Difference between revisions of "2024 AMC 8 Problems/Problem 3"
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| − | ==Problem== | + | ==Problem 3 == |
| − | ==Solution 1== | + | Four squares of side length <math>4, 7, 9,</math> and <math>10</math> are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units? |
| + | <asy> | ||
| + | size(150); | ||
| + | filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1)); | ||
| + | filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1)); | ||
| + | filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1)); | ||
| + | filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1)); | ||
| + | draw((11,0)--(11,4),linewidth(1)); | ||
| + | draw((11,6)--(11,10),linewidth(1)); | ||
| + | label("$10$",(11,5),fontsize(14pt)); | ||
| + | draw((10.75,0)--(11.25,0),linewidth(1)); | ||
| + | draw((10.75,10)--(11.25,10),linewidth(1)); | ||
| + | draw((0,11)--(3,11),linewidth(1)); | ||
| + | draw((5,11)--(9,11),linewidth(1)); | ||
| + | draw((0,11.25)--(0,10.75),linewidth(1)); | ||
| + | draw((9,11.25)--(9,10.75),linewidth(1)); | ||
| + | label("$9$",(4,11),fontsize(14pt)); | ||
| + | draw((-1,0)--(-1,1),linewidth(1)); | ||
| + | draw((-1,3)--(-1,7),linewidth(1)); | ||
| + | draw((-1.25,0)--(-0.75,0),linewidth(1)); | ||
| + | draw((-1.25,7)--(-0.75,7),linewidth(1)); | ||
| + | label("$7$",(-1,2),fontsize(14pt)); | ||
| + | draw((0,-1)--(1,-1),linewidth(1)); | ||
| + | draw((3,-1)--(4,-1),linewidth(1)); | ||
| + | draw((0,-1.25)--(0,-.75),linewidth(1)); | ||
| + | draw((4,-1.25)--(4,-.75),linewidth(1)); | ||
| + | label("$4$",(2,-1),fontsize(14pt)); | ||
| + | </asy> | ||
| + | <math>\textbf{(A)}\ 42 \qquad \textbf{(B)}\ 45\qquad \textbf{(C)}\ 49\qquad \textbf{(D)}\ 50\qquad \textbf{(E)}\ 52</math> | ||
| + | |||
| + | ==Solution 1== | ||
| + | We work inwards. The area of the outer shaded square is the area of the whole square minus the area of the second largest square. The area of the inner shaded region is the area of the third largest square minus the area of the smallest square. The sum of these areas is | ||
| + | <cmath>10^2 - 9^2 + 7^2 - 4^2 = 100 - 81 + 49 - 16 = 19 + 33 = \boxed{\textbf{(E)}\ 52}</cmath> | ||
| + | |||
| + | This problem appears multiple times in various math competitions including the AMC and MATHCOUNTS. | ||
| + | |||
| + | -Benedict (countmath1) ~Nivaar and anabel.disher | ||
| + | ==Solution 2== | ||
| + | In solution 1, we can use [[Difference of squares]] to get the answer, rather than calculating the squares of the numbers: | ||
| + | <cmath>10^2 - 9^2 + 7^2 - 4^2 = (10 - 9)(10 + 9) - (7 - 4)(7 + 4) = 1\cdot19 + 3\cdot11 = 19 + 33 = \boxed{\textbf{(E)}\ 52}</cmath> | ||
| + | |||
| + | |||
| + | ~anabel.disher | ||
| + | ==Solution 3== | ||
| + | |||
| + | Solve for the areas for each square inside the larger one , and then subtract the areas of the white regions from the areas of each square: | ||
| + | <cmath>(10^2 + 7^2) - (9^2 + 4^2) = 149 - 97 = \boxed{\textbf{(E)}\ 52}</cmath> | ||
| + | |||
| + | -MrTechguyPCMT and anabel.disher | ||
| + | |||
| + | ==Solution 4== | ||
| + | We can calculate it as the sum of the areas of <math>2</math> smaller trapezoids and <math>2</math> larger trapezoids. | ||
| + | <cmath>2\left(\cfrac{(7+4)(7-4)}{2}\right)+2\left(\cfrac{(10+9)(10-9)}{2}\right)=10^2 - 9^2 + 7^2 - 4^2 = 19 + 33 = \boxed{\textbf{(E)}\ 52}</cmath> | ||
| + | |||
| + | -SahanWijetunga | ||
| + | |||
| + | ==Video Solution (A Clever Explanation You’ll Get Instantly)== | ||
| + | https://youtu.be/q64dg8YAJbE | ||
| + | |||
| + | ~mr_mathman | ||
| + | |||
| + | ==Video Solution by Math-X (First fully understand the problem!!!)== | ||
| + | https://youtu.be/BaE00H2SHQM?si=F37qz7vyfRgZm-1u&t=469 | ||
| + | |||
| + | ~Math-X | ||
| + | |||
| + | ==Video Solution (A Clever Explanation You’ll Get Instantly)== | ||
| + | https://youtu.be/5ZIFnqymdDQ?si=vSpf507m-7QMFkbZ&t=238 | ||
| + | |||
| + | ~hsnacademy | ||
| + | |||
| + | ==Video Solution (easy to digest) by Power Solve== | ||
| + | https://youtu.be/HE7JjZQ6xCk?si=39xd5CKI9nx-7lyV&t=118 | ||
| + | |||
| + | ==Video Solution by Parshwa== | ||
| + | https://www.youtube.com/watch?v=DQbe_BNfPYw | ||
| + | |||
| + | ==Video Solution by NiuniuMaths (Easy to understand!)== | ||
| + | https://www.youtube.com/watch?v=Ylw-kJkSpq8 | ||
| + | |||
| + | ~NiuniuMaths | ||
| + | |||
| + | ==Video Solution by SpreadTheMathLove== | ||
| + | https://www.youtube.com/watch?v=L83DxusGkSY | ||
| + | |||
| + | == Video Solution by CosineMethod [🔥Fast and Easy🔥]== | ||
| + | |||
| + | https://youtu.be/sPzsce8FQtY?si=IS1hVC0cMd-0CYj5 | ||
| + | ==Video Solution by Interstigation== | ||
| + | https://youtu.be/ktzijuZtDas&t=158 | ||
| + | |||
| + | ==Video Solution by Daily Dose of Math (Understandable, Quick, and Speedy)== | ||
| + | |||
| + | https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR | ||
| + | |||
| + | ~Thesmartgreekmathdude | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/653x6HwexjY | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2024|num-b=2|num-a=4}} | ||
| + | {{MAA Notice}} | ||
Revision as of 16:49, 21 December 2024
Contents
- 1 Problem 3
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Video Solution (A Clever Explanation You’ll Get Instantly)
- 7 Video Solution by Math-X (First fully understand the problem!!!)
- 8 Video Solution (A Clever Explanation You’ll Get Instantly)
- 9 Video Solution (easy to digest) by Power Solve
- 10 Video Solution by Parshwa
- 11 Video Solution by NiuniuMaths (Easy to understand!)
- 12 Video Solution by SpreadTheMathLove
- 13 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 14 Video Solution by Interstigation
- 15 Video Solution by Daily Dose of Math (Understandable, Quick, and Speedy)
- 16 Video Solution by WhyMath
- 17 See Also
Problem 3
Four squares of side length 
and 
are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?
![[asy] size(150); filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1)); filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1)); draw((11,0)--(11,4),linewidth(1)); draw((11,6)--(11,10),linewidth(1)); label("$10$",(11,5),fontsize(14pt)); draw((10.75,0)--(11.25,0),linewidth(1)); draw((10.75,10)--(11.25,10),linewidth(1)); draw((0,11)--(3,11),linewidth(1)); draw((5,11)--(9,11),linewidth(1)); draw((0,11.25)--(0,10.75),linewidth(1)); draw((9,11.25)--(9,10.75),linewidth(1)); label("$9$",(4,11),fontsize(14pt)); draw((-1,0)--(-1,1),linewidth(1)); draw((-1,3)--(-1,7),linewidth(1)); draw((-1.25,0)--(-0.75,0),linewidth(1)); draw((-1.25,7)--(-0.75,7),linewidth(1)); label("$7$",(-1,2),fontsize(14pt)); draw((0,-1)--(1,-1),linewidth(1)); draw((3,-1)--(4,-1),linewidth(1)); draw((0,-1.25)--(0,-.75),linewidth(1)); draw((4,-1.25)--(4,-.75),linewidth(1)); label("$4$",(2,-1),fontsize(14pt)); [/asy]](http://latex.artofproblemsolving.com/1/5/e/15e0c32b9dd52758b61dd87c199e00db7b42fbd1.png)
Solution 1
We work inwards. The area of the outer shaded square is the area of the whole square minus the area of the second largest square. The area of the inner shaded region is the area of the third largest square minus the area of the smallest square. The sum of these areas is
![\[10^2 - 9^2 + 7^2 - 4^2 = 100 - 81 + 49 - 16 = 19 + 33 = \boxed{\textbf{(E)}\ 52}\]](http://latex.artofproblemsolving.com/9/5/6/95633fa0d1a0b80eefa9abbed7e848048d4cad4b.png)
This problem appears multiple times in various math competitions including the AMC and MATHCOUNTS.
-Benedict (countmath1) ~Nivaar and anabel.disher
Solution 2
In solution 1, we can use Difference of squares to get the answer, rather than calculating the squares of the numbers:
![\[10^2 - 9^2 + 7^2 - 4^2 = (10 - 9)(10 + 9) - (7 - 4)(7 + 4) = 1\cdot19 + 3\cdot11 = 19 + 33 = \boxed{\textbf{(E)}\ 52}\]](http://latex.artofproblemsolving.com/e/5/0/e5014be578cf02380c1a172750a5a207a7e785ce.png)
~anabel.disher
Solution 3
Solve for the areas for each square inside the larger one , and then subtract the areas of the white regions from the areas of each square:
![\[(10^2 + 7^2) - (9^2 + 4^2) = 149 - 97 = \boxed{\textbf{(E)}\ 52}\]](http://latex.artofproblemsolving.com/f/0/f/f0f77a5413873f171a139517f2351ffe0aad7887.png)
-MrTechguyPCMT and anabel.disher
Solution 4
We can calculate it as the sum of the areas of 
smaller trapezoids and larger trapezoids.
![\[2\left(\cfrac{(7+4)(7-4)}{2}\right)+2\left(\cfrac{(10+9)(10-9)}{2}\right)=10^2 - 9^2 + 7^2 - 4^2 = 19 + 33 = \boxed{\textbf{(E)}\ 52}\]](http://latex.artofproblemsolving.com/7/0/0/700260a89bb96e26af11b7f5bfb448d4fa1499c4.png)
-SahanWijetunga
Video Solution (A Clever Explanation You’ll Get Instantly)
~mr_mathman
Video Solution by Math-X (First fully understand the problem!!!)
https://youtu.be/BaE00H2SHQM?si=F37qz7vyfRgZm-1u&t=469
~Math-X
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/5ZIFnqymdDQ?si=vSpf507m-7QMFkbZ&t=238
~hsnacademy
Video Solution (easy to digest) by Power Solve
https://youtu.be/HE7JjZQ6xCk?si=39xd5CKI9nx-7lyV&t=118
Video Solution by Parshwa
https://www.youtube.com/watch?v=DQbe_BNfPYw
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=Ylw-kJkSpq8
~NiuniuMaths
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=L83DxusGkSY
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://youtu.be/sPzsce8FQtY?si=IS1hVC0cMd-0CYj5
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=158
Video Solution by Daily Dose of Math (Understandable, Quick, and Speedy)
https://youtu.be/bSPWqeNO11M?si=HIzlxPjMfvGM5lxR
~Thesmartgreekmathdude
Video Solution by WhyMath
See Also
| 2024 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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. 
