Difference between revisions of "2006 AMC 8 Problems/Problem 18"
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A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? | A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? | ||
| − | {{ | + | <math> \textbf{(A)}\ \frac{1}{9}\qquad\textbf{(B)}\ \frac{1}{4}\qquad\textbf{(C)}\ \frac{4}{9}\qquad\textbf{(D)}\ \frac{5}{9}\qquad\textbf{(E)}\ \frac{19}{27} </math> |
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| + | ==Solution== | ||
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| + | The surface area of the cube is <math>6(3)(3)=54</math>. Each of the eight black cubes has 3 faces on the outside, making <math>3(8)=24</math> black faces. Therefore there are <math>54-24=30</math> white faces. To find the ratio, we evaluate <math>\frac{30}{54}= \boxed{\textbf{(D)}\ \frac{5}{9}}</math>. | ||
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| + | ==Solution 2== | ||
| + | We can notice that each face is the same, so each face is <math>\boxed{\textbf{(D)}\ \frac{5}{9}}</math> white. | ||
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| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/nDkpGeJcTd8 | ||
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| + | ==See Also== | ||
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| + | {{AMC8 box|year=2006|num-b=17|num-a=19}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 17:08, 8 November 2024
Problem
A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?
Solution
The surface area of the cube is 
. Each of the eight black cubes has 3 faces on the outside, making 
black faces. Therefore there are 
white faces. To find the ratio, we evaluate 
.
Solution 2
We can notice that each face is the same, so each face is 
white.
Video Solution by WhyMath
See Also
| 2006 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
