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2010 AMC 8

Problem 10

Problem 10 · 2010 AMC 8 Medium
Geometry & Measurement area-ratio-of-circles

Six pepperoni circles will exactly fit across the diameter of a 12-inch pizza when placed. If a total of 24 circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?

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Answer: B — 2/3.
Show hints
Hint 1 of 2
Six pepperonis span the diameter, so each is 1/6 as wide as the pizza. But area doesn't shrink by 1/6 — it shrinks by 1/6 squared. That squaring is the whole trick.
Still stuck? Show hint 2 →
Hint 2 of 2
For any two similar shapes, the ratio of areas is the square of the ratio of lengths. π never has to appear — it cancels.
Show solution
Approach: areas scale with the square of length
  1. Each pepperoni is 1/6 the width of the pizza, so its area is (1/6)2 = 1/36 of the pizza — squaring the length ratio. No need to compute any actual area.
  2. 24 pepperonis cover 24 · (1/36) = 24/36 = 2/3 of the pizza.
  3. Why this transfers: ‘length ratio k ⇒ area ratio k2’ (and volume ratio k3) lets you skip π and radii entirely in scaling problems. Spot the length ratio, square it, done.
Another way — plug in real numbers:
  1. Pizza diameter 12 ⇒ pepperoni diameter 2, radius 1, area π. Pizza radius 6, area 36π.
  2. 24 pepperonis: 24π / 36π = 24/36 = 2/3; the π cancels, matching the shortcut above.
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