Problem 9 · 2003 AMC 8
Medium
Ratios, Rates & Proportions
proportionratio
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.
Art's cookies sell for 60 cents each. To bring in the same total from one batch, how much should one of Roger's cookies cost, in cents?
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Answer: C — 40 cents.
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Hint 1 of 2
Both bakers earn the same total from the same dough, so every square inch of cookie is worth the same money — a small cookie should cost proportionally less.
Still stuck? Show hint 2 →
Hint 2 of 2
Price per cookie scales with cookie area: take Roger's area ÷ Art's area, then apply that fraction to 60¢.
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Approach: price per cookie scales with cookie area
- Same dough and same total revenue means the price is really being charged per square inch of cookie. So a cookie's price is proportional to its area — you can scale straight from area to price.
- Roger's cookie is 8 in², Art's is 12 in²: Roger's is 8/12 = 2/3 the size.
- So Roger charges 2/3 of Art's price: 60 × 2/3 = 40 cents.
- You'll see this again: spotting the hidden constant rate (here, cents per in²) turns a multi-step count into one proportion.
Another way — count the cookies:
- Art: 12 cookies at 60¢ = 720¢ per batch, using 12 × 12 = 144 in² of dough.
- Roger's 8 in² cookies: 144 ÷ 8 = 18 per batch.
- 720 ÷ 18 = 40 cents each.
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