Difference between revisions of "2005 AMC 12A Problems/Problem 10"

Latest revision as of 14:50, 1 July 2025

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
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

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 == Problem ==
-A wooden [[cube]] <math>n</math> units on a side is painted red on all six faces and then cut into <math>n^3</math> unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is <math>n</math>?
+A wooden cube <math>n</math> units on a side is painted red on all six faces and then cut into <math>n^3</math> unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is <math>n</math>?
 <math>

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Difference between revisions of "2005 AMC 12A Problems/Problem 10"

Latest revision as of 14:50, 1 July 2025

Problem

Solution

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