https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&feed=atom&action=history2009 AMC 10B Problems/Problem 23 - Revision history2024-03-29T08:37:35ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=157231&oldid=prevMobius247: /* Solution */2021-07-02T17:23:32Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:23, 2 July 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11" >Line 11:</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;"></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>== Solution ==</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>== Solution ==</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes, Robert will have completed <math>7</math> laps and be <math>40</math> seconds from completing his eighth lap.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> seconds and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes, Robert will have completed <math>7</math> laps and be <math>40</math> seconds from completing his eighth lap.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> seconds and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> <ins class="diffchange diffchange-inline">seconds </ins>and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>==Solution 2 (Video solution)==</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>==Solution 2 (Video solution)==</div></td></tr>
</table>Mobius247https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=157230&oldid=prevMobius247: /* Solution */2021-07-02T17:23:16Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:23, 2 July 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11" >Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</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>== Solution ==</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>== Solution ==</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes, Robert will have completed <math>7</math> laps and be <math>40</math> seconds from completing his eighth lap.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes, Robert will have completed <math>7</math> laps and be <math>40</math> seconds from completing his eighth lap.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> <ins class="diffchange diffchange-inline">seconds </ins>and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>==Solution 2 (Video solution)==</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>==Solution 2 (Video solution)==</div></td></tr>
</table>Mobius247https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=157228&oldid=prevMobius247: /* Solution */2021-07-02T17:22:16Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:22, 2 July 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11" >Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</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>== Solution ==</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>== Solution ==</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes, Robert will have completed <math>7</math> laps and <del class="diffchange diffchange-inline">will </del>be <math>40</math> seconds <del class="diffchange diffchange-inline">past the starting line</del>.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes, Robert will have completed <math>7</math> laps and be <math>40</math> seconds <ins class="diffchange diffchange-inline">from completing his eighth lap</ins>.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>==Solution 2 (Video solution)==</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>==Solution 2 (Video solution)==</div></td></tr>
</table>Mobius247https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=157226&oldid=prevMobius247: /* Solution */2021-07-02T17:13:46Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:13, 2 July 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11" >Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</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>== Solution ==</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>== Solution ==</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes Robert will have completed <math>7</math> laps and will be <math>40</math> seconds past the starting line.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes<ins class="diffchange diffchange-inline">, </ins>Robert will have completed <math>7</math> laps and will be <math>40</math> seconds past the starting line.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>==Solution 2 (Video solution)==</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>==Solution 2 (Video solution)==</div></td></tr>
</table>Mobius247https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=139532&oldid=prevDabobwholikemath1234: /* Solution 2 (Video solution) */2020-12-13T15:26:52Z<p><span dir="auto"><span class="autocomment">Solution 2 (Video solution)</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:26, 13 December 2020</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>==Solution 2 (Video solution)==</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>==Solution 2 (Video solution)==</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>Video: https://youtu.be/eZjJ5MQV47o</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>Video: https://youtu.be/eZjJ5MQV47o</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>~<del class="diffchange diffchange-inline">Dabob</del></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>~<ins class="diffchange diffchange-inline">DaBobWhoLikeMath</ins></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>Dabobwholikemath1234https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=139531&oldid=prevDabobwholikemath1234: /* Solution */2020-12-13T15:26:08Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:26, 13 December 2020</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>== Solution ==</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>== Solution ==</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes Robert will have completed <math>7</math> laps and will be <math>40</math> seconds past the starting line.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</math>.</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>After <math>10</math> minutes <math>(600</math> seconds<math>),</math> Rachel will have completed <math>6</math> laps and be <math>30</math> seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <math>22.5</math> seconds, she will be in the picture between <math>18.75</math> seconds and <math>41.25</math> seconds of the tenth minute.  After 10 minutes Robert will have completed <math>7</math> laps and will be <math>40</math> seconds past the starting line.  Because Robert runs one-fourth of a lap in <math>20</math> seconds, he will be in the picture between <math>30</math> and <math>50</math> seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <math>30</math> and <math>41.25</math> seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</math>.</div></td></tr>
<tr><td colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;">==Solution 2 (Video solution)==</ins></div></td></tr>
<tr><td colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;">Video: https://youtu.be/eZjJ5MQV47o</ins></div></td></tr>
<tr><td colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;">~Dabob</ins></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>Dabobwholikemath1234https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=114691&oldid=prevHeeeeeeeheeeee: /* Solution */2020-01-13T22:17:47Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:17, 13 January 2020</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;"></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>== Solution ==</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>== Solution ==</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>After 10 minutes (600 seconds), Rachel will have completed 6 laps and be 30 seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in 22.5 seconds, she will be in the picture between 18.75 seconds and 41.25 seconds of the tenth minute.  After 10 minutes Robert will have completed 7 laps and will be 40 seconds past the starting line.  Because Robert runs one-fourth of a lap in 20 seconds, he will be in the picture between 30 and 50 seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between 30 and 41.25 seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>After <ins class="diffchange diffchange-inline"><math></ins>10<ins class="diffchange diffchange-inline"></math> </ins>minutes <ins class="diffchange diffchange-inline"><math></ins>(600<ins class="diffchange diffchange-inline"></math> </ins>seconds<ins class="diffchange diffchange-inline"><math></ins>),<ins class="diffchange diffchange-inline"></math> </ins>Rachel will have completed <ins class="diffchange diffchange-inline"><math></ins>6<ins class="diffchange diffchange-inline"></math> </ins>laps and be <ins class="diffchange diffchange-inline"><math></ins>30<ins class="diffchange diffchange-inline"></math> </ins>seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in <ins class="diffchange diffchange-inline"><math></ins>22.5<ins class="diffchange diffchange-inline"></math> </ins>seconds, she will be in the picture between <ins class="diffchange diffchange-inline"><math></ins>18.75<ins class="diffchange diffchange-inline"></math> </ins>seconds and <ins class="diffchange diffchange-inline"><math></ins>41.25<ins class="diffchange diffchange-inline"></math> </ins>seconds of the tenth minute.  After 10 minutes Robert will have completed <ins class="diffchange diffchange-inline"><math></ins>7<ins class="diffchange diffchange-inline"></math> </ins>laps and will be <ins class="diffchange diffchange-inline"><math></ins>40<ins class="diffchange diffchange-inline"></math> </ins>seconds past the starting line.  Because Robert runs one-fourth of a lap in <ins class="diffchange diffchange-inline"><math></ins>20<ins class="diffchange diffchange-inline"></math> </ins>seconds, he will be in the picture between <ins class="diffchange diffchange-inline"><math></ins>30<ins class="diffchange diffchange-inline"></math> </ins>and <ins class="diffchange diffchange-inline"><math></ins>50<ins class="diffchange diffchange-inline"></math> </ins>seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between <ins class="diffchange diffchange-inline"><math></ins>30<ins class="diffchange diffchange-inline"></math> </ins>and <ins class="diffchange diffchange-inline"><math></ins>41.25<ins class="diffchange diffchange-inline"></math> </ins>seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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>Heeeeeeeheeeeehttps://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=54615&oldid=prevNathan wailes at 16:54, 4 July 20132013-07-04T16:54:30Z<p></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;"><div>[[Category:Introductory Combinatorics Problems]]</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>[[Category:Introductory Combinatorics Problems]]</div></td></tr>
<tr><td colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;">{{MAA Notice}}</ins></div></td></tr>
</table>Nathan waileshttps://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=31219&oldid=prevQntty at 05:31, 11 April 20092009-04-11T05:31:25Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 05:31, 11 April 2009</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>{{AMC12 box|year=2009|ab=B|num-b=17|num-a=19}}</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>{{AMC12 box|year=2009|ab=B|num-b=17|num-a=19}}</div></td></tr>
<tr><td colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;">[[Category:Introductory Combinatorics Problems]]</ins></div></td></tr>
</table>Qnttyhttps://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_23&diff=30496&oldid=prevVelaDabant at 22:58, 26 February 20092009-02-26T22:58:21Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:58, 26 February 2009</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13" >Line 13:</td>
<td colspan="2" class="diff-lineno">Line 13:</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>After 10 minutes (600 seconds), Rachel will have completed 6 laps and be 30 seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in 22.5 seconds, she will be in the picture between 18.75 seconds and 41.25 seconds of the tenth minute.  After 10 minutes Robert will have completed 7 laps and will be 40 seconds past the starting line.  Because Robert runs one-fourth of a lap in 20 seconds, he will be in the picture between 30 and 50 seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between 30 and 41.25 seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</math>.</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>After 10 minutes (600 seconds), Rachel will have completed 6 laps and be 30 seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in 22.5 seconds, she will be in the picture between 18.75 seconds and 41.25 seconds of the tenth minute.  After 10 minutes Robert will have completed 7 laps and will be 40 seconds past the starting line.  Because Robert runs one-fourth of a lap in 20 seconds, he will be in the picture between 30 and 50 seconds of the tenth minute.  Hence both Rachel and Robert will be in the picture if it is taken between 30 and 41.25 seconds of the tenth minute.  So the probability that both runners are in the picture is <math>\frac {41.25 - 30} {60} = \boxed{\frac {3}{16}}</math>.  The answer is <math>\mathrm{(C)}</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 colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;">== See also ==</ins></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>{{AMC10 box|year=2009|ab=B|num-b=22|num-a=24}}</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>{{AMC10 box|year=2009|ab=B|num-b=22|num-a=24}}</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>{{AMC12 box|year=2009|ab=B|num-b=17|num-a=19}}</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>{{AMC12 box|year=2009|ab=B|num-b=17|num-a=19}}</div></td></tr>
</table>VelaDabant