https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_10B_Problems/Problem_5&feed=atom&action=history2007 AMC 10B Problems/Problem 5 - Revision history2024-03-28T18:58:39ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_10B_Problems/Problem_5&diff=88839&oldid=prevA87321dc0: /* Solution */2017-12-09T04:52:22Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It may be easier to visualize this by drawing some sort of diagram. From the first statement, you can draw an Arog circle inside of the Braf circle, since all Arogs are Brafs, but <del class="diffchange diffchange-inline">no </del>all Brafs are Arogs. Ignore the second statement for now, and draw a Dramp circle in the Arog circle and a Crup circle in the Dramp circle. You can see the second statement is already true because all Crups are Arogs. As you can see, the only statement that is true is <math>\boxed{\mathrm{(D)}}</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It may be easier to visualize this by drawing some sort of diagram. From the first statement, you can draw an Arog circle inside of the Braf circle, since all Arogs are Brafs, but <ins class="diffchange diffchange-inline">not </ins>all Brafs are Arogs. Ignore the second statement for now, and draw a Dramp circle in the Arog circle and a Crup circle in the Dramp circle. You can see the second statement is already true because all Crups are Arogs. As you can see, the only statement that is true is <math>\boxed{\mathrm{(D)}}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td></tr>
</table>A87321dc0https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_10B_Problems/Problem_5&diff=79417&oldid=prevSlester at 17:27, 18 July 20162016-07-18T17:27:16Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:27, 18 July 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C) } \text{All Arogs are Crups and are Dramps.}\\</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C) } \text{All Arogs are Crups and are Dramps.}\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D) } \text{All Crups are Arogs and are Brafs.}\\</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D) } \text{All Crups are Arogs and are Brafs.}\\</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E) } \text{All Arogs are Dramps and some Arogs may not be <del class="diffchange diffchange-inline">Crumps</del>.}</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E) } \text{All Arogs are Dramps and some Arogs may not be <ins class="diffchange diffchange-inline">Crups</ins>.}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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</table>Slesterhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_10B_Problems/Problem_5&diff=54547&oldid=prevNathan wailes at 16:19, 4 July 20132013-07-04T16:19:49Z<p></p>
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</table>Nathan waileshttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_10B_Problems/Problem_5&diff=39210&oldid=prevGina: Created page with "==Problem== In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these fa..."2011-06-05T19:08:27Z<p>Created page with "==Problem== In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these fa..."</p>
<p><b>New page</b></p><div>==Problem==<br />
<br />
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?<br />
<br />
<math>\textbf{(A) } \text{All Dramps are Brafs and are Crups.}\\<br />
\textbf{(B) } \text{All Brafs are Crups and are Dramps.}\\<br />
\textbf{(C) } \text{All Arogs are Crups and are Dramps.}\\<br />
\textbf{(D) } \text{All Crups are Arogs and are Brafs.}\\<br />
\textbf{(E) } \text{All Arogs are Dramps and some Arogs may not be Crumps.}</math><br />
<br />
==Solution==<br />
<br />
<center><asy><br />
unitsize(6mm);<br />
defaultpen(linewidth(.8pt)+fontsize(9pt));<br />
dotfactor=4;<br />
<br />
real r1=1, r2=2, r3=3, r4=4;<br />
pair O1=(0,0), O2=(0,-0.5), O3=(0,-1), O4=(0,-1.5);<br />
path circleA=Circle(O1,r1); draw(circleA);<br />
path circleB=Circle(O2,r2); draw(circleB);<br />
path circleC=Circle(O3,r3); draw(circleC);<br />
path circleD=Circle(O4,r4); draw(circleD);<br />
<br />
label("$Crups$",(0,-.5));<br />
label("$Dramps$",(0,-2));<br />
label("$Arogs$",(0,-3.5));<br />
label("$Brafs$",(0,-5));<br />
</asy></center><br />
<br />
It may be easier to visualize this by drawing some sort of diagram. From the first statement, you can draw an Arog circle inside of the Braf circle, since all Arogs are Brafs, but no all Brafs are Arogs. Ignore the second statement for now, and draw a Dramp circle in the Arog circle and a Crup circle in the Dramp circle. You can see the second statement is already true because all Crups are Arogs. As you can see, the only statement that is true is <math>\boxed{\mathrm{(D)}}</math><br />
<br />
==See Also==<br />
<br />
{{AMC10 box|year=2007|ab=B|num-b=4|num-a=6}}</div>Gina