AMC 8 · 2003 · #9

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

ratio-proportionarea-rectanglesarea-trianglesfraction-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: area-rectanglesmulti-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:

Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Four friends bake cookies of equal thickness using equal amounts of dough. Art's $12$ trapezoid cookies (bases $5$ in and $3$ in, height $3$ in) sell at $60$ cents each. Roger's cookies are $4 \text{ in} \times 2 \text{ in}$ rectangles. What price per Roger cookie makes Roger's whole batch earn the same total as Art's batch?

Givens: Art's cookies are trapezoids with parallel bases $5$ in and $3$ in, height $3$ in; Roger's cookies are rectangles $4$ in by $2$ in; Art makes $12$ cookies; Art sells each cookie for $60$ cents; Each friend uses the same amount of dough; all cookies are the same thickness; Answer choices: (A) $18$, (B) $25$, (C) $40$, (D) $75$, (E) $90$

Unknowns: The price in cents Roger should charge per cookie

Understand

Plan

Primary tool: #7 Break into Subproblems

Secondary: #1 Draw a Diagram

The final price hides behind a chain of small questions, so Tool #7 (Break into Subproblems) is the cleanest fit. Split the work into four sub-jobs: (a) area of one Art cookie, (b) total dough area for Art's batch, (c) area of one Roger cookie, and (d) how many cookies Roger gets from the same dough. Tool #1 (Draw a Diagram) is already done for us by the problem's figures — we just read off the bases, heights, and side lengths to feed each area formula. Once Roger's cookie count is known, matching Art's revenue is one division.

Execute — Answer: C

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

Find the area of one Art cookie.
The figure shows a trapezoid with parallel bases of $5$ in and $3$ in and height $3$ in (the right-angle mark tells us the $3$-in vertical side is the height).

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

💡 Grade 6 area-of-trapezoid formula: average the two parallel bases, then multiply by the height.

#7 Break into Subproblems 6.G.A.1 Step 2

Find the total dough area for Art's batch.
Art makes $12$ cookies of $12$ in$^2$ each.

$$T = 12 \times 12 = 144 \text{ in}^2$$

💡 Equal thickness means dough volume is proportional to total surface area, so $144$ in$^2$ is the shared dough budget for every friend.

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

Find the area of one Roger cookie.
The figure shows a $4 \text{ in} \times 2 \text{ in}$ rectangle.

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

💡 Grade 6 rectangle area: length times width.

#7 Break into Subproblems 6.RP.A.3 Step 4

Find how many cookies Roger makes by dividing the shared dough area by one cookie's area.

$$N_{\text{Roger}} = \dfrac{144}{8} = 18 \text{ cookies}$$

💡 Same dough, smaller cookies, so Roger gets more pieces — $18 > 12$ is the expected direction.

#7 Break into Subproblems 6.RP.A.3 Step 5

Match revenue.
Art earns $12 \times 60 = 720$ cents.
To match that, Roger's $18$ cookies must average that same total, so each one costs $720 \div 18$.

$$\text{price} = \dfrac{720}{18} = 40 \text{ cents} \;\Rightarrow\; \textbf{(C)}$$

💡 Roger sells more cookies than Art, so each must cost less than $60$ cents — $40$ fits.

[1] #1 6.G.A.1 Find the area of one Art cookie. The figure shows a trapezoid with parallel base

[2] #7 6.G.A.1 Find the total dough area for Art's batch. Art makes $12$ cookies of $12$ in$^2$

[3] #1 6.G.A.1 Find the area of one Roger cookie. The figure shows a $4 \text{ in} \times 2 \te

[4] #7 6.RP.A.3 Find how many cookies Roger makes by dividing the shared dough area by one cooki

[5] #7 6.RP.A.3 Match revenue. Art earns $12 \times 60 = 720$ cents. To match that, Roger's $18$

Review

Reasonableness: Direction check: Roger's cookie ($8$ in$^2$) is smaller than Art's ($12$ in$^2$), so the same dough yields more Roger cookies, and each must sell for less than $60$ cents. The price $40$ cents satisfies both. Cross-check the totals: Roger's batch earns $18 \times 40 = 720$ cents, exactly Art's $12 \times 60 = 720$ cents. Trap choices fit common slips: (A) $18$ is Roger's cookie count mistaken for the price, (D) $75$ is $60 \times \tfrac{12}{8}$ that swaps the area ratio (using areas as if more area meant more money), (E) $90$ multiplies $60$ by $\tfrac{12}{8}$ as $\tfrac{3}{2}$ in the wrong direction.

Alternative: Tool #11 (Find an Invariant) via price per square inch. Same dough means each friend earns the same total, so the earnings per square inch is invariant across friends. Art earns $720$ cents on $144$ in$^2$, giving $720/144 = 5$ cents per in$^2$. Roger's cookie covers $8$ in$^2$, so it must sell for $5 \times 8 = 40$ cents. Same answer (C) without ever counting Roger's cookies.

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 trapezoid area $12$ in$^2$ for one Art cookie and the rectangle area $8$ in$^2$ for one Roger cookie.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Translating equal dough into equal total area, dividing $144 \div 8$ to get Roger's $18$ cookies, and dividing $720 \div 18$ to find the $40$-cent price.)

⭐ When everyone uses the same dough, swap cookies for square inches. Convert Art's batch to area, see how many of Roger's smaller cookies fit, then split Art's money across them — each Roger cookie is $40$ cents.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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