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<title>Directly Proportional and Inversely Proportional</title>
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<meta name="description" content="Directly proportional: as one amount increases another amount increases at the same rate.">
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Directly Proportional<br>
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and Inversely Proportional</h1>
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<p class="center"><br></p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/proportional-dogs.jpg" alt="proportional dogs" height="104" width="203"></p>
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<p><br>Directly proportional: as one amount increases, another amount increases at the same rate.</p>
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<div style="clear:both">
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</div>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td class="huge">∝</td>
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<td width="20"> </td>
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<td>The symbol for<span class="center"> "directly proportional"</span> is <span class="large">∝</span><br>
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(Don't confuse it with the symbol for <a href="../numbers/infinity.html">infinity</a> <span class="large">∞</span>)</td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="example">
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<h3>Example: you are paid $20 an hour</h3>
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<p>How much you earn is <b>directly proportional</b> to how many hours you work</p>
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<p>Work more hours, get more pay; in direct proportion.</p>
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<p>This could be written:</p>
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<p class="center larger">Earnings <span class="times">∝</span> Hours worked</p>
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<ul>
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<li>If you work 2 hours you get paid $40</li>
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<li>If you work 3 hours you get paid $60</li>
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<li>etc ...</li>
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</ul>
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</div>
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<h2>Constant of Proportionality</h2>
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<p>The "constant of proportionality" is the value that relates the two amounts</p>
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<div class="example">
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<h3>Example: you are paid $20 an hour (continued)</h3>
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<p>The constant of proportionality is <b>20</b> because:</p>
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<p class="center larger">Earnings = <span class="hilite">20</span> × Hours worked</p>
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</div>
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<p>This can be written:</p>
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<p class="center larger">y = kx</p>
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<p class="center">Where <b>k</b> is the constant of proportionality</p>
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<div class="example">
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<h3>Example: y is directly proportional to x, and when x=3 then y=15. <br>
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What is the constant of proportionality?</h3>
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<p>They are directly proportional, so:</p>
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<div class="so"> y = kx</div>
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<p>Put in what we know (y=15 and x=3):</p>
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<div class="so"> 15 = k × 3</div>
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<p>Solve (by dividing both sides by 3):</p>
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<div class="so"> 15/3 = k × 3/3</div>
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<div class="so"> 5 = k × 1</div>
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<div class="so"> k = 5</div>
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<p>The constant of proportionality is 5:</p>
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<p class="center larger">y = 5x</p>
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</div>
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<p>When we know the <span class="center">constant of proportionality</span> we can then answer other questions</p>
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<div class="example">
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<h3>Example: (continued)</h3>
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<p>What is the value of y when x = 9?</p>
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<div class="so">y = 5 × 9 = 45<br>
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</div>
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<p>What is the value of x when y = 2?</p>
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<div class="so">2 = 5x <br>
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</div>
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<div class="so">x = 2/5 = 0.4<br>
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</div>
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</div>
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<h2><span class="center">Inversely Proportional</span></h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td> </td>
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<td class="larger"><span class="center"><b>Inversely</b> Proportional</span>: when one value <b>decreases</b> at the same rate that the other increases.</td>
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</tr>
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</tbody></table>
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<div class="example">
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<h3>Example: speed and travel time</h3>
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<p>Speed and travel time are <span class="center">Inversely Proportional</span> because the faster we go the shorter the time.</p>
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<ul>
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<li>As speed goes up, travel time goes down</li>
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<li>And as speed goes down, travel time goes up</li>
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</ul>
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</div>
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<div class="tbl">
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<div class="row"><span class="left">This:</span><span class="right"><b>y is inversely proportional to x</b></span></div>
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<div class="row"><span class="left">Is the same thing as:</span><span class="right"><b>y is <span class="hilite">directly</span> proportional to <span class="hilite">1/x</span></b></span></div>
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<div class="row"><span class="left">Which can be written:</span><span class="right">
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<p class="center large">y = <span class="intbl">
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<em>k</em><strong>x</strong></span></p></span></div>
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</div>
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<p> </p>
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<div class="example"><img style="float:right; margin: 10px;" src="images/fence.jpg" alt="fence" height="112" width="150">
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<h3>Example: 4 people can paint a fence in 3 hours. <br>
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<br>
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How long will it take 6 people to paint it? <br>
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<br>
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(Assume everyone works at the same rate)</h3>
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<p>It is an Inverse Proportion:</p>
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<ul>
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<li>As the number of people goes up, the painting time goes down.</li>
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<li>As the number of people goes down, the painting time goes up.</li>
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</ul>
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<p>We can use:</p>
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<p class="center larger">t = k/n</p>
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<p>Where:</p>
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<ul>
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<li>t = number of hours</li>
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<li>k = constant of proportionality</li>
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<li>n = number of people</li>
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</ul>
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<p>"4 people can paint a fence in 3 hours" means that t = 3 when n = 4</p>
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<div class="so">
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3 = k/4 </div>
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<div class="so">
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3 × 4 = k × 4 / 4</div>
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<div class="so">
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12 = k</div>
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<div class="so">
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k = 12</div>
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<p>So now we know:</p>
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<div class="so"><span class="center larger">t = 12/n</span></div>
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<p>And when n = 6:</p>
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<div class="so"><span class="center larger">t = 12/6 = 2 hours</span></div>
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<p>So 6 people will take 2 hours to paint the fence.</p>
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<p> </p>
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<h3>How many people are needed to complete the job in half an hour?</h3>
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<div class="so">½ = 12/n </div>
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<div class="so">n = 12 / ½ = 24</div>
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<p>So it needs 24 people to complete the job in half an hour.<br>
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(Assuming they don't all get in each other's way!)</p>
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</div>
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<p> </p>
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<h2><span class="center"> Proportional</span> to ...</h2>
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<p>It is also possible to be proportional to a square, a cube, an exponential, or other function!</p>
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<div class="example">
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<h3>Example: Proportional to x<sup>2</sup></h3>
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<img style="float:left; margin: 10px;" src="images/stone.jpg" alt="stone" height="75" width="70"> <div class="example2"></div>
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<p>A stone is dropped from the top of a high tower.</p>
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<p>The distance it falls is <b>proportional to the square</b> of the time of fall.</p>
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<p>The stone falls 19.6 m after 2 seconds, how far does it fall after 3 seconds?</p>
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<p> </p>
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<p>We can use:</p>
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<p class="center larger">d = kt<sup>2</sup></p>
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<p>Where:</p>
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<ul>
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<li>d is the distance fallen and</li>
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<li>t is the time of fall</li>
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</ul>
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<p> </p>
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<p>When d = 19.6 then t = 2</p>
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<div class="so"> 19.6 = k × 2<sup>2</sup></div>
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<div class="so"> 19.6 = 4k</div>
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<div class="so"> k = 4.9</div>
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<p>So now we know:</p>
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<p class="center larger">d = 4.9t<sup>2</sup></p>
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<p>And when t = 3:</p>
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<div class="so"> d = 4.9 × 3<sup>2</sup> </div>
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<div class="so"> d = 44.1</div>
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<p>So it has fallen 44.1 m after 3 seconds.</p>
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</div>
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<h2>Inverse Square</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/proportion-inverse-square.svg" alt="inverse square"></p>
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<p class="larger"><b>Inverse Square</b>: when one value <b>decreases</b> as the <b>square</b> of the other value.</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: light and distance</h3>
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<p>The further away we are from a light, the less bright it is.</p>
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<p class="center" style="margin:-25px 0 -25px -40px;"><img src="../physics/images/inverse-square-law.svg" alt="inverse square law"></p>
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<p>In fact the brightness decreases as the <b>square</b> of the distance. Because the light is spreading out in all directions.</p>
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<p>So a brightness of "1" at 1 meter is only "0.25" at 2 meters (double the distance leads to a quarter of the brightness), and so on.</p>
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</div><p> </p>
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<div class="questions">
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<script>getQ(8943, 8945, 8947, 8948, 8951, 8952, 7005, 8064, 8065, 8067, 8072);</script>
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</div>
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<div class="related">
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<a href="proportions.html">Proportions</a>
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<a href="../numbers/ratio.html">Ratios</a>
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<a href="index.html">Algebra Index</a>
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Copyright © 2020 MathsIsFun.com
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