Difference between revisions of "2019 AMC 10A Problems/Problem 10"
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− | Draw | + | Draw a diagram (optionally with grid paper and/or a ruler). We can cut the floor into 4 pieces and draw a diagram for only one piece that is <math>5</math> feet by <math>8.5</math> feet. |
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<asy> | <asy> |
Revision as of 22:01, 23 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 a diagram (optionally with grid paper and/or a ruler). We can cut the floor into 4 pieces and draw a diagram for only one piece that is feet by
feet.
While it appears that the line we drew comes very close to several points, we know that since and
are relatively prime, it will not actually pass through any of these points, so the total squares will be the same regardless of which side we count. If we sum the diagram, we get
squares, for a total of
.
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.