Difference between revisions of "2008 AMC 10B Problems/Problem 23"
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− | Because the | + | Because the painted part of the floor covers half the area, then the unpainted rectangle covers half the area as well. Since the border width is 1 foot, the dimensions of the rectangle are <math>a-2</math> by <math>b-2</math>. With this information we can make the equation: |
<cmath> | <cmath> |
Revision as of 16:15, 9 October 2020
Problem
A rectangular floor measures by
feet, where
and
are positive integers and
. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width
foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair
?
Solution
Because the painted part of the floor covers half the area, then the unpainted rectangle covers half the area as well. Since the border width is 1 foot, the dimensions of the rectangle are by
. With this information we can make the equation:
Applying Simon's Favorite Factoring Trick, we get
Since , then we have the possibilities
and
, or
and
. This allows for 2 possibilities:
or
which gives us
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 |
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