2005 AMC 12A Problems/Problem 10

Problem

A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$ ?

$(\mathrm {A}) \ 3 \qquad (\mathrm {B}) \ 4 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 6 \qquad (\mathrm {E})\ 7$

Solution

There are $6n^3$ sides total on the unit cubes, and $6n^2$ are painted red.

$\dfrac{6n^2}{6n^3}=\dfrac{1}{4} \Rightarrow n=4 \rightarrow \mathrm {B}$

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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2005 AMC 12A Problems/Problem 10

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