Difference between revisions of "2024 AMC 8 Problems/Problem 3"
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− | ==Problem== | + | ==Problem 3 == |
− | What is the | + | 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)--(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> | ||
+ | <math>\textbf{(A)}\ 42 \qquad \textbf{(B)}\ 45\qquad \textbf{(C)}\ 49\qquad \textbf{(D)}\ 50\qquad \textbf{(E)}\ 52</math> | ||
+ | |||
+ | [[2024 AMC 8 Problems/Problem 3|Solution]] | ||
==Solution 1== | ==Solution 1== |
Revision as of 12:27, 26 January 2024
Contents
[hide]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?
Solution 1
We can simplify this expression into . Now, taking the common denominator, we get
~Dreamer1297
Solution 2
We convert each of the fractions to . After adding up the values, we get .
-ILoveMath31415926535