2008 AMC 10B Problems/Problem 23
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 unpainted part of the floor covers half the area, then the painted 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 | |
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All AMC 10 Problems and Solutions |
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