Competition · AMC preparation · step 4 of 4
AMC 8 · 2003 · #9
Grade 6 rate-ratio
Pick an answer.
AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find one Art cookie's area
One Art cookie is a trapezoid with bases 5 in and 3 in and height 3 in, so its area is 12 in².
Grade 6 area-of-trapezoid formula: average the two parallel bases, then multiply by the height.
6.G.A.1Draw A DiagramFind the total dough area
Art bakes 12 such cookies, so his dough covers 144 in² of area.
Equal thickness means dough volume is proportional to total surface area, so 144 in² is the shared dough budget for every friend.
6.G.A.1Identify SubproblemsFind one Roger cookie's area
One Roger cookie is a 4 in × 2 in rectangle, so its area is 8 in².
Grade 6 rectangle area: length times width.
6.G.A.1Draw A DiagramCount Roger's cookies
The same 144 in² split into 8 in² pieces gives Roger 18 cookies.
Same dough, smaller cookies, so Roger gets more pieces — 18 ＞ 12 is the expected direction.
Roger's share of dough comes out to 18 cookies.
▸ Why?
Roger uses the same dough as Art and rolls it to the same thickness, so his cookies must cover the same total surface Art's do, namely 144 square inches.
▸ Why?
A rolled slab fills a volume equal to its flat area times its thickness, so when the thickness is the same for both, the same amount of dough (the same volume) can only spread over the same area.
▸ Why?
That shared surface is 144 square inches because Art's twelve 12-square-inch cookies lie side by side with no gaps or overlaps, so their areas add back up to the whole batch.
▸ Why?
Each of Roger's cookies is a 4-by-2 rectangle, which is 4 rows of 2 unit squares, so one cookie covers 8 square inches.
▸ Why?
Splitting 144 square inches into equal 8-square-inch cookies asks how many 8s make 144, and that count is exactly the number which, times 8, gives 144 back.
Divide the earnings by the count
Art earns 12 × 60 = 720 cents, so Roger's 18 cookies must each cost 720 ÷ 18 = 40 cents.
Roger sells more cookies than Art, so each must cost less than 60 cents — 40 fits.
6.RP.A.3Identify SubproblemsWhen 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.
- Find one Art cookie's area
- Find the total dough area
- Find one Roger cookie's area
- Count Roger's cookies
- Divide the earnings by the count
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