Difference between revisions of "2008 AMC 10B Problems/Problem 23"

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==Problem==
 
==Problem==
A rectangular floor measures a by b feet, where a and b are positive integers and b > a. 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 1 foot around the painted rectangle and occupied half the area of the whole floor. How many possibilities are there for the ordered pair (a,b)?
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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) 1 B) 2 C) 3 D) 4 E) 5
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<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math>
  
 
==Solution==
 
==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 a-2 by b-2. With this information we can make the equation:
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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 <math>a-2</math> by <math>b-2</math>. With this information we can make the equation:
 
 
ab = 2[(a-2)(b-2)]
 
 
 
ab = 2ab - 4a - 4b + 8
 
  
ab - 4a - 4b + 16 = 8
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<cmath>
 +
\begin{eqnarray*}
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ab &=& 2\left((a-2)(b-2)\right) \\
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ab &=& 2ab - 4a - 4b + 8 \\
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ab - 4a - 4b + 8 &=& 0
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\end{eqnarray*}
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</cmath>
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Applying [[Simon's Favorite Factoring Trick]], we get
  
(a-4)(b-4) = 8
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<cmath>\begin{eqnarray*}ab - 4a - 4b + 16 &=& 8 \\ (a-4)(b-4) &=& 8 \end{eqnarray*}</cmath>
  
Since b > a, then we have the possibilities (a-4) = 1 and (b-4) = 8, or (a-4) = 2 and (b-4) = 4. This gives 2 (B) possibilities: (5,12) or (6,8).
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Since <math>b > a</math>, then we have the possibilities <math>(a-4) = 1</math> and <math>(b-4) = 8</math>, or <math>(a-4) = 2</math> and <math>(b-4) = 4</math>. This allows for 2 possibilities: <math>(5,12)</math> or <math>(6,8)</math> which gives us <math>\boxed{\textbf{(B)} \: 2}</math>
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=B|num-b=22|num-a=24}}
 
{{AMC10 box|year=2008|ab=B|num-b=22|num-a=24}}
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{{MAA Notice}}

Latest revision as of 23:47, 17 October 2023

Problem

A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. 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 $1$ foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair $(a,b)$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

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 $a-2$ by $b-2$. With this information we can make the equation:

\begin{eqnarray*} ab &=& 2\left((a-2)(b-2)\right) \\ ab &=& 2ab - 4a - 4b + 8 \\ ab - 4a - 4b + 8 &=& 0  \end{eqnarray*} Applying Simon's Favorite Factoring Trick, we get

\begin{eqnarray*}ab - 4a - 4b + 16 &=& 8 \\ (a-4)(b-4) &=& 8 \end{eqnarray*}

Since $b > a$, then we have the possibilities $(a-4) = 1$ and $(b-4) = 8$, or $(a-4) = 2$ and $(b-4) = 4$. This allows for 2 possibilities: $(5,12)$ or $(6,8)$ which gives us $\boxed{\textbf{(B)} \: 2}$

See also

2008 AMC 10B (ProblemsAnswer KeyResources)
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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