Difference between revisions of "2008 AMC 10B Problems/Problem 23"
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A rectangular floor measures <math>a</math> by <math>b</math> feet, where <math>a</math> and <math>b</math> are positive integers and <math>b > a</math>. 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 <math>1</math> foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair <math>(a,b)</math>? | A rectangular floor measures <math>a</math> by <math>b</math> feet, where <math>a</math> and <math>b</math> are positive integers and <math>b > a</math>. 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 <math>1</math> foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair <math>(a,b)</math>? | ||
− | <math>\ | + | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> |
==Solution== | ==Solution== |
Revision as of 19:14, 1 May 2021
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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