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<!DOCTYPE html>
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
<!-- Mirrored from www.mathsisfun.com/activity/random-add-multiply.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:57:57 GMT -->
<head>
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
<!-- #BeginEditable "doctitle" -->
<title>Activity: Random, or Not?</title>
<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
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<!-- #BeginEditable "Body" -->
<h1 class="center">Random, or Not?</h1>
<p>Before you begin this activity, you might lke to read these definitions:</p>
<div class="center80">
<ul>
<li><a href="../definitions/whole-number.html"><strong>Whole
numbers</strong></a><strong><br>
</strong>Whole numbers are the
numbers {0, 1, 2, 3, ...} etc.<br>
There
is no fractional or decimal part. And no negatives.<br>
</li>
<li><a href="../definitions/random.html"><strong>Random</strong></a><strong><br>
</strong>Random means: Without order. Not able to be predicted. Happening by
chance.<br>
</li>
<li><b>"Equally likely"</b> means that
each possible outcome from an
experiment has the same chance of occurring (example: when you toss a <a href="../geometry/fair-dice.html"><strong>fair
die</strong></a>,
each of the six faces is equally likely to land face up).</li>
</ul>
</div>
<h2>Add or Multiply Two Whole Numbers Together</h2>
<p>Have you ever thought about what result you get:</p>
<div class="bigul">
<ul>
<li>When you add two whole numbers together?&nbsp;</li>
<li>Or when you multiply two
whole numbers together?</li>
</ul>
</div>
<h3>In particular, are all <b>last digits</b> equally likely?</h3>
<div class="example">
<p>Example:</p>
<p><strong>39 + 57 = 96</strong> has last
digit <strong>6</strong></p>
<p><strong>38 × 45 =
1,710</strong> has last digit <strong>0</strong>.</p>
</div>
<h3>So are the digits 0 to 9 all equally likely?</h3>
<p>What's your guess?</p>
<h3><br>
<strong>Adding.</strong> Tick one of
the following:</h3>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<td><strong>When you add two
randomly selected whole numbers</strong></td>
<td align="center" width="50"><strong>Tick</strong></td>
</tr>
<tr>
<td style="text-align:right;">Yes, last digits are all equally likely</td>
<td height="30"><br></td>
</tr>
<tr>
<td style="text-align:right;">No, last digits are not all equally likely</td>
<td height="30"><br></td>
</tr>
</tbody>
</table><br>
<br>
<h3><strong>Multipling.</strong> Tick one of
the following:</h3>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<td><strong>When you multiply two
randomly selected whole numbers</strong></td>
<td align="center" width="50"><strong>Tick</strong></td>
</tr>
<tr>
<td style="text-align:right;">Yes, last digits are all equally likely</td>
<td height="30"><br></td>
</tr>
<tr>
<td style="text-align:right;">No, last digits are not all equally likely</td>
<td height="30"><br></td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>
<h3>Let's see whether you guessed correctly ...</h3>
<p>(Note: we give answers to the tables at the bottom of the page ... but only check them when you are done, or this wouldn't be an <b>activity</b> would it?)</p>
<h2>Addition</h2>
<p>Think about:</p>
<ul>
<li>13 + 18 = 31,</li>
<li>23 + 78 = 101,</li>
<li>53 + 68 = 121,
and</li>
<li>83 + 58 = 141</li>
</ul>
<p>You
will see that they all end in the digit <strong>1</strong>.</p>
<h3>So what do they have in common?</h3>
<p>They
are all sums of whole
numbers whose last digits are <strong>3</strong> and <strong>8</strong> respectively. When we add a
number ending in <strong>3</strong> to a number ending in <strong>8</strong>, we always get a number
ending in <strong>1</strong>.</p>
<p>So all we need to
consider are <b>the last digits of the two numbers</b> we are adding
together.</p>
<h3>We can do this by completing a table.</h3>
<p><strong> The following table is incomplete. Can you fill in the missing numbers?</strong></p>
<p>Remember: just the last digit after addition, so with 6+7=13, we want the "3"</p>
<table align="center" border="1">
<tbody>
<tr>
<th align="center" height="30" width="30">+</th>
<th align="center" width="30">0</th>
<th align="center" width="30">1</th>
<th align="center" width="30">2</th>
<th align="center" width="30">3</th>
<th align="center" width="30">4</th>
<th align="center" width="30">5</th>
<th align="center" width="30">6</th>
<th align="center" width="30">7</th>
<th align="center" width="30">8</th>
<th align="center" width="30">9</th>
</tr>
<tr>
<th align="center" height="30">0</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">5</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
</tr>
<tr>
<th align="center" height="30">1</th>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">2</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">4</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">7</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
</tr>
<tr>
<th align="center" height="30">2</th>
<td style="text-align:center;">2</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">5</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">7</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1 </td>
</tr>
<tr>
<th align="center" height="30">3</th>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">4</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8 </td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1 </td>
<td style="text-align:center;"><br></td>
</tr>
<tr>
<th align="center" height="30">4</th>
<td style="text-align:center;">4</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8 </td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">3</td>
</tr>
<tr>
<th align="center" height="30">5</th>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0 </td>
<td style="text-align:center;">1</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">3 </td>
<td style="text-align:center;"><br></td>
</tr>
<tr>
<th align="center" height="30">6</th>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0 </td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3 </td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
</tr>
<tr>
<th align="center" height="30">7</th>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">3</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">6</td>
</tr>
<tr>
<th align="center" height="30">8</th>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">3</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">5 </td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
</tr>
<tr>
<th align="center" height="30">9</th>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1 </td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">3</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
</tr>
</tbody>
</table>
<p><strong>You can now tally the numbers and complete a frequency table:</strong></p>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<td align="center" width="60"><strong>Last
digit</strong></td>
<td align="center" width="120"><strong>Tally</strong></td>
<td align="center" width="80"><strong>Frequency</strong></td>
<td align="center" width="80"><strong>Relative<br>
frequency</strong></td>
</tr>
<tr>
<td align="center" height="30">0</td>
<td style="text-align:center;"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"></td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td align="center" height="30">1</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">2</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">3</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">4</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">5</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">6</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">7</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">8</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">9</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>
<h3>Did you find all last digits are equally likely this time?</h3>
<p><strong>The answer is YES.<br>
</strong>Each value <strong>0</strong> to<strong> 9 </strong>occurs exactly <strong>10</strong> times out of <strong>100</strong>.<br>
So they are all equally likely, just like when you throw a <a href="../geometry/fair-dice.html">die</a>.</p>
<h3>Relative frequencies</h3>
<p><strong>Can you complete the last column of the table with the <a href="../data/relative-frequency.html">relative
frequencies</a> for each last digit?</strong></p>
<div class="example">
<h3>Example:<br>
<br>
<strong>0 </strong>occurs <strong>10</strong> times out of <strong>100</strong>, so the relative frequency
for <strong>0</strong> is <strong>10/100 = 0.1</strong></h3>
</div>
<p>&nbsp;</p>
<h2>Multiplication</h2>
<p>Think about:</p>
<ul>
<li>12 × 19 = 228,</li>
<li>22 × 79 = 1,738,</li>
<li>52 × 49 = 2,548
and</li>
<li>82 × 39 = 3,198</li>
</ul>
<p>You
will see that they all end in the digit&nbsp;8.</p>
<h3>So what do they have in common?</h3>
<p>They
are all products of whole
numbers whose last digits are <strong>2</strong> and <strong>9</strong> respectively. When
we muliply a number ending in <strong>2</strong> with a number ending in <strong>9</strong>, we always
get a number ending in <strong>8</strong>.</p>
<p>So
all we need to
consider are the last digits of the two numbers we are multiplying
together.</p>
<p><strong>The following table is incomplete. Can you fill in the missing numbers?</strong></p>
<p>Remember: just the last digit after multiplication, so with 3×6=18, we want the "8".</p>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<th height="30" width="30">×</th>
<th width="30">0</th>
<th width="30">1</th>
<th width="30">2</th>
<th width="30">3</th>
<th width="30">4</th>
<th width="30">5</th>
<th width="30">6</th>
<th width="30">7</th>
<th width="30">8</th>
<th width="30">9</th>
</tr>
<tr>
<th height="30">0</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
</tr>
<tr>
<th height="30">1</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">9</td>
</tr>
<tr>
<th height="30">2</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
</tr>
<tr>
<th height="30">3</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">7</td>
</tr>
<tr>
<th height="30">4</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">6</td>
</tr>
<tr>
<th height="30">5</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">0 </td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">0 </td>
<td style="text-align:center;">5</td>
</tr>
<tr>
<th height="30">6</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
</tr>
<tr>
<th height="30">7</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">3</td>
</tr>
<tr>
<th height="30">8</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">2</td>
</tr>
<tr>
<th height="30">9</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">6</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;"><br></td>
<td style="text-align:center;">1</td>
</tr>
</tbody>
</table>
<p><strong>You can now tally the numbers and complete a frequency table:</strong></p>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<td align="center" width="60"><strong>Last
digit</strong></td>
<td align="center" width="120"><strong>Tally</strong></td>
<td align="center" width="80"><strong>Frequency</strong></td>
<td align="center" width="80"><strong>Relative<br>
frequency</strong></td>
</tr>
<tr>
<td align="center" height="30">0</td>
<td style="text-align:center;"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally2.gif" alt="tally 2" height="20" width="14"></td>
<td style="text-align:center;">27</td>
<td style="text-align:center;">0.27</td>
</tr>
<tr>
<td align="center" height="30">1</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">2</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">3</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">4</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">5</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">6</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">7</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">8</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
<tr>
<td align="center" height="30">9</td>
<td><br></td>
<td><br></td>
<td><br></td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>
<h3>Did you find all last digits are equally likely this time?</h3>
<p><strong>The answer is still NO.</strong><br>
Last digit <strong>0</strong> occurs <strong>27</strong> times out of <strong>100</strong>,
but last digit
<strong>7</strong> only occurs four times:</p>
<blockquote><strong>1 × 7</strong>, <strong>3 × 9</strong>, <strong>7 × 1</strong> and <strong>9 × 3</strong> </blockquote>
<h3>Relative frequencies</h3>
<p><strong>Can you complete the last column of the table with the <a href="../data/relative-frequency.html">relative
frequencies</a> for each last digit?</strong></p>
<div class="example">
<h3>Example</h3>
<p><strong>0</strong> occurs <strong>27</strong> times out of <strong>100</strong>, so the relative frequency
for <strong>0</strong> is <strong>27/100 = 0.27</strong></p>
</div>
<p>&nbsp;</p>
<h2>Conclusions</h2>
<p><strong>Did you predict the results correctly? </strong></p>
<p><br>
Addition gives equally likely results but multiplication does not ... how about that!</p>
<p>Could the relative frequences be useful somehow?</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p class="center larger large">... Don't look past here until you have completed the activity! ...</p>
<p>&nbsp;</p>
<h2>Completed Tables</h2>
<p>Here are the answers:</p>
<h3>Addition</h3>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<th height="20" width="20">+</th>
<th width="20">0</th>
<th width="20">1</th>
<th width="20">2</th>
<th width="20">3</th>
<th width="20">4</th>
<th width="20">5</th>
<th width="20">6</th>
<th width="20">7</th>
<th width="20">8</th>
<th width="20">9</th>
</tr>
<tr>
<th height="20">0</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">9</td>
</tr>
<tr>
<th height="20">1</th>
<td style="text-align:center;">1</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">6 </td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">9 </td>
<td style="text-align:center;">0</td>
</tr>
<tr>
<th height="20">2</th>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3 </td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">6 </td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8 </td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1 </td>
</tr>
<tr>
<th height="20">3</th>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5 </td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8 </td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1 </td>
<td style="text-align:center;">2</td>
</tr>
<tr>
<th height="20">4</th>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5 </td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8 </td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1 </td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3</td>
</tr>
<tr>
<th height="20">5</th>
<td style="text-align:center;">5</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7 </td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">0 </td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3 </td>
<td style="text-align:center;">4</td>
</tr>
<tr>
<th height="20">6</th>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7 </td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">0 </td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3 </td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5 </td>
</tr>
<tr>
<th height="20">7</th>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">9 </td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">2 </td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5 </td>
<td style="text-align:center;">6</td>
</tr>
<tr>
<th height="20">8</th>
<td style="text-align:center;">8</td>
<td style="text-align:center;">9 </td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">2 </td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5 </td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7</td>
</tr>
<tr>
<th height="20">9</th>
<td style="text-align:center;">9</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1 </td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4 </td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7 </td>
<td style="text-align:center;">8</td>
</tr>
</tbody>
</table><br>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<td style="text-align:center;"><strong>Last
digit</strong></td>
<td style="text-align:center;"><strong>Frequency</strong></td>
<td style="text-align:center;"><strong>Relative<br>
frequency</strong></td>
</tr>
<tr>
<td style="text-align:center;">0</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">1</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">2</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">3</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">4</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">5</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">6</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">7</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">8</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">9</td>
<td style="text-align:center;">10</td>
<td style="text-align:center;">0.1</td>
</tr>
<tr>
<td style="text-align:center;">Total</td>
<td style="text-align:center;">100</td>
<td style="text-align:center;">1.0</td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>
<h3>Multiplication</h3>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<th height="20" width="20">×</th>
<th width="20">0</th>
<th width="20">1</th>
<th width="20">2</th>
<th width="20">3</th>
<th width="20">4</th>
<th width="20">5</th>
<th width="20">6</th>
<th width="20">7</th>
<th width="20">8</th>
<th width="20">9</th>
</tr>
<tr>
<th height="20">0</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">0</td>
</tr>
<tr>
<th height="20">1</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">9</td>
</tr>
<tr>
<th height="20">2</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">8</td>
</tr>
<tr>
<th height="20">3</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">7</td>
</tr>
<tr>
<th height="20">4</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">6</td>
</tr>
<tr>
<th height="20">5</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">5</td>
</tr>
<tr>
<th height="20">6</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">4</td>
</tr>
<tr>
<th height="20">7</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">1</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">3</td>
</tr>
<tr>
<th height="20">8</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">0</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">2</td>
</tr>
<tr>
<th height="20">9</th>
<td style="text-align:center;">0</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">8</td>
<td style="text-align:center;">7</td>
<td style="text-align:center;">6</td>
<td style="text-align:center;">5</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">3</td>
<td style="text-align:center;">2</td>
<td style="text-align:center;">1</td>
</tr>
</tbody>
</table><br>
<table align="center" cellspacing="2" cellpadding="2" border="1">
<tbody>
<tr>
<td style="text-align:center;"><strong>Last
digit</strong></td>
<td style="text-align:center;"><strong>Frequency</strong></td>
<td style="text-align:center;"><strong>Relative<br>
frequency</strong></td>
</tr>
<tr>
<td style="text-align:center;">0</td>
<td style="text-align:center;">27</td>
<td style="text-align:center;">0.27</td>
</tr>
<tr>
<td style="text-align:center;">1</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">0.04</td>
</tr>
<tr>
<td style="text-align:center;">2</td>
<td style="text-align:center;">12</td>
<td style="text-align:center;">0.12</td>
</tr>
<tr>
<td style="text-align:center;">3</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">0.04</td>
</tr>
<tr>
<td style="text-align:center;">4</td>
<td style="text-align:center;">12</td>
<td style="text-align:center;">0.12</td>
</tr>
<tr>
<td style="text-align:center;">5</td>
<td style="text-align:center;">9</td>
<td style="text-align:center;">0.09</td>
</tr>
<tr>
<td style="text-align:center;">6</td>
<td style="text-align:center;">12</td>
<td style="text-align:center;">0.12</td>
</tr>
<tr>
<td style="text-align:center;">7</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">0.04</td>
</tr>
<tr>
<td style="text-align:center;">8</td>
<td style="text-align:center;">12</td>
<td style="text-align:center;">0.12</td>
</tr>
<tr>
<td style="text-align:center;">9</td>
<td style="text-align:center;">4</td>
<td style="text-align:center;">0.04</td>
</tr>
<tr>
<td style="text-align:center;">Total</td>
<td style="text-align:center;">100</td>
<td style="text-align:center;">1.00</td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>
<div class="related">
<a href="../data/relative-frequency.html">Relative Frequencies</a>
<a href="index.html">Activity Index</a>
</div>
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