https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_12&feed=atom&action=history1988 AHSME Problems/Problem 12 - Revision history2024-03-28T13:25:25ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_12&diff=175252&oldid=prevPlainoldnumbertheory: /* Solution (needs to be checked) */2022-06-24T12:59:35Z<p><span dir="auto"><span class="autocomment">Solution (needs to be checked)</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;"><div>We can draw a sample space diagram, and find that of the <math>9^2 = 81</math> possibilities, <math>9</math> of them give a sum of <math>0</math>, and each other sum mod <math>10</math> (from <math>1</math> to <math>9</math>) is given by <math>8</math> of the possibilities (and indeed we can check that <math>9 + 8 \times 9 = 81</math>). Thus <math>0</math> is the most likely, so the answer is <math>\boxed{\text{A}}</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>We can draw a sample space diagram, and find that of the <math>9^2 = 81</math> possibilities, <math>9</math> of them give a sum of <math>0</math>, and each other sum mod <math>10</math> (from <math>1</math> to <math>9</math>) is given by <math>8</math> of the possibilities (and indeed we can check that <math>9 + 8 \times 9 = 81</math>). Thus <math>0</math> is the most likely, so the answer is <math>\boxed{\text{A}}</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="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>==Solution (<del class="diffchange diffchange-inline">needs to be checked</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>==Solution (<ins class="diffchange diffchange-inline">alternative explanation</ins>)==  </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>The largest sum that can be formed from two slips of these papers is 18</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>The largest sum that can be formed from two slips of these papers is 18<ins class="diffchange diffchange-inline">, and if we list the composition of each of the numbers greater than 1 and less than 19, we can find that there are 9 combinations in total which make a sum of 10 (<math>1+9</math>, <math>2+8</math>, etc). Notice that the order in which either number appears doesn't matter since they all count toward the number of possibilities.</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>Plainoldnumbertheoryhttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_12&diff=175251&oldid=prevPlainoldnumbertheory: /* Solution */2022-06-24T12:53:16Z<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 12:53, 24 June 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14" >Line 14:</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>We can draw a sample space diagram, and find that of the <math>9^2 = 81</math> possibilities, <math>9</math> of them give a sum of <math>0</math>, and each other sum mod <math>10</math> (from <math>1</math> to <math>9</math>) is given by <math>8</math> of the possibilities (and indeed we can check that <math>9 + 8 \times 9 = 81</math>). Thus <math>0</math> is the most likely, so the answer is <math>\boxed{\text{A}}</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>We can draw a sample space diagram, and find that of the <math>9^2 = 81</math> possibilities, <math>9</math> of them give a sum of <math>0</math>, and each other sum mod <math>10</math> (from <math>1</math> to <math>9</math>) is given by <math>8</math> of the possibilities (and indeed we can check that <math>9 + 8 \times 9 = 81</math>). Thus <math>0</math> is the most likely, so the answer is <math>\boxed{\text{A}}</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;">==Solution (needs to be checked)== </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;">The largest sum that can be formed from two slips of these papers is 18</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>Plainoldnumbertheoryhttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_12&diff=92389&oldid=prevHapaxoromenon: Added a box around the answer2018-02-26T22:15:29Z<p>Added a box around the answer</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:15, 26 February 2018</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>We can draw a sample space diagram, and find that of the <math>9^2 = 81</math> possibilities, <math>9</math> of them give a sum of <math>0</math>, and each other sum mod <math>10</math> (from <math>1</math> to <math>9</math>) is given by <math>8</math> of the possibilities (and indeed we can check that <math>9 + 8 \times 9 = 81</math>). Thus <math>0</math> is the most likely, so the answer is <math>A</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>We can draw a sample space diagram, and find that of the <math>9^2 = 81</math> possibilities, <math>9</math> of them give a sum of <math>0</math>, and each other sum mod <math>10</math> (from <math>1</math> to <math>9</math>) is given by <math>8</math> of the possibilities (and indeed we can check that <math>9 + 8 \times 9 = 81</math>). Thus <math>0</math> is the most likely, so the answer is <math><ins class="diffchange diffchange-inline">\boxed{\text{</ins>A<ins class="diffchange diffchange-inline">}}</ins></math>.</div></td></tr>
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</table>Hapaxoromenonhttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_12&diff=92388&oldid=prevHapaxoromenon: Added a solution with explanation2018-02-26T22:14:30Z<p>Added a solution with explanation</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:14, 26 February 2018</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> </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">We can draw a sample space diagram, and find that of the <math>9^2 = 81</math> possibilities, <math>9</math> of them give a sum of <math>0</math>, and each other sum mod <math>10</math> (from <math>1</math> to <math>9</math>) is given by <math>8</math> of the possibilities (and indeed we can check that <math>9 + 8 \times 9 = 81</math>). Thus <math>0</math> is the most likely, so the answer is <math>A</math>.</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>
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</table>Hapaxoromenonhttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_12&diff=65576&oldid=prevTimneh: Created page with "==Problem== Each integer <math>1</math> through <math>9</math> is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at ra..."2014-10-23T05:53:52Z<p>Created page with "==Problem== Each integer <math>1</math> through <math>9</math> is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at ra..."</p>
<p><b>New page</b></p><div>==Problem==<br />
<br />
Each integer <math>1</math> through <math>9</math> is written on a separate slip of paper and all nine slips are put into a hat. <br />
Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. <br />
Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?<br />
<br />
<math>\textbf{(A)}\ 0\qquad<br />
\textbf{(B)}\ 1\qquad<br />
\textbf{(C)}\ 8\qquad<br />
\textbf{(D)}\ 9\qquad<br />
\textbf{(E)}\ \text{each digit is equally likely} </math><br />
<br />
==Solution==<br />
<br />
<br />
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== See also ==<br />
{{AHSME box|year=1988|num-b=11|num-a=13}} <br />
<br />
[[Category: Introductory Number Theory Problems]]<br />
{{MAA Notice}}</div>Timneh