AMC 8 · 2003 · #8

Grade 6 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

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

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

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 × 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?

Pick an answer.

(A)

Art

(B)

Roger

(C)

Paul

(D)

Trisha

(E)

There is a tie for fewest.

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

How to solve

Strategy Draw a Diagram

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."

1STEP 1

Art's trapezoid: average the parallel sides $\frac{3+5}{2}$ , times height 3 = 12 in².

A_Art =

\frac{3 + 5}{2}

× 3 = 4 × 3 = 12 in²

2STEP 2

Roger's rectangle: 4 × 2 = 8 in².

A_Roger = 4 × 2 = 8 in²

3STEP 3

Paul's parallelogram: base 3 × perpendicular height 2 = 6 in².

A_Paul = 3 × 2 = 6 in²

4STEP 4

Trisha's right triangle: the legs are base and height, so $\frac{1}{2}$ × 3 × 4 = 6 in².

A_Trisha =

\frac{1}{2}

× 3 × 4 = 6 in²

5STEP 5

Same total area, so fewer cookies means a bigger cookie: 12 ＞ 8 ＞ 6 = 6, so the largest cookie makes the fewest.

12 ＞ 8 ＞ 6 = 6 → Art's cookie is largest → (A)

Answer

Art

Pin down the constant. Art makes 12 cookies of area 12 in² each, so the total top area per batch is 12 × 12 = 144 in². Then Roger bakes

\frac{144}{8}

= 18 cookies, Paul bakes

\frac{144}{6}

= 24, and Trisha bakes

\frac{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.

💡Key takeaway

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.

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