e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
270 lines
11 KiB
HTML
270 lines
11 KiB
HTML
<!doctype html>
|
|
<html lang="en">
|
|
<!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
|
<!-- Mirrored from www.mathsisfun.com/algebra/rational-numbers-operations.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:01:42 GMT -->
|
|
<head>
|
|
<meta http-equiv="content-type" content="text/html; charset=utf-8">
|
|
<!-- #BeginEditable "doctitle" -->
|
|
<title>Using Rational Numbers</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." />
|
|
<script src="../images/player.js" type="text/javascript"></script>
|
|
<link rel="stylesheet" type="text/css" href="../stylejs.css" />
|
|
<!-- #EndEditable -->
|
|
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
|
<meta name="HandheldFriendly" content="true">
|
|
<meta name="referrer" content="always">
|
|
<link rel="stylesheet" type="text/css" href="../style3.css">
|
|
<script src="../main3.js"></script>
|
|
</head>
|
|
|
|
<body id="bodybg">
|
|
<div class="bg">
|
|
<div id="stt"></div>
|
|
<header>
|
|
<div id="hdr"></div>
|
|
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
|
|
<div id="gtran">
|
|
<script>document.write(getTrans());</script>
|
|
</div>
|
|
<div id="adTopOuter" class="centerfull noprint">
|
|
<div id="adTop">
|
|
<script>document.write(getAdTop());</script>
|
|
</div>
|
|
</div>
|
|
<div id="adHide">
|
|
<div id="showAds1"><a href="javascript:showAds()">Show Ads</a></div>
|
|
<div id="hideAds1"><a href="javascript:hideAds()">Hide Ads</a><br>
|
|
<a href="../about-ads.html">About Ads</a></div>
|
|
</div>
|
|
</header>
|
|
<nav>
|
|
<div id="menuWide" class="menu">
|
|
<script>document.write(getMenu(0));</script>
|
|
</div>
|
|
<div id="linkto">
|
|
<div id="linktort">
|
|
<script>document.write(getLinks());</script>
|
|
</div>
|
|
</div>
|
|
<div id="search" role="search">
|
|
<script>document.write(getSearch());</script>
|
|
</div>
|
|
<div id="menuSlim" class="menu">
|
|
<script>document.write(getMenu(1));</script>
|
|
</div>
|
|
<div id="menuTiny" class="menu">
|
|
<script>document.write(getMenu(2));</script>
|
|
</div>
|
|
</nav>
|
|
<div id="extra"></div>
|
|
</div>
|
|
<article id="content" role="main">
|
|
<!-- #BeginEditable "Body" -->
|
|
<h1 class="center">Using Rational Numbers</h1>
|
|
<p class="center">How to add, subtract, multiply and divide rational numbers</p>
|
|
<p>A <a href="../rational-numbers.html">rational number</a> is a number that can be written as a simple fraction (i.e. as a <b>ratio</b>).</p>
|
|
<h3>Examples:</h3>
|
|
<div class="simple">
|
|
<table width="400" border="0" align="center">
|
|
<tr align="center">
|
|
<th>Number</th>
|
|
<th>As a Fraction</th>
|
|
</tr>
|
|
<tr align="center">
|
|
<td height="9">5</td>
|
|
<td height="9">5/1</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td height="9">1.75</td>
|
|
<td height="9">7/4</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>.001</td>
|
|
<td>1/1000</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>0.111...</td>
|
|
<td>1/9</td>
|
|
</tr>
|
|
</table>
|
|
<h2>In general ...</h2> </div>
|
|
<p>So a rational number looks like this:</p>
|
|
<p class="center large"><span class="intbl"><em>p</em><strong>q</strong></span></p>
|
|
<p>But q cannot be zero, as that is <a href="../numbers/dividing-by-zero.html">dividing by zero</a>.</p>
|
|
<h2>How to Add, Subtract, Multiply and Divide</h2>
|
|
<p>When the rational number is something simple like <b>3</b>, or <b>0.001</b>, then just use mental arithmetic, or a calculator!</p>
|
|
<p><b>But what about when it is in <span class="center large"><span class="intbl"><em>p</em><strong>q</strong></span></span> form? </b></p>
|
|
<div class="beach">
|
|
<table style="border: 0; margin:auto;">
|
|
<tr>
|
|
<td align="center" valign="top"><span class="huge"><br>
|
|
½</span></td>
|
|
<td> </td>
|
|
<td>
|
|
<p>Well, a rational number is a <a href="../fractions-menu.html">fraction</a>, so we can use:</p>
|
|
<ul>
|
|
<li><a href="../fractions_addition.html">Adding Fractions</a>,</li>
|
|
<li><a href="../fractions_subtraction.html">Subtracting Fractions</a>,</li>
|
|
<li><a href="../fractions_multiplication.html">Multiplying Fractions</a> and</li>
|
|
<li><a href="../fractions_division.html">Dividing Fractions</a> </li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
<p><b>Here we will see those operations in a more general <a href="index.html">Algebra</a> style.</b></p>
|
|
<p>You might also like to read <a href="fractions-algebra.html">Fractions in Algebra</a>.</p>
|
|
<p>Let us start with multiplication, as that is the easiest.</p>
|
|
<h2>Multiplication</h2>
|
|
<p>To multiply two rational numbers <b>multiply the tops and bottoms separately</b>, like this:</p>
|
|
<script>
|
|
playerMain("images/rational-mult-a.txt", 600, 140);
|
|
</script>
|
|
<p>Here is an example:</p>
|
|
<script>
|
|
playerMain("../numbers/images/fractions-mult-a.txt", 600, 160);
|
|
</script>
|
|
<h2>Division</h2>
|
|
<p>To divide two rational numbers, first flip the second number over (make it a reciprocal) and then do a multiply like above:</p>
|
|
<script>
|
|
playerMain("images/rational-div-a.txt", 600, 200);
|
|
</script>
|
|
<p>Here is an example:</p>
|
|
<script>
|
|
playerMain("../numbers/images/fractions-div-a.txt", 630, 140);
|
|
</script>
|
|
<h2>Addition and Subtraction</h2>
|
|
<p>We will cover Addition and Subtraction in one go, as they are the same method.</p>
|
|
<p>Before we add or subtract, the rational numbers should have the <b>same bottom number</b> (called a <a href="../numbers/common-denominator.html">Common Denominator</a>).</p>
|
|
<p>The easiest way to do this is to</p>
|
|
<p class="center"><span class="large">Multiply both parts of each number by the bottom part of the other</span></p>
|
|
<p>Like this (note that the dot <span class="large"> · </span> means multiply):</p>
|
|
<script>
|
|
playerMain("images/rational-add-a.txt", 620, 220);
|
|
</script>
|
|
<p>Here is an example of addition:</p>
|
|
<script>
|
|
playerMain("images/rational-add-b.txt", 600, 200);
|
|
</script>
|
|
<p>And an example of subtraction (the middle step is skipped to make it quicker):</p>
|
|
<script>
|
|
playerMain("images/rational-add-c.txt", 620, 140);
|
|
</script>
|
|
<h2>Simplest Form</h2>
|
|
<p>Sometimes we have a rational number like this:</p>
|
|
<p class="center large"><span class="intbl"><em>10</em><strong>15</strong></span></p>
|
|
<p>But that is not as simple as it can be!</p>
|
|
<p>We can divide both top and bottom by 5 to get:</p>
|
|
<table style="border: 0; margin:auto;">
|
|
<tr>
|
|
<td align="center" class="large">÷ 5</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center"><img src="../images/style/right-over-arrow.gif" width="75" height="25" alt="right over arrow" /></td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center"><span class="large"><span class="intbl"><em>10</em><strong>15</strong></span> = <span class="intbl"><em>2</em><strong>3</strong></span></span>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center"><img src="../images/style/right-under-arrow.gif" width="75" height="25" alt="right under arrow" /></td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center" class="large">÷ 5</td>
|
|
</tr>
|
|
</table>
|
|
<p>Now it is in "simplest form", which is how most people want it!</p>
|
|
<h2>Be Careful With "Mixed Fractions"</h2>
|
|
<p>We may be tempted to write an <a href="../improper-fractions.html">Improper Fraction</a> (a fraction that is "top-heavy", i.e. where the top number is bigger then the bottom number) as a <a href="../mixed-fractions.html">Mixed Fraction</a>:</p>
|
|
<p>For example <span class="frac"><sup>7</sup>/<sub>4</sub></span> = 1 <span class="frac"><sup>3</sup>/<sub>4</sub></span>, shown here:<br></p>
|
|
<table style="border: 0; margin:auto;">
|
|
<tr align="center">
|
|
<td><span class="large"><img src="../images/style/yes.svg" alt="yes" width="48" /><br>
|
|
Improper Fraction</span></td>
|
|
<td> </td>
|
|
<td><span class="large"><img src="../images/style/no.svg" alt="not" width="48" /><br>
|
|
Mixed Fraction</span></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><span class="frac-large"><sup>7</sup>/<sub>4</sub></span> </td>
|
|
<td width="40"> </td>
|
|
<td><span class="large">1</span> <span class="frac-large"><sup>3</sup>/<sub>4</sub></span> </td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><img src="../images/fractions/pie-4-4.jpg" alt="4/4" width="120" height="120" /><img src="../images/fractions/pie-3-4.jpg" alt="3/4" width="120" height="120" /></td>
|
|
<td width="40" class="large">=</td>
|
|
<td><img src="../images/fractions/pie-1-1.jpg" width="120" height="120" alt="pie full" /><img src="../images/fractions/pie-3-4.jpg" alt="3/4" width="120" height="120" /></td>
|
|
</tr>
|
|
</table>
|
|
<p>But for mathematics the "Improper" form (such as <span class="frac"><sup>7</sup>/<sub>4</sub></span>) is actually <b>better</b>.</p>
|
|
<p>Because Mixed fractions (such as 1 <span class="frac"><sup>3</sup>/<sub>4</sub></span>) can be confusing when we write them down in a formula, as it can look like a <b>multiplication</b>:</p>
|
|
<table style="border: 0; margin:auto;">
|
|
<tr>
|
|
<th align="right">Mixed Fraction: </th>
|
|
<th> </th>
|
|
<td align="right">What is: </td>
|
|
<td>1 + 2<span class="intbl"><em>1</em><strong>4</strong></span> ?</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right">Is it: </td>
|
|
<td>1 + 2<span class="hilite"> + </span> <span class="intbl"><em>1</em><strong>4</strong></span> = 3<span class="intbl"><em>1</em><strong>4</strong></span> <b class="large">?</b></td>
|
|
</tr>
|
|
<tr>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"><i>Or</i> is it: </td>
|
|
<td>1 + 2<span class="hilite"> × </span> <span class="intbl"><em>1</em><strong>4</strong>
|
|
</span> = 1<span class="intbl"><em>1</em><strong>2</strong></span> <b class="large">?</b></td>
|
|
</tr>
|
|
<tr>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
</tr>
|
|
<tr>
|
|
<th align="right">Improper Fraction: </th>
|
|
<th> </th>
|
|
<td align="right">What is: </td>
|
|
<td>1 + <span class="intbl"><em>9</em><strong>4</strong></span> ?</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right">It is: </td>
|
|
<td><span class="intbl"><em>4</em><strong>4</strong></span> + <span class="intbl"><em>9</em><strong>4</strong></span> = <span class="intbl"><em>13</em><strong>4</strong></span> <img src="../images/style/yes.svg" alt="yes" /></td>
|
|
</tr>
|
|
</table>
|
|
<p>So try to use the <b>Improper Fraction</b> when doing mathematics.</p>
|
|
<p> </p>
|
|
<div class="questions">
|
|
<script>
|
|
getQ(437, 430, 436, 433, 1060, 1061, 1062, 1063, 3985, 3986);
|
|
</script> </div>
|
|
|
|
<div class="related">
|
|
<a href="fractions-algebra.html">Fractions in Algebra</a>
|
|
<a href="index.html">Algebra Index</a>
|
|
</div>
|
|
<!-- #EndEditable -->
|
|
|
|
</article>
|
|
<div id="adend" class="centerfull noprint">
|
|
<script>document.write(getAdEnd());</script>
|
|
</div>
|
|
<footer id="footer" class="centerfull noprint">
|
|
<script>document.write(getFooter());</script>
|
|
</footer>
|
|
<div id="copyrt">
|
|
Copyright © 2020 MathsIsFun.com
|
|
</div>
|
|
<script>document.write(getBodyEnd());</script>
|
|
</body>
|
|
<!-- #EndTemplate -->
|
|
|
|
<!-- Mirrored from www.mathsisfun.com/algebra/rational-numbers-operations.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:01:44 GMT -->
|
|
</html>
|