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<!DOCTYPE html>
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
<!-- Mirrored from www.mathsisfun.com/1089algebra.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:01:49 GMT -->
<head>
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<!-- #BeginEditable "doctitle" -->
<title>Why the Answer is Always 1089</title>
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<h1 class="center">Why is it always 1089</h1>
<p><i><br>
</i></p>
<p><i>This is the reasoning behind <a href="1089.html">Always End With 1089</a>. </i></p>
<p>We will need a little bit of <a href="algebra/index.html">algebra</a>, but stick with it, it is not difficult.</p>
<p><br></p>
<p class="center">Show the number as <b>ABC</b></p>
<p class="center"><img src="images/arrow3.gif" alt="down arrow" height="30" width="30"></p>
<p class="center">Reverse this to get <b>CBA</b></p>
<p class="center"><img src="images/arrow3.gif" alt="down arrow" height="30" width="30"></p>
<p class="center">Remember that A is Hundreds B
is tens and C is units<br>
Take <b>CBA</b> from <b>ABC</b> like this:</p>
<p class="center"><img src="images/arrow3.gif" alt="down arrow" height="30" width="30"></p>
<div class="simple">
<table style="margin:auto;">
<tbody>
<tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center; width:80px;"><b>Hundreds</b> </td>
<td style="text-align:center; width:80px;"><b>Tens</b> </td>
<td style="text-align:center; width:80px;"><b>Units</b> </td>
</tr><tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center;">A </td>
<td style="text-align:center;">B </td>
<td style="text-align:center;">C </td>
</tr>
<tr>
<td style="text-align:center;">Subtract:</td>
<td style="text-align:center;">C </td>
<td style="text-align:center;">B </td>
<td style="text-align:center;">A </td>
</tr><tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center;"><b>A&minus;C </b></td>
<td style="text-align:center;"><b>0 </b></td>
<td style="text-align:center;"><b>C&minus;A </b></td>
</tr>
</tbody></table>
</div>
<p class="center"><img src="images/arrow3.gif" alt="down arrow" height="30" width="30"></p>
<p class="center">Now here is the trick:<br>
Subtract 1 Hundred, and add 9 Tens and 10 Ones<br>
(&minus;100, +90, +10 = 0, so won't change answer):</p>
<div class="simple">
<table style="margin:auto;">
<tbody>
<tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center; width:80px;"><b>Hundreds</b> </td>
<td style="text-align:center; width:80px;"><b>Tens</b> </td>
<td style="text-align:center; width:80px;"><b>Units</b> </td>
</tr><tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center;">A<span class="hilite">&minus;1</span></td>
<td style="text-align:center;">B<span class="hilite">+9</span></td>
<td style="text-align:center;">C<span class="hilite">+10</span></td>
</tr>
<tr>
<td style="text-align:center;">Subtract:</td>
<td style="text-align:center;">C </td>
<td style="text-align:center;">B </td>
<td style="text-align:center;">A </td>
</tr><tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center;"><b>A&minus;1&minus;C </b></td>
<td style="text-align:center;"><b>9 </b></td>
<td style="text-align:center;"><b>10+C&minus;A </b></td>
</tr>
</tbody></table>
</div>
<p class="center"><img src="images/arrow3.gif" alt="down arrow" height="30" width="30"></p>
<p class="center">Last Step: Reverse the answer and<br>
add the
two numbers together.</p>
<div class="simple">
<table style="margin:auto;">
<tbody>
<tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center; width:80px;"><b>Hundreds</b> </td>
<td style="text-align:center; width:80px;"><b>Tens</b> </td>
<td style="text-align:center; width:80px;"><b>Units</b> </td>
</tr><tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center; width:80px;">A&minus;1&minus;C </td>
<td style="text-align:center; width:80px;">9 </td>
<td style="text-align:center; width:80px;">10+C&minus;A </td>
</tr>
<tr>
<td style="text-align:center;">Add:</td>
<td style="text-align:center;">10+C&minus;A </td>
<td style="text-align:center;">9 </td>
<td style="text-align:center;">A&minus;1&minus;C </td>
</tr><tr>
<td style="text-align:center;"><br>
</td>
<td style="text-align:center;">9 </td>
<td style="text-align:center;">18 </td>
<td style="text-align:center;">9 </td>
</tr><tr>
<td style="text-align:center;">Simplify:
</td>
<td style="text-align:center;">10 </td>
<td style="text-align:center;">8 </td>
<td style="text-align:center;">9 </td>
</tr>
</tbody></table>
</div>
<p class="center"><span class="large">As predicted the answer was 1089</span></p>
<p class="center">&nbsp;</p>
<div class="related">
<a href="1089.html">Always End With 1089</a>
<a href="puzzles/index.html">Puzzles Index</a>
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