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Difference between revisions of "2010 AMC 10A Problems/Problem 2"

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== Problem 2 ==
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Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
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<center><asy>
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unitsize(8mm);
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defaultpen(linewidth(.8pt));
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draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
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draw((0,3)--(0,4)--(1,4)--(1,3)--cycle);
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draw((1,3)--(1,4)--(2,4)--(2,3)--cycle);
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draw((2,3)--(2,4)--(3,4)--(3,3)--cycle);
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draw((3,3)--(3,4)--(4,4)--(4,3)--cycle);
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</asy></center>
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<math>
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\mathrm{(A)}\ \dfrac{5}{4}
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\qquad
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\mathrm{(B)}\ \dfrac{4}{3}
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\qquad
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\mathrm{(C)}\ \dfrac{3}{2}
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\qquad
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\mathrm{(D)}\ 2
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\qquad
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\mathrm{(E)}\ 3
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</math>
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==Solution==
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Let the length of the small square be <math>x</math>, intuitively, the length of the big square is <math>4x</math>. It can be seen that the width of the rectangle is <math>3x</math>. Thus, the length of the rectangle is <math>4x/3x = 4/3</math> times large as the width. The answer is <math>\boxed{B}</math>.

Revision as of 15:35, 20 December 2010

Problem 2

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?

[asy] unitsize(8mm); defaultpen(linewidth(.8pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((0,3)--(0,4)--(1,4)--(1,3)--cycle); draw((1,3)--(1,4)--(2,4)--(2,3)--cycle); draw((2,3)--(2,4)--(3,4)--(3,3)--cycle); draw((3,3)--(3,4)--(4,4)--(4,3)--cycle); [/asy]

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

Solution

Let the length of the small square be $x$, intuitively, the length of the big square is $4x$. It can be seen that the width of the rectangle is $3x$. Thus, the length of the rectangle is $4x/3x = 4/3$ times large as the width. The answer is $\boxed{B}$.

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