<p><img alt="[asy] draw((0,0)--(10,20*sqrt(3)/2)--(20,0)--cycle,black+linewidth(.75)); draw((20,0)--(20,12)--(32,12)--(32,0)--cycle,black+linewidth(.75)); draw((32,0)--(37,10*sqrt(3)/2)--(42,0)--cycle,black+linewidth(.75)); MP("I",(10,0),N);MP("II",(26,0),N);MP("III",(37,0),N); MP("A",(0,0),S);MP("B",(20,0),S);MP("C",(32,0),S);MP("D",(42,0),S); [/asy]" height="118" src="https://latex.artofproblemsolving.com/7/2/d/72d20f34a21f4d35e2f011ff1523d4b9e962577c.png" width="252" /></p> <p>In this diagram, not drawn to scale, Figures <img alt="$I$" height="12" src="https://latex.artofproblemsolving.com/0/2/7/027f4a11d6090f9eac0ce2488df6384dad1263ea.png" width="9" /> and <img alt="$III$" height="12" src="https://latex.artofproblemsolving.com/4/0/a/40a16dd7d1d1fedb353096f50c74ce0da7d1d093.png" width="28" /> are equilateral triangular regions with respective areas of <img alt="$32\sqrt{3}$" height="18" src="https://latex.artofproblemsolving.com/b/b/0/bb091bfa9be998f89ca94c53b39b17e531a62d76.png" width="43" /> and <img alt="$8\sqrt{3}$" height="18" src="https://latex.artofproblemsolving.com/8/5/3/8534d43cb0583d4a61bb00cb328d4538d971ab7a.png" width="33" /> square inches. Figure <img alt="$II$" height="14" src="https://latex.artofproblemsolving.com/a/f/3/af3906651d16a548230a504127bc5b308a64068f.png" width="21" /> is a square region with area <img alt="$32$" height="12" src="https://latex.artofproblemsolving.com/4/f/4/4f4b8590f8aa9f9894b3891dd5dede0e354fa04e.png" width="17" /> square inches. Let the length of segment <img alt="$AD$" height="13" src="https://latex.artofproblemsolving.com/b/c/5/bc56862b1fdb71204894da278e42ef75de73a40b.png" width="28" /> be decreased by <img alt="$12\tfrac{1}{2}$" height="21" src="https://latex.artofproblemsolving.com/7/9/c/79c0ebdb35902c248213cd96d2c4d538061a6a03.png" width="27" /> % of itself, while the lengths of <img alt="$AB$" height="13" src="https://latex.artofproblemsolving.com/5/7/e/57ee5125358c0606c9b588580ddfa66f83e607b7.png" width="27" /> and <img alt="$CD$" height="12" src="https://latex.artofproblemsolving.com/0/9/4/0948661e822f2a953c43a57ac9e40b2734476de4.png" width="29" /> remain unchanged. The percent decrease in the area of the square is:</p>

Question No: 313

$[asy] draw((0,0)--(10,20*sqrt(3)/2)--(20,0)--cycle,black+linewidth(.75)); draw((20,0)--(20,12)--(32,12)--(32,0)--cycle,black+linewidth(.75)); draw((32,0)--(37,10*sqrt(3)/2)--(42,0)--cycle,black+linewidth(.75)); MP("I",(10,0),N);MP("II",(26,0),N);MP("III",(37,0),N); MP("A",(0,0),S);MP("B",(20,0),S);MP("C",(32,0),S);MP("D",(42,0),S); [/asy]$

In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\tfrac{1}{2}$ % of itself, while the lengths of $AB$ and $CD$ remain unchanged. The percent decrease in the area of the square is:

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Tags: Algebra Geometry

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