Difference between revisions of "2011 AMC 8 Problems/Problem 3"
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One way of approaching this is drawing the next circle of boxes around the current square. | One way of approaching this is drawing the next circle of boxes around the current square. | ||
<asy> | <asy> | ||
| + | filldraw((-1,-1)--(6,-1)--(6,6)--(-1,6)--cycle,mediumgray,black); | ||
| + | filldraw((0,0)--(5,0)--(5,5)--(0,5)--cycle,white,black); | ||
| + | filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle,mediumgray,black); | ||
| + | filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,white,black); | ||
| + | draw((4,-1)--(4,6)); | ||
| + | draw((3,-1)--(3,6)); | ||
| + | draw((2,-1)--(2,6)); | ||
| + | draw((1,-1)--(1,6)); | ||
| + | draw((-1,4)--(6,4)); | ||
| + | draw((-1,3)--(6,3)); | ||
| + | draw((-1,2)--(6,2)); | ||
| + | draw((-1,1)--(6,1)); | ||
| + | draw((0,-1)--(0,6)); | ||
| + | draw((-1,5)--(6,5)); | ||
| + | draw((-1,0)--(6,0)); | ||
| + | draw((5,-1)--(5,6)); | ||
</asy> | </asy> | ||
We can now count the number of black and white tiles; 32 black tiles and 17 white tiles. This means the answer is <math>\textbf{(D) }32:17</math>. | We can now count the number of black and white tiles; 32 black tiles and 17 white tiles. This means the answer is <math>\textbf{(D) }32:17</math>. | ||
Revision as of 18:11, 1 December 2011
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
![[asy] filldraw((0,0)--(5,0)--(5,5)--(0,5)--cycle,white,black); filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle,mediumgray,black); filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,white,black); draw((4,0)--(4,5)); draw((3,0)--(3,5)); draw((2,0)--(2,5)); draw((1,0)--(1,5)); draw((0,4)--(5,4)); draw((0,3)--(5,3)); draw((0,2)--(5,2)); draw((0,1)--(5,1)); [/asy]](http://latex.artofproblemsolving.com/a/2/4/a24e23a268700a40b52b1b449dac9c6d2ec80a4e.png)
Solution
One way of approaching this is drawing the next circle of boxes around the current square.
![[asy] filldraw((-1,-1)--(6,-1)--(6,6)--(-1,6)--cycle,mediumgray,black); filldraw((0,0)--(5,0)--(5,5)--(0,5)--cycle,white,black); filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle,mediumgray,black); filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,white,black); draw((4,-1)--(4,6)); draw((3,-1)--(3,6)); draw((2,-1)--(2,6)); draw((1,-1)--(1,6)); draw((-1,4)--(6,4)); draw((-1,3)--(6,3)); draw((-1,2)--(6,2)); draw((-1,1)--(6,1)); draw((0,-1)--(0,6)); draw((-1,5)--(6,5)); draw((-1,0)--(6,0)); draw((5,-1)--(5,6)); [/asy]](http://latex.artofproblemsolving.com/c/7/d/c7d5e03ab002d178fce4ca257ee3d4fff4e3c31f.png)
We can now count the number of black and white tiles; 32 black tiles and 17 white tiles. This means the answer is 
.
See Also
| 2011 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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 | ||
