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363 lines
14 KiB
HTML
363 lines
14 KiB
HTML
<!doctype html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/simplifying-fractions.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:36:00 GMT -->
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<head>
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<!-- #BeginEditable "doctitle" -->
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<title>Simplifying Fractions</title>
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<script src="video.js" type="text/javascript"></script>
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<script language="JavaScript" type="text/javascript">
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<!-- Begin
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function go() {
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var input1 = eval(document.form1.text1.value);
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var input2 = eval(document.form1.text2.value);
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hcfval = hcf(input1,input2);
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// document.form1.answer.value = hcfval;
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document.form1.ans1.value = (input1/hcfval);
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document.form1.ans2.value = (input2/hcfval);
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ex = "The Greatest Common Factor is " + hcfval + "<br><br>";
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if (hcfval == 1) {
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ex += "(No reduction possible)";
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} else {
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ex += "The fraction simplifies like this: ";
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ex += "<span class=\"frac-large\"><sup>" + input1 + "</sup>/<sub>" + input2 + "</sub></span>";
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ex += " »» ";
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ex += "<span class=\"frac-large\"><sup>" + (input1/hcfval) + "</sup>/<sub>" + (input2/hcfval) + "</sub></span>";
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}
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var example = document.getElementById("example");
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example.innerHTML = ex;
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}
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function hcf1(text1,text2){
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var gcd=1;
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if (text1>text2) {text1=text1+text2; text2=text1-text2; text1=text1-text2;}
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if ((text2==(Math.round(text2/text1))*text1)) {gcd=text1}else {
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for (var i = Math.round(text1/2) ; i > 1; i=i-1) {
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if ((text1==(Math.round(text1/i))*i))
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if ((text2==(Math.round(text2/i))*i)) {gcd=i; i=-1;}
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}
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}
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return gcd;
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}
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function hcf(text1, text2) {
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text1 = text1 << 0;
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text2 = text2 << 0;
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if (text1 == 0 || text2 == 0) return 0;
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if (text1 > text2) {
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text1 = text1 + text2;
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text2 = text1 - text2;
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text1 = text1 - text2;
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}
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var remainder = text2 % text1;
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while (remainder != 0) {
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text2 = text1;
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text1 = remainder;
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remainder = text2 % text1;
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}
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return text1;
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}
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function lcm(t1,t2){
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var cm=1;
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var f=hcf(t1,t2);
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cm=t1*t2/f;
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return cm;
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}
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// End -->
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</script>
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</div>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 id="title" align="center">Simplifying Fractions</h1>
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<span id="video"></span>
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<script type="text/javascript">initVideo('aNQXhknSwrI');</script>
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<p> </p>
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<span style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/cancel.svg" alt="Cancel 2/10 to 1/5" /></span> <p><i>To simplify a fraction, divide the top and bottom by the <b>highest number</b> that<br />
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can divide into both numbers exactly.</i></p>
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<h2>Simplifying Fractions</h2>
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<p>Simplifying (or <i>reducing</i>) fractions means to make the fraction as simple as possible. </p>
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<p>Why say four-eighths (<span class="intbl"><em>4</em><strong>8</strong>
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</span>)
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when we really mean half (<span class="intbl">
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<em>1</em><strong>2</strong></span>) ?</p>
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<table border="0" align="center">
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<tr align="center" class="large">
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<td><span class="intbl"><em>4</em><strong>8</strong></span></td>
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<td>=</td>
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<td><span class="intbl"><em>2</em><strong>4</strong></span></td>
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<td>=</td>
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<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
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</tr>
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<tr align="center">
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<td>(Four-Eighths)</td>
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<td> </td>
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<td>(Two-Quarters)</td>
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<td> </td>
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<td>(One-Half)</td>
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</tr>
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<tr align="center">
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<td><img src="images/fractions/pie-4-8.jpg" width="120" height="120" alt="pie 4/8" /></td>
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<td class="large">=</td>
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<td><img src="images/fractions/pie-2-4.jpg" width="120" height="120" alt="pie 2/4" /></td>
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<td class="large">=</td>
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<td><img src="images/fractions/pie-1-2.jpg" width="120" height="120" alt="pie 1/2" /></td>
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</tr>
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</table>
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<p style="float:right; margin: 0 0 5px 10px;"> </p>
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<h2>How do I Simplify a Fraction ?</h2>
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<p>There are two ways to simplify a fraction:</p>
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<h2>Method 1</h2>
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<p>Try to <b>exactly divide</b> (only whole number answers) both the top and bottom of the fraction by 2, 3, 5, 7 ,... etc, until we can't go any further.<br />
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</p>
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<div class="example">
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<h3>Example: Simplify the fraction <span class="intbl">
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<em>24</em>
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<strong>108</strong>
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</span> :</h3>
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<div class="clear">
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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>÷ 2</td>
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<td> </td>
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<td>÷ 2</td>
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<td> </td>
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<td>÷ 3</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="7"><img src="images/style/right-over-arrow.gif" width="75" height="25" alt="right over arrow" /> <img src="images/style/right-over-arrow.gif" width="75" height="25" alt="right over arrow" /> <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 colspan="7"><span class="intbl"><em>24</em><strong>108</strong></span> = <span class="intbl"><em>12</em><strong>54</strong></span> = <span class="intbl"><em>6</em><strong>27</strong></span> = <span class="intbl"><em>2</em><strong>9</strong></span></td>
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</tr>
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<tr align="center">
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<td colspan="7"><img src="images/style/right-under-arrow.gif" width="75" height="25" alt="right under arrow" /> <img src="images/style/right-under-arrow.gif" width="75" height="25" alt="right under arrow" /> <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>÷ 2</td>
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<td> </td>
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<td>÷ 2</td>
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<td> </td>
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<td>÷ 3</td>
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<td> </td>
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</tr>
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</table>
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<p>That is as far as we can go. The fraction simplifies to <span class="intbl">
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<em>2</em>
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<strong>9</strong>
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</span></p>
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</div>
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</div>
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<div class="example">
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<h3>Example: Simplify the fraction <span class="intbl">
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<em>10</em>
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<strong>35</strong>
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</span> :</h3>
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<p>Dividing by <b>2</b> doesn't work because <b>35 can't be exactly divided by 2</b> (35/2 = 17½)</p>
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<p>Likewise we can't divide exactly by <b>3</b> (10/3 = 3<span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> and also 35/3=11<span class="intbl">
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<em>2</em>
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<strong>3</strong>
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</span>)</p>
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<p>No need to check <b>4</b> (we checked 2 already, and 4 is just 2×2).</p>
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<p>But <b>5</b> does work!</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>÷ 5</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 colspan="3"><span class="intbl"><em>10</em><strong>35</strong></span> = <span class="intbl"><em>2</em><strong>7</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>÷ 5</td>
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<td> </td>
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</tr>
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</table>
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<p>That is as far as we can go. The fraction simplifies to <span class="intbl">
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<em>2</em>
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<strong>7</strong>
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</span></p>
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</div>
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<p>Notice that after checking 2 we didn't need to check 4 (4 is 2×2)? </p>
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<p>We also don't need to check 6 when we have checked 2 and 3 (6 is 2x3). </p>
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<p>In fact, when checking from smallest to largest we use <a href="prime_numbers.html">prime numbers</a>:</p>
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<p class="center larger">2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...</p>
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<h2>Method 2</h2>
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<p>Divide both the top and bottom of the fraction by the <a href="greatest-common-factor.html">Greatest Common Factor</a> (you have to work it out first!).
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<br />
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<br />
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</p>
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<div class="example">
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<h3>Example: Simplify the fraction <span class="intbl"><em>8</em><strong>12</strong></span> :</h3>
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<p>
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The largest number that goes exactly into both 8 and 12 is 4, so <i>the Greatest Common Factor is 4</i>.
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</p>
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<p>Divide both top and bottom by 4: </p>
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<div class="clear">
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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 class="Larger">÷ 4</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" class="Larger"><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 colspan="3"><span class="intbl"><em>8</em><strong>12</strong></span> = <span class="intbl"><em>2</em><strong>3</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>÷ 4</td>
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<td> </td>
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</tr>
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</table>
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</div><p>That is as far as we can go. The fraction simplifies to <span class="intbl">
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<em>2</em>
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<strong>3</strong>
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</span></p>
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</div>
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<h2>Simplifying Fractions Automatically</h2>
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<p>OK, there is a <i>third</i> method, use this tool: </p>
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<div class="center">
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<form name="form1" id="form1">
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<div class="clear">
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<table border="0" cellspacing="5" cellpadding="0" align="center">
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<tr>
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<td><input type="text" name="text1" size="3" style="text-align:center; font: 18px Verdana;" /></td>
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<td rowspan="3"><b>
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<input type="button" value="Simplify ⇒" onClick="go()" name="button" />
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</b></td>
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<td><input type="text" name="ans1" size="3" style="text-align:center; font: 18px Verdana;" /></td>
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</tr>
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<tr style="border-top: 1px solid black;">
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<td><input type="text" name="text2" size="3" style="text-align:center; font: 18px Verdana;" /></td>
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<td><input type="text" name="ans2" size="3" style="text-align:center; font: 18px Verdana;" /></td>
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</tr>
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</table>
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</div>
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<br /><span id="example"> </span>
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</form>
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</div>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><a href="numbers/fraction-number-line.html"><img src="numbers/images/fraction-number-line-thumb.gif" width="189" height="94" border="0" alt="fraction number line" /></a></p>
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<h2>Fractions on the Number Line</h2>
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<p>Also ... see an animation of <a href="numbers/fraction-number-line.html">Fractions on the Number Line</a> where you can see many common fractions and their simpler version.</p>
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<p>We also have a <a href="numbers/fraction-decimal-chart.html">chart of fractions</a> with the simplest fraction highlighted.</p>
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<p> </p>
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<p> </p>
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<div class="questions">
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