2024 AMC 8 Problems/Problem 3
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)--(4,11),linewidth(1)); draw((6,11)--(10,11),linewidth(1)); draw((0,11.25)--(0,10.75),linewidth(1)); draw((10,11.25)--(10,10.75),linewidth(1)); label("$9$",(5,11),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/6/e/a/6ea6b3881a8bf974d8354925f4beacf0bf020745.png)
Solution 1
We can simplify this expression into 
. Now, taking the common denominator, we get ![\[\frac{200}{50}+\frac{125}{50}+\frac{2}{50}\]](http://latex.artofproblemsolving.com/f/a/f/fafe7fec668c75cd9b9f12bf444aec1e6c167719.png)
![\[= \frac{200+125+2}{50}\]](http://latex.artofproblemsolving.com/a/0/3/a03873103972cbf654a34dfd7bcf97b5db9413e1.png)
![\[= \frac{327}{50}\]](http://latex.artofproblemsolving.com/8/8/b/88b748fc048fd4ff8456c910e852f3e485a9832a.png)
![\[= \frac{654}{100}\]](http://latex.artofproblemsolving.com/b/e/f/bef73b25eb9085283b60990d080efe56451865ec.png)
![\[= \boxed{\textbf{(C) }6.54}\]](http://latex.artofproblemsolving.com/a/2/4/a2435d588ba21f0e8429dc29067aafe587a8ec58.png)
~Dreamer1297
Solution 2
We convert each of the fractions to 
. After adding up the values, we get 
.
-ILoveMath31415926535
