Competition · AMC preparation · step 4 of 4
AMC 8 · 2003 · #8
Grade 6 geometry-2d
Pick an answer.
AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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."
Find Art's cookie area
Art's trapezoid: average the parallel sides , times height 3 = 12 in².
Grade 6 area of a trapezoid: average the two parallel sides, then multiply by the height.
6.G.A.1Draw A DiagramFind Roger's cookie area
Roger's rectangle: 4 × 2 = 8 in².
Grade 6 area of a rectangle: length times width.
6.G.A.1Draw A DiagramFind Paul's cookie area
Paul's parallelogram: base 3 × perpendicular height 2 = 6 in².
Grade 6 area of a parallelogram: base times perpendicular height.
6.G.A.1Draw A DiagramFind Trisha's cookie area
Trisha's right triangle: the legs are base and height, so × 3 × 4 = 6 in².
Grade 6 area of a right triangle: half of base times height.
6.G.A.1Draw A DiagramCompare the four areas
Same total area, so fewer cookies means a bigger cookie: 12 ＞ 8 ＞ 6 = 6, so the largest cookie makes the fewest.
Constant total area split into cookies of size a gives count = (total)/a. Bigger a means smaller count.
The friend whose single cookie covers the greatest top area is the one who bakes the fewest cookies from one batch of dough.
▸ Why?
Each friend uses the same amount of dough at the same thickness, so every friend's cookies together cover the same total top area.
▸ Why?
The dough is a flat slab whose amount equals its total top area times its thickness, so when the thickness is the same for everyone the only way the amounts can match is for the total areas to match.
▸ Why?
A slab of uniform thickness is a stack of equal layers, one for each unit of thickness, and every layer covers the whole top area, so the amount of dough is that top area counted once per layer.
▸ Why?
Dividing each friend's equal amount of dough by the shared thickness gives back the top area, and division undoes that multiplication, so equal amounts at one thickness can only come from equal areas.
▸ Why?
With that shared total area fixed, the number of cookies is the total area divided by one cookie's area, so splitting the same total into larger cookies leaves fewer of them.
▸ Why?
One friend's cookies are equal pieces that fill the total area with no gaps or overlaps, so their count multiplied by one cookie's area rebuilds the whole total area.
▸ Why?
Because the count times one cookie's area equals the fixed total, dividing that total by a larger single-cookie area returns a smaller count — division undoes the multiplication, and a bigger share per cookie means fewer 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.
- Find Art's cookie area
- Find Roger's cookie area
- Find Paul's cookie area
- Find Trisha's cookie area
- Compare the four areas
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