twice as many cookies: 2 × 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

\frac{1}{2}

in the triangle area (

\frac{144}{3 \cdot 4}

= 12 then adjusted wrongly). Only (E) survives.

💡Key takeaway

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 × 12 = 24.

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