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469 lines
19 KiB
HTML
469 lines
19 KiB
HTML
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 id="title" align="center">Least Common Denominator</h1>
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<span id="video"></span>
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<script type="text/javascript">initVideo('Ip4uIKA7uTs');</script>
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<h2>What is a ...<br>
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... Denominator?</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/fraction.svg" alt="fraction" /></p>
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<p>The <b>denominator</b> is the bottom number in a fraction.</p>
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<p><i>It shows how many equal parts the item is divided into</i></p>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/common-denominator.svg" alt="2/5 + 1/5 common denominator" /> </p>
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<h2>... Common Denominator?</h2>
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<p>When the denominators of two or more fractions are the <b>same</b>, they have <b>Common Denominators</b>.</p><div style="clear:both"></div>
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<h2>... Least Common Denominator?</h2>
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<p>it is the <b>smallest</b> of all the common denominators. </p>
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<h2>Why?</h2>
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<p>Why do we want common denominators?</p>
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<p>Because we <b>can't</b> add fractions with different denominators:</p>
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<table border="0" align="center">
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<tr align="center">
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<td><span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span></td>
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<td width="40" class="larger">+</td>
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<td><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></td>
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<td width="40" class="larger">=</td>
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<td>?</td>
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</tr>
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<tr align="center">
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<td><img src="images/fractions/pizza-1-3.jpg" alt="1/3" width="111" height="111" /></td>
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<td class="large"> </td>
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<td><img src="images/fractions/pizza-1-6.jpg" alt="1/6" width="112" height="111" /></td>
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<td class="large"> </td>
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<td><img src="images/fractions/pie-huh.jpg" width="120" height="120" alt="pie huh" /></td>
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</tr>
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</table>
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<p>Before we can add them we must make the <b>denominators the same</b>.</p>
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<h2>Finding a Common Denominator</h2>
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<p>But what should the new denominator be?</p>
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<p>One simple answer is to multiply the current denominators together: </p>
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<p align="center"><span class="large">3 × 6 = 18</span></p>
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<p>So instead of having 3 or 6 slices, we will make <b>both</b> of them have <b>18 slices</b>.</p>
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<p>The pizzas now look like this:</p>
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<table border="0" align="center">
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<tr align="center">
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<td><span class="intbl">
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<em>6</em>
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<strong>18</strong>
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</span></td>
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<td width="40">+</td>
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<td><span class="intbl">
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<em>3</em>
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<strong>18</strong>
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</span></td>
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<td width="40">=</td>
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<td><span class="intbl">
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<em>9</em>
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<strong>18</strong>
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</span></td>
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</tr>
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<tr align="center">
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<td><img src="images/fractions/pizza-6-18.jpg" alt="1/3" width="114" height="114" /></td>
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<td class="large"> </td>
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<td><img src="images/fractions/pizza-3-18.jpg" alt="1/6" width="114" height="113" /></td>
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<td class="large"> </td>
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<td><img src="images/fractions/pizza-9-18.jpg" width="114" height="114" alt="pizza 9/18" /></td>
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</tr>
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</table>
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<p>They now have common denominators (but not the <i>least</i> common denominator)</p>
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<p><i>(Read more about <a href="numbers/common-denominator.html">Common Denominators</a>.)</i></p>
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<h2>Least Common Denominator</h2>
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<p>That is all fine, but 18 is a lot of slices ... can we do it with <b>fewer slices</b>?</p>
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<p>Here is how to find out:</p>
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<table border="0" align="center">
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<tr>
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<td width="60" align="center"><span class="large"><span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span></span></td>
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<td align="right">List multiples of 3: </td>
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<td> </td>
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<td class="larger">3, 6, 9, 12, 15, 18, 21, ...</td>
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</tr>
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<tr>
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<td align="center"><span class="large"><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></span></td>
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<td align="right">List multiples of 6: </td>
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<td> </td>
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<td class="larger">6, 12, 18, 24, ...</td>
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</tr>
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</table>
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<p>Now find the <b>smallest number</b> that is the same:</p>
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<table border="0" align="center">
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<tr>
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<td align="right">multiples of 3: </td>
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<td width="15"> </td>
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<td class="larger">3, <b class="hilite">6</b>, 9, 12, 15, 18, 21, ...</td>
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</tr>
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<tr>
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<td align="right"> multiples of 6: </td>
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<td> </td>
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<td class="larger"><b class="hilite">6</b>, 12, 18, 24, ...</td>
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</tr>
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</table>
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<p> The answer is <span class="large">6</span>, and that is the <span class="larger"><b>Least</b> Common Denominator</span>.</p>
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<p>So let us try using it! </p>
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<p>We want both fractions to have 6 slices:</p>
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<ul>
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<li>When we multiply top and bottom of <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> by 2 we get <span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span> </li>
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<li><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span> already has a denominator of 6</li>
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</ul>
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<p>And our question now looks like: </p>
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<table border="0" align="center">
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<tr align="center">
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<td width="124"><span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span></td>
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<td width="143">+</td>
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<td width="143"><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></td>
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<td width="145">=</td>
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<td width="145"><span class="intbl">
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<em>3</em>
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<strong>6</strong>
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</span></td>
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<td width="145"> </td>
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<td width="145"> </td>
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</tr>
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<tr align="center">
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<td width="124"><img src="images/fractions/pizza-2-6.jpg" alt="2/6" width="112" height="112" /></td>
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<td width="143" class="large"> </td>
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<td width="143"><img src="images/fractions/pizza-1-6.jpg" alt="1/6" width="112" height="111" /></td>
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<td width="145" class="large"> </td>
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<td width="145"><img src="images/fractions/pizza-3-6.jpg" alt="3/6" width="112" height="111" /></td>
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<td width="145" class="large"> </td>
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<td width="145"><img src="images/fractions/pie-blank.jpg" width="120" height="120" alt="empty" /></td>
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</tr>
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</table>
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<p>One last step is to <a href="simplifying-fractions.html">simplify the fraction</a> (if possible). In this case 3/6 is simpler as 1/2:</p>
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<table border="0" align="center">
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<tr align="center">
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<td width="124"><span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span></td>
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<td width="143">+</td>
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<td width="143"><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></td>
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<td width="145">=</td>
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<td width="145"><span class="intbl">
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<em>3</em>
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<strong>6</strong>
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</span></td>
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<td width="145">=</td>
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<td width="145"><span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span></td>
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</tr>
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<tr align="center">
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<td width="124"><img src="images/fractions/pizza-2-6.jpg" alt="2/6" width="112" height="112" /></td>
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<td width="143" class="large"> </td>
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<td width="143"><img src="images/fractions/pizza-1-6.jpg" alt="1/6" width="112" height="111" /></td>
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<td width="145" class="large"> </td>
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<td width="145"><img src="images/fractions/pizza-3-6.jpg" alt="3/6" width="112" height="111" /></td>
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<td width="145" class="large"> </td>
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<td width="145"><img src="images/fractions/pizza-1-2.jpg" alt="3/6" width="112" height="111" /></td>
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</tr>
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</table>
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<p> </p>
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<p align="center">And that is what the <span class="larger">Least Common Denominator</span> is all about.</p>
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<p align="center">It lets us add (or subtract) fractions using the least number of slices.</p>
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<div class="clear"></div>
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<h2>What Did We Do?</h2>
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<p>The trick was to list the multiples of each denominator, then find the <a href="least-common-multiple.html">Least Common Multiple</a></p>
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<p> In the previous example the Least Common Multiple of 3 and 6
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was <span class="large">6</span>. </p>
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<p>In other words the <b>Least Common Denominator</b> of <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> and <span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span> is <b class="large">6</b>.</p>
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<p>Here are the steps to follow:</p>
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<table border="0">
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<tr>
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<td><img src="images/art/sage-looking-ear-down.gif" width="77" height="100" alt="dog looking ear down" /></td>
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<td><ul>
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<div class="bigul">
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<li>Find the <b>Least Common Multiple</b> of the denominators (which is called the <b>Least Common Denominator</b>).</li>
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<li>Change each fraction (using <a href="equivalent_fractions.html">equivalent fractions</a>) to make their denominators the same as the least common denominator</li>
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<li>Then add (or subtract) the fractions, as we wish!</li>
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</div>
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</ul></td>
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</tr>
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</table>
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<div class="example">
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<h3>Example: What is <span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span> + <span class="intbl">
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<em> 7</em>
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<strong>15</strong>
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</span> ? </h3>
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<p>The Denominators are 6 and 15:</p>
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<table border="0" align="center">
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<tr>
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<td align="right">multiples of 6: </td>
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<td> </td>
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<td class="larger">6, 12, 18, 24, <b>30</b>, 36, ...</td>
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</tr>
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<tr>
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<td align="right"> multiples 15: </td>
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<td> </td>
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<td class="larger">15,<b> 30</b>, 45, 60, ...</td>
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</tr>
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</table>
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<p>So the <b>Least Common Multiple</b> of 6 and 15 is <b>30</b>. </p>
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<p> </p>
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<p>Now let's try to make the denominators the same. </p>
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<p align="center" class="larger">Note: what we do to the bottom of the fraction, <br />
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we must also do to the top</p>
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<p>For the first fraction we can multiply top and bottom by 5 to get a denominator of 30:</p>
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<table align="center" border="0" cellspacing="3" cellpadding="0">
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<tr align="center">
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<td> </td>
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<td><b>× 5</b></td>
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<td> </td>
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</tr>
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<tr align="center">
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<td colspan="3"><img src="images/style/right-over-arrow.gif" width="75" height="25" alt="right over arrow" /></td>
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</tr>
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<tr align="center" class="large">
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<td><span class="intbl"><em>1</em><strong>6</strong></span></td>
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<td width="45">=</td>
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<td><span class="intbl"><em>5</em><strong>30</strong></span></td>
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</tr>
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<tr align="center">
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<td colspan="3"><img src="images/style/right-under-arrow.gif" width="75" height="25" alt="right under arrow" /></td>
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</tr>
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<tr align="center">
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<td> </td>
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<td><b>× 5</b></td>
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<td> </td>
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</tr>
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</table>
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<p> </p>
|
|
<p>For the second fraction we can multiply top and bottom by 2 to get a denominator of 30:</p>
|
|
<table align="center" border="0" cellspacing="3" cellpadding="0">
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td><b>× 2</b></td>
|
|
<td> </td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td colspan="3"><img src="images/style/right-over-arrow.gif" width="75" height="25" alt="right over arrow" /></td>
|
|
</tr>
|
|
<tr align="center" class="large">
|
|
<td><span class="intbl"><em>7</em><strong>15</strong></span></td>
|
|
<td width="45">=</td>
|
|
<td><span class="intbl"><em>14</em><strong>30</strong></span></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td colspan="3"><img src="images/style/right-under-arrow.gif" width="75" height="25" alt="right under arrow" /></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td><b>× 2</b></td>
|
|
<td> </td>
|
|
</tr>
|
|
</table>
|
|
<p> </p>
|
|
<p>Now we can do the addition by adding the top numbers:</p>
|
|
<p align="center"> <span class="intbl">
|
|
<em>5</em>
|
|
<strong>30</strong>
|
|
</span> + <span class="intbl">
|
|
<em>14</em>
|
|
<strong>30</strong>
|
|
</span> = <span class="intbl">
|
|
<em>19</em>
|
|
<strong>30</strong>
|
|
</span></p>
|
|
<p>The fraction is already as simple as it can be, so that is the answer.</p>
|
|
</div>
|
|
<h3> </h3>
|
|
<h2>Least Common Multiple Tool</h2>
|
|
To find the least common denominator automatically use the <a href="least-common-multiple-tool.html">Least Common Multiple Tool</a>. Just put in the denominators, press the button, and the least common denominator is shown.
|
|
<h2>One More Example</h2>
|
|
<div class="example">
|
|
<h3>Example: What is <span class="intbl">
|
|
<em>3</em>
|
|
<strong>8</strong>
|
|
</span> + <span class="intbl">
|
|
<em>5</em>
|
|
<strong>12</strong>
|
|
</span>? </h3>
|
|
<p>List the multiples of 8 and 12</p>
|
|
<table border="0" align="center">
|
|
<tr>
|
|
<td align="right">multiples of 8: </td>
|
|
<td> </td>
|
|
<td class="larger">8, 16, <b>24</b>, 32, 40, ...</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="right"> multiples 12: </td>
|
|
<td> </td>
|
|
<td class="larger">12,<b> 24</b>, 36, 48, ...</td>
|
|
</tr>
|
|
</table>
|
|
|
|
<p>The Least Common Multiple is <b>24</b></p>
|
|
<p>For the first fraction we can multiply top and bottom by 3 to get a denominator of 24:</p>
|
|
<table align="center" border="0" cellspacing="3" cellpadding="0">
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td><b>× 3</b></td>
|
|
<td> </td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td colspan="3"><div align="center"><img src="images/style/right-over-arrow.gif" width="75" height="25" alt="right over arrow" /></div></td>
|
|
</tr>
|
|
<tr align="center" class="large">
|
|
<td><span class="intbl"><em>3</em><strong>8</strong></span></td>
|
|
<td width="45">=</td>
|
|
<td><span class="intbl"><em>9</em><strong>24</strong></span></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td colspan="3"><div align="center"><img src="images/style/right-under-arrow.gif" width="75" height="25" alt="right under arrow" /></div></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td><b>× 3</b></td>
|
|
<td> </td>
|
|
</tr>
|
|
</table>
|
|
<p> </p>
|
|
<p>For the second fraction we can multiply top and bottom by 2 to get a denominator of 24:</p>
|
|
<table align="center" border="0" cellspacing="3" cellpadding="0">
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td><b>× 2</b></td>
|
|
<td> </td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td colspan="3"><div align="center"><img src="images/style/right-over-arrow.gif" width="75" height="25" alt="right over arrow" /></div></td>
|
|
</tr>
|
|
<tr align="center" class="large">
|
|
<td><span class="intbl"><em>5</em><strong>12</strong></span></td>
|
|
<td width="45">=</td>
|
|
<td><span class="intbl"><em>10</em><strong>24</strong></span></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td colspan="3"><div align="center"><img src="images/style/right-under-arrow.gif" width="75" height="25" alt="right under arrow" /></div></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td><b>× 2</b></td>
|
|
<td> </td>
|
|
</tr>
|
|
</table>
|
|
<p> </p>
|
|
<p> Now we can do the addition:</p>
|
|
<p align="center"> <span class="intbl">
|
|
<em>9</em>
|
|
<strong>24</strong>
|
|
</span> + <span class="intbl">
|
|
<em>10</em>
|
|
<strong>24</strong>
|
|
</span> = <span class="intbl">
|
|
<em>19</em>
|
|
<strong>24</strong>
|
|
</span></p>
|
|
<p>The fraction is already as simple as it can be, so that is the answer.</p>
|
|
</div>
|
|
<p> </p>
|
|
<div class="questions">
|
|
<script type="text/javascript">getQ(928, 929, 1395, 1396, 1397, 1398, 3560, 3561, 3562, 3563);</script> </div>
|
|
<div class="related"> <a href="fractions.html">Introduction to Fractions</a> <a href="comparing-fractions.html">Comparing Fractions</a> <a href="least-common-multiple.html">Least Common Multiple</a> <a href="least-common-multiple-tool.html">Least Common Multiple Tool</a> <a href="fractions_addition.html">Adding Fractions</a> <a href="fractions_subtraction.html">Subtracting Fractions</a> <a href="fractions-menu.html">Fractions Index</a></div>
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