2010 AMC 8 Problem 10

Below is the professionally curated solution for Problem 10 of the 2010 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 8 solutions, or check the answer key.

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Concepts:circle area area ratio power scaling of length, area, and volume

Difficulty rating: 1030

10.

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

$\ \dfrac 12$

$\ \dfrac 23$

$\ \dfrac 34$

$\ \dfrac 56$

$\ \dfrac 78$

Solution:

Each circle has $\dfrac{1}{6}$ the diameter of the large circle, so it has $(\frac 16)^2 = \frac 1{36}$ of the total area.

Since there are $24$ pepperoni, they take up $24 * \dfrac 1{36} = \frac 23$ of the area.

Therefore, the answer is B .

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