Difference between revisions of "2019 AMC 10A Problems/Problem 10"
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<math>\textbf{(A) } 17 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 26 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 28</math> | <math>\textbf{(A) } 17 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 26 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 28</math> | ||
| − | ==Solution== | + | ==Solution 1== |
The number of tiles the bug visits is equal to <math>1</math> plus the number of times it crosses a horizontal or vertical line. As it must cross <math>16</math> horizontal lines and <math>9</math> vertical lines, it must be that the bug visits a total of <math>16+9+1 = \boxed{\textbf{(C) }26}</math> squares. | The number of tiles the bug visits is equal to <math>1</math> plus the number of times it crosses a horizontal or vertical line. As it must cross <math>16</math> horizontal lines and <math>9</math> vertical lines, it must be that the bug visits a total of <math>16+9+1 = \boxed{\textbf{(C) }26}</math> squares. | ||
| − | Note: The general formula for this is <math>a+b-\gcd(a,b)</math> | + | ''Note'': The general formula for this is <math>a+b-\gcd(a,b)</math>, because it is the number of vertical/horizontal lines crossed minus the number of corners crossed (to avoid double counting). In this particular problem, it was <math>16 + 9 - 1</math> (since <math>\text{gcd}(16,9) = 1</math>), which is <math>24</math>, but then you add <math>2</math> because the first tile and the last tile are counted, which in the general formula are not counted. |
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==Solution 2 (Draw it out)== | ==Solution 2 (Draw it out)== | ||
| − | Draw it out using grid paper and a ruler. Carefully counting the squares gives us 26. | + | Draw it out using grid paper and a ruler. Carefully counting the squares gives us <math>26</math>. |
==See Also== | ==See Also== | ||
Revision as of 20:11, 17 February 2019
Problem
A rectangular floor that is 
feet wide and 
feet long is tiled with 
one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
Solution 1
The number of tiles the bug visits is equal to 
plus the number of times it crosses a horizontal or vertical line. As it must cross 
horizontal lines and 
vertical lines, it must be that the bug visits a total of 
squares.
Note: The general formula for this is 
, because it is the number of vertical/horizontal lines crossed minus the number of corners crossed (to avoid double counting). In this particular problem, it was 
(since 
), which is 
, but then you add 
because the first tile and the last tile are counted, which in the general formula are not counted.
Solution 2 (Draw it out)
Draw it out using grid paper and a ruler. Carefully counting the squares gives us 
.
See Also
| 2019 AMC 10A (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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
