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Difference between revisions of "2020 AMC 8 Problems/Problem 25"

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==Solution 4==
 
==Solution 4==
 
Assuming that the problem is well-posed, it should be true in the case where <math>S_1 \cong S_3</math>. Let the side length of square <math>S_1</math> be <math>x</math> and the side length of square <math>S_2</math> be <math>y</math>. We then have the system <cmath>\begin{dcases}2x-y =2020 \\2x+y =3322\end{dcases}</cmath> and we solve it to determine that <math>y=\boxed{\textbf{(A) }651}</math>.
 
Assuming that the problem is well-posed, it should be true in the case where <math>S_1 \cong S_3</math>. Let the side length of square <math>S_1</math> be <math>x</math> and the side length of square <math>S_2</math> be <math>y</math>. We then have the system <cmath>\begin{dcases}2x-y =2020 \\2x+y =3322\end{dcases}</cmath> and we solve it to determine that <math>y=\boxed{\textbf{(A) }651}</math>.
 +
 +
==Solution 5==
 +
Let the side length of <math>S_2</math> be s, and the shorter side length of <math>R_1</math> and <math>R_2</math> be <math>r</math>. We have
 +
 +
<asy>
 +
draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0));
 +
draw((3,0)--(3,1)--(0,1));
 +
draw((3,1)--(3,2)--(5,2));
 +
draw((3,2)--(2,2)--(2,1)--(2,3));
 +
label("$R_1$",(3/2,1/2));
 +
label("$S_3$",(4,1));
 +
label("$S_2$",(5/2,3/2));
 +
label("$S_1$",(1,2));
 +
label("$R_2$",(7/2,5/2));
 +
label("$r$",(5.2,5/2));
 +
label("$r$",(3.2,1/2));
 +
label("$s$",(3.2,3/2));
 +
</asy>
 +
@
 +
From this diagram, it is evident that <math>r+s+r=2020</math>. Also, the side length of <math>S_1</math> and <math>S_3</math> is <math>r+s</math>. Then, <math>r+s+s+r+s=3322</math>. Now, we have 2 systems of equations.
 +
 +
<cmath>\begin{align*}r+s+r &= 2020 \\ r+s+r+s+s &= 3322 \\ \end{align*}</cmath>
 +
 +
We can see an <math>r+s+r</math> in the 2nd equation, so substituting that in gives us <math>2020+2s=3322 \Rightarrow 2s= 1302 \Rightarrow s=\boxed{\textbf{(A) }651}</math>
 +
 +
~MrThinker
  
 
==Video Solution==
 
==Video Solution==

Revision as of 11:29, 29 October 2022

Problem

Rectangles $R_1$ and $R_2,$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label("$R_1$",(3/2,1/2)); label("$S_3$",(4,1)); label("$S_2$",(5/2,3/2)); label("$S_1$",(1,2)); label("$R_2$",(7/2,5/2)); [/asy]

$\textbf{(A) }651 \qquad \textbf{(B) }655 \qquad \textbf{(C) }656 \qquad \textbf{(D) }662 \qquad \textbf{(E) }666$

Solution 1

Let the side length of each square $S_k$ be $s_k$. Then, from the diagram, we can line up the top horizontal lengths of $S_1$, $S_2$, and $S_3$ to cover the top side of the large rectangle, so $s_{1}+s_{2}+s_{3}=3322$. Similarly, the short side of $R_2$ will be $s_1-s_2$, and lining this up with the left side of $S_3$ to cover the vertical side of the large rectangle gives $s_{1}-s_{2}+s_{3}=2020$. We subtract the second equation from the first to obtain $2s_{2}=1302$, and thus $s_{2}=\boxed{\textbf{(A) }651}$.

Solution 2

Assuming that the problem is well-posed, it should be true in the particular case where $S_1 \cong S_3$ and $R_1 \cong R_2$. Let the sum of the side lengths of $S_1$ and $S_3$ be $x$, and let the length of rectangle $S_2$ be $y$. We then have the system \[\begin{dcases}x+y =3322 \\x-y=2020\end{dcases}\] which we solve to determine $y=\boxed{\textbf{(A) }651}$.

Solution 3 (faster version of Solution 1)

Since, for each pair of rectangles, the side lengths have a sum of $3322$ or $2020$ and a difference of $S_2$, the answer must be $\dfrac{3322 - 2020}{2} = \dfrac{1302}{2} = \boxed{\textbf{(A) }651}$.

Solution 4

Assuming that the problem is well-posed, it should be true in the case where $S_1 \cong S_3$. Let the side length of square $S_1$ be $x$ and the side length of square $S_2$ be $y$. We then have the system \[\begin{dcases}2x-y =2020 \\2x+y =3322\end{dcases}\] and we solve it to determine that $y=\boxed{\textbf{(A) }651}$.

Solution 5

Let the side length of $S_2$ be s, and the shorter side length of $R_1$ and $R_2$ be $r$. We have

[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label("$R_1$",(3/2,1/2)); label("$S_3$",(4,1)); label("$S_2$",(5/2,3/2)); label("$S_1$",(1,2)); label("$R_2$",(7/2,5/2)); label("$r$",(5.2,5/2)); label("$r$",(3.2,1/2)); label("$s$",(3.2,3/2)); [/asy] @ From this diagram, it is evident that $r+s+r=2020$. Also, the side length of $S_1$ and $S_3$ is $r+s$. Then, $r+s+s+r+s=3322$. Now, we have 2 systems of equations.

\begin{align*}r+s+r &= 2020 \\ r+s+r+s+s &= 3322 \\ \end{align*}

We can see an $r+s+r$ in the 2nd equation, so substituting that in gives us $2020+2s=3322 \Rightarrow 2s= 1302 \Rightarrow s=\boxed{\textbf{(A) }651}$

~MrThinker

Video Solution

https://www.youtube.com/watch?v=a3Z7zEc7AXQ

Video Solution by WhyMath

https://youtu.be/LebVAuPkpcg

~savannahsolver

Video Solution by The Learning Royal

https://youtu.be/JAZXFv1fFGo

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=1639

~Interstigation

See also

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
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All AJHSME/AMC 8 Problems and Solutions

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