Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m
m (→‎Solution 3 (faster version of Solution 1))
 
(91 intermediate revisions by 44 users not shown)
Line 1: Line 1:
 +
 +
==Problem==
 +
 
Rectangles <math>R_1</math> and <math>R_2,</math> and squares <math>S_1,\,S_2,\,</math> and <math>S_3,</math> shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of <math>S_2</math> in units?
 
Rectangles <math>R_1</math> and <math>R_2,</math> and squares <math>S_1,\,S_2,\,</math> and <math>S_3,</math> shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of <math>S_2</math> in units?
  
Line 16: Line 19:
  
 
==Solution 1==
 
==Solution 1==
−
For each square <math>S_{k}</math>, let the sidelength of this square be denoted by <math>s_{k}</math>.
+
Let the side length of each square <math>S_k</math> be <math>s_k</math>. Then, from the diagram, we can line up the top horizontal lengths of <math>S_1</math>, <math>S_2</math>, and <math>S_3</math> to cover the top side of the large rectangle, so <math>s_{1}+s_{2}+s_{3}=3322</math>. Similarly, the short side of <math>R_2</math> will be <math>s_1-s_2</math>, and lining this up with the left side of <math>S_3</math> to cover the vertical side of the large rectangle gives <math>s_{1}-s_{2}+s_{3}=2020</math>. We subtract the second equation from the first to obtain <math>2s_{2}=1302</math>, and thus <math>s_{2}=\boxed{\textbf{(A) }651}</math>.
  
−
As the diagram shows, <math>s_{1}+s_{2}+s_{3}=3322, s_{1}-s_{2}+s_{3}=2020.</math> We subtract the second equation from the first, getting <math>2s_{2}=1302</math>, and thus <math>s_{2}=651</math>, so the answer is <math>\boxed{\textbf{(A)}\text{ }651}</math> ~icematrix, edits by starrynight7210
+
==Solution 2==
 +
Assuming that the problem is well-posed, it should be true in the particular case where <math>S_1 \cong S_3</math> and <math>R_1 \cong R_2</math>. Let the sum of the side lengths of <math>S_1</math> and <math>S_3</math> be <math>x</math>, and let the length of square <math>S_2</math> be <math>y</math>. We then have the system <cmath>\begin{dcases}x+y =3322 \\x-y=2020\end{dcases}</cmath> which we solve to determine <math>y=\boxed{\textbf{(A) }651}</math>.
  
−
==Solution 2==
+
==Solution 3 (faster version of Solution 1)==
 +
Since, for each pair of rectangles, the side lengths have a sum of <math>3322</math> or <math>2020</math> and a difference of <math>S_2</math>, the answer must be <math>\dfrac{3322 - 2020}{2} = \dfrac{1302}{2} = \boxed{\textbf{(A) }651}</math>.
 +
 
 +
==Solution 4==
 +
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>.
 +
 
 +
==Video Solution by Math-X (First understand the problem!!!)==
 +
https://youtu.be/UnVo6jZ3Wnk?si=ryNc6kwFiy7YkEbc&t=6127
 +
 
 +
~Math-X
 +
 
 +
==Video Solution(🚀Just 1 min🚀)==
 +
https://youtu.be/2yiZ1Mx2P1M
 +
 
 +
~<i>Education, the Study of Everything</i>
 +
 
 +
==Video Solution==
 +
https://youtu.be/wAUam5A-jcA
 +
 
 +
Please like and subscribe!
 +
 
 +
https://www.youtube.com/watch?v=gJXMZq2Rbwg  ~David
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/jhJifWaoUI8?t=441
 +
 
 +
~ pi_is_3.14
 +
 
 +
==Video Solution by WhyMath==
 +
https://youtu.be/LebVAuPkpcg
 +
 
 +
~savannahsolver
 +
 
 +
==Video Solution by The Learning Royal==
 +
https://youtu.be/JAZXFv1fFGo
  
−
WLOG, assume that <math>S_1=S_3</math> and <math>R_1=R_2</math>. Let the sum of the lengths of <math>S_1</math> and <math>S_2</math> be <math>x</math> and let the length of <math>S_2</math> be <math>y</math>. We have the system <cmath>x+y =3322</cmath> <cmath>x-y=2020</cmath>
+
==Video Solution by Interstigation==
 +
https://youtu.be/YnwkBZTv5Fw?t=1639
  
−
which we solve to find that <math>y=\textbf{(A) }651</math>.
+
~Interstigation
  
−
-franzliszt
+
==Video Solution by STEMbreezy==
 +
https://youtu.be/wq8EUCe5oQU?t=588
  
−
==Solution 3==
+
~STEMbreezy
−
Since each pair of boxes has a sum of <math>3322</math> or <math>2020</math> and a difference of <math>S_2</math>, we see that the answer is <math>\dfrac{3322 - 2020}{2} = \dfrac{1302}{2} = \boxed{(\text{A}) 651}.</math>
 
  
−
-A_MatheMagician. Note this is just a more quicker way to do it to get \boxed{(\text{A}) 651}.$
+
==See also==
−
https://artofproblemsolving.com/community/my-aops
 
−
==See also==  
 
 
{{AMC8 box|year=2020|num-b=24|after=Last Problem}}
 
{{AMC8 box|year=2020|num-b=24|after=Last Problem}}
 +
 +
[[Category:Introductory Geometry Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 21:26, 18 November 2024

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 square $S_2$ 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

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 $2020+2s=3322 \Rightarrow 2s= 1302 \Rightarrow s=\boxed{\textbf{(A) }651}$.

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/UnVo6jZ3Wnk?si=ryNc6kwFiy7YkEbc&t=6127

~Math-X

Video Solution(🚀Just 1 min🚀)

https://youtu.be/2yiZ1Mx2P1M

~Education, the Study of Everything

Video Solution

https://youtu.be/wAUam5A-jcA

Please like and subscribe!

https://www.youtube.com/watch?v=gJXMZq2Rbwg ~David

Video Solution by OmegaLearn

https://youtu.be/jhJifWaoUI8?t=441

~ pi_is_3.14

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

Video Solution by STEMbreezy

https://youtu.be/wq8EUCe5oQU?t=588

~STEMbreezy

See also

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2020_AMC_8_Problems/Problem_25&oldid=234964"