Problem 10 · 2003 AMC 8
Medium
Arithmetic & Operations
divisionproportion
Art, Roger, Paul, and Trisha bake cookies that are all the same thickness, in the shapes shown below (dimensions in inches). Each friend uses the same amount of dough, and Art's batch makes exactly 12 cookies.
How many cookies will be in one batch of Trisha's cookies?
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Answer: E — 24 cookies.
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Hint 1 of 2
Skip computing the dough total — just compare cookie sizes: Trisha's vs. Art's.
Still stuck? Show hint 2 →
Hint 2 of 2
Half the size means double the count, because the same dough is being split into pieces.
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Approach: compare Trisha's cookie to Art's directly
- Art makes 12 cookies of 12 in² each. Trisha's triangle is ½(3)(4) = 6 in² — exactly half of Art's.
- Same dough split into half-size pieces makes twice as many pieces: 12 × 2 = 24.
- Worth keeping: for a fixed total, count and size are inversely proportional — halve the size, double the count; third the size, triple the count. Reaching for the "144 in² total" works too, but the size comparison is one step.
Another way — through the dough total:
- A batch is 12 of Art's 12 in² cookies = 144 in² of dough.
- Trisha's cookies are 6 in²: 144 ÷ 6 = 24.
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