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362 lines
15 KiB
HTML
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HTML
<!DOCTYPE html>
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<title>Proportions</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Proportions</h1>
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<p><span class="center">Proportion</span> says that two <a href="../numbers/ratio.html">ratios</a> (or fractions) are equal.</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center"><img src="images/proportion.svg" alt="proportion 1/3 : 2/6" height="" width=""></p>
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<p>We see that <b>1-out-of-3</b> is equal to <b>2-out-of-6</b></p>
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<p>The ratios are the same, so they are in proportion.</p>
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</div>
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<div class="example">
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<h3>Example: Rope</h3>
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<p>A rope's <b>length</b> and <b>weight</b> are in proportion.</p>
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<p>When <b>20m</b> of rope weighs <b>1kg</b>, then:</p>
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<ul>
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<li><b>40m</b> of that rope weighs <b>2kg</b></li>
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<li><b>200m</b> of that rope weighs <b>10kg</b></li>
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<li>etc.</li>
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</ul>
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<p class="center"><img src="images/proportional-rope.svg" alt="rope 20m / 1kg : 40m / 2kg" height="" width=""></p>
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<p>So:</p>
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<p class="center large"><span class="intbl">
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<em>20</em>
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<strong>1</strong>
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</span> = <span class="intbl">
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<em>40</em>
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<strong>2</strong>
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</span></p>
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</div>
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<h3>Sizes</h3>
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<p>When shapes are "in proportion" their relative sizes are the same.</p>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td>
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<p>Here we see that the ratios of head length to body length are the same in both drawings.</p>
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<p>So they are <b>proportional</b>.</p>
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<p>Making the head too long or short would look bad!</p></td>
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<td><img src="images/proportion-1a.gif" alt="proportion 10/20 : 15/30" height="184" width="261"></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: <a href="../geometry/paper-sizes.html">International paper sizes</a> (like A3, A4, A5, etc) all have the same proportions:</h3>
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<p class="center"><img src="../geometry/images/paper-size-resize.svg" alt="paper size resize" height="223" width="487"></p>
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<p>So any artwork or document can be resized to fit on any sheet. Very neat.</p>
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</div>
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<h2>Working With Proportions</h2>
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<p>NOW, how do we use this?</p>
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<div class="example">
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<h3>Example: you want to draw the dog's head ... how long should it be?</h3>
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<p class="center"><img src="images/proportion-2.gif" alt="proportion dog" height="146" width="242"></p>
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<p>Let us write the proportion with the help of the 10/20 ratio from above:</p>
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<p class="center larger"><span class="intbl">
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<em>?</em>
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<strong>42</strong>
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</span> = <span class="intbl">
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<em>10</em>
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<strong>20</strong>
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</span></p>
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<p>Now
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we solve it using a special method:</p>
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<p class="center"><img src="images/proportion-3.svg" alt="?/42 : 10/20" height="68" width="189"></p>
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<p class="center larger">Multiply across the known corners,<br>
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then divide by the third number</p>
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<p>And we get this:</p>
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<p class="center large">? = (42 × 10) / 20<br>
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= 420 / 20<br>
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= <b>21</b></p>
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<p>So you should draw the head <b>21</b> long.</p>
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</div>
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<p> </p>
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<h2>Using Proportions to Solve Percents</h2>
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<p>A percent is actually a ratio! Saying "25%" is actually saying "25 per 100":</p>
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<p class="center large">25% = <span class="intbl"><em>25</em><strong>100</strong></span></p>
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<p>We can use proportions to solve questions involving percents.</p>
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<p>The trick is to put what we know into this form:</p>
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<p class="center large"><span class="intbl"><em>Part</em><strong>Whole</strong></span> = <span class="intbl"><em>Percent</em><strong>100</strong></span></p>
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<p> </p>
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<div class="example">
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<h3>Example: what is 25% of 160 ?</h3>
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<p>The percent is 25, the whole is 160, and we want to find the "part":</p>
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<p class="center large"><span class="intbl"><em>Part</em><strong>160</strong></span> = <span class="intbl"><em>25</em><strong>100</strong></span></p>
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<p>Multiply across the known corners, then divide by the third number:</p>
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<p class="center"><img src="images/proportion-6.svg" alt="Part/160 : 25/100" height="68" width="219"></p>
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<p class="center large">Part = (160 × 25) / 100<br>
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= 4000 / 100<br>
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= <b>40</b></p>
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<p><b>Answer: 25% of 160 is 40.</b></p>
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<p> </p>
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<p>Note: we could have also solved this by doing the divide first, like this:</p>
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<p class="center large">Part = 160 × (25 / 100)<br>
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= 160 × 0.25<br>
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= <b>40</b></p>
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<p>Either method works fine.</p>
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</div>
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<p>We can also find a Percent:</p>
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<div class="example">
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<h3>Example: what is $12 as a percent of $80 ?</h3>
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<p>Fill in what we know:</p>
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<p class="center large"><span class="intbl"><em>$12</em><strong>$80</strong></span> = <span class="intbl"><em>Percent</em><strong>100</strong></span></p>
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<p>Multiply across the known corners, then divide by the third number. This time the known corners are top left and bottom right:</p>
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<p class="center"><img src="images/proportion-7.svg" alt="$12/$80 : Percent/100" height="66" width="245"></p>
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<p class="center large">Percent = ($12 × 100) / $80<br>
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= 1200 / 80<br>
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= <b>15%</b></p>
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<p>Answer: $12 is <b>15%</b> of $80</p>
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</div>
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<p>Or find the Whole:</p>
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<div class="example">
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<h3>Example: The sale price of a phone was $150, which was only 80% of normal price. What was the normal price?</h3>
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<p>Fill in what we know:</p>
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<p class="center large"><span class="intbl"><em>$150</em><strong>Whole</strong></span> = <span class="intbl"><em>80</em><strong>100</strong></span></p>
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<p>Multiply across the known corners, then divide by the third number:</p>
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<p class="center"><img src="images/proportion-8.svg" alt="$150/whole : 80/100" height="63" width="236"></p>
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<p class="center large">Whole = ($150 × 100) / 80<br>
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= 15000 / 80<br>
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= <b>187.50</b></p>
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<p>Answer: the phone's normal price was <b>$187.50</b></p>
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</div>
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<h2>Using Proportions to Solve Triangles</h2>
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<p>We can use proportions to solve similar triangles.</p>
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<div class="example">
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<h3>Example: How tall is the Tree?</h3>
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<p>Sam tried using a ladder, tape measure, ropes and various other things, but still couldn't work out how tall the tree was.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/proportion-4.jpg" alt="proportion tree " height="198" width="273"></p>
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<p>But then Sam has a clever idea ... similar triangles!</p>
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<p>Sam measures a stick and its shadow (in meters), and also the shadow of the tree, and this is what he gets:</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="images/proportion-5.gif" alt="proportion " height="143" width="222"></p>
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<p>Now Sam makes a sketch of the triangles, and writes down the "Height to Length" ratio for both triangles:</p>
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<p class="center larger"><span class="intbl">
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<em>Height:</em>
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<strong>Shadow Length:</strong></span> <span class="intbl"><em>h</em>
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<strong>2.9 m</strong></span> = <span class="intbl">
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<em>2.4 m</em>
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<strong>1.3 m</strong></span></p>
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<p>Multiply across the known corners, then divide by the third number:</p>
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<p class="center large">h = (2.9 × 2.4) / 1.3<br>
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= 6.96 / 1.3<br>
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= <b>5.4 m</b> (to nearest 0.1)</p>
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<p><b>Answer: the tree is 5.4 m tall.</b></p>
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<p>And he didn't even need a ladder!</p>
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</div>
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<p>The "Height" could have been at the bottom, so long as it was on the bottom for BOTH ratios, like this:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/proportion-5.gif" alt="proportion " height="143" width="222"></p>
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<p>Let us try the ratio of "Shadow Length to Height":</p>
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<p class="center larger"><span class="intbl">
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<em>Shadow Length:</em>
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<strong>Height:</strong></span> <span class="intbl">
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<em>2.9 m</em>
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<strong>h</strong></span> = <span class="intbl">
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<em>1.3 m</em>
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<strong>2.4 m</strong></span></p>
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<p>Multiply across the known corners, then divide by the third number:</p>
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<p class="center large">h = (2.9 × 2.4) / 1.3<br>
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= 6.96 / 1.3<br>
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= <b>5.4 m</b> (to nearest 0.1)</p>
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<p><b>It is the same calculation as before.</b></p>
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</div>
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<h2>A "Concrete" Example</h2>
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<p>Ratios can have <b>more than two numbers</b>!</p>
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<p>For example concrete is made by mixing cement, sand, stones and water.</p>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="../numbers/images/concrete-pouring.jpg" alt="concrete pouring" height="150" width="200"></p>
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<p>A typical mix of cement, sand and stones is written as a ratio, such as <span class="large">1:2:6</span>.</p>
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<p>We can multiply all values by the same amount and still have the same ratio.</p>
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<p class="center large">10:20:60 is the same as 1:2:6</p>
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<p>So when we use 10 buckets of cement, we should use 20 of sand and 60 of stones.</p>
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<div class="example">
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<h3>Example: you have just put 12 buckets of stones into a mixer, how much cement and how much sand should you add to make a <span class="large">1:2:6</span> mix?</h3>
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<p>Let us lay it out in a table to make it clearer:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th width="160"> </th>
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<th width="80">Cement</th>
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<th width="80">Sand</th>
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<th width="80">Stones</th>
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</tr>
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<tr style="text-align:center;">
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<th width="160">Ratio Needed:</th>
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<td class="large" width="80">1</td>
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<td class="large" width="80">2</td>
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<td class="large" width="80">6</td>
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</tr>
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<tr style="text-align:center;">
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<th width="160">You Have:</th>
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<td class="large bga1" width="80" > </td>
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<td class="large bga1" width="80" > </td>
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<td class="large" width="80">12</td>
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</tr>
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</tbody></table>
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<p>You have 12 buckets of stones but the ratio says 6.</p>
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<p>That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of <b>everything</b> to keep the ratio.</p>
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<p>Here is the solution:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th width="160"> </th>
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<th width="80">Cement</th>
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<th width="80">Sand</th>
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<th width="80">Stones</th>
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</tr>
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<tr style="text-align:center;">
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<th width="160">Ratio Needed:</th>
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<td class="large" width="80">1</td>
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<td class="large" width="80">2</td>
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<td class="large" width="80">6</td>
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</tr>
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<tr style="text-align:center;">
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<th width="160">You Have:</th>
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<td class="large bga1" width="80">2</td>
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<td class="large bga1" width="80">4</td>
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<td class="large" width="80">12</td>
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</tr>
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</tbody></table>
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<p>And the ratio 2:4:12 is the same as 1:2:6 (because they show the same <i><b>relative</b></i> sizes)</p>
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<p>So the answer is: add 2 buckets of Cement and 4 buckets of Sand. <i>(You will also need water and a lot of stirring....)</i></p>
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</div>
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<p><i>Why are they the same ratio?</i> <i>Well, the <b>1:2:6</b> ratio says to have</i>:</p>
|
||
<ul>
|
||
<li><i> twice as much Sand as Cement (<span class="hilite">1</span>:<span class="hilite">2</span>:6)</i></li>
|
||
<li><i>6 times as much Stones as Cement (<span class="hilite">1</span>:2:<span class="hilite">6</span>)</i></li>
|
||
</ul>
|
||
<p><i>In our mix we have:</i></p>
|
||
<ul>
|
||
<li><i> twice as much Sand as Cement (<span class="hilite">2</span>:<span class="hilite">4</span>:12)</i></li>
|
||
<li><i>6 times as much Stones as Cement (<span class="hilite">2</span>:4:<span class="hilite">12</span>)</i></li>
|
||
</ul>
|
||
<p><i>So it should be just right!</i></p>
|
||
<p>That is the good thing about ratios. You can make the amounts bigger or smaller and so long as the <b>relative</b> sizes are the same then the ratio is the same.</p>
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<p> </p>
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<p> </p>
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<div class="questions">5674, 5676, 5678, 5680, 5682, 5684, 5686, 5688, 5690, 5692</div>
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<div class="related">
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<a href="../numbers/ratio.html">Ratios</a>
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<a href="directly-inversely-proportional.html">Directly Proportional and Inversely Proportional</a>
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<a href="index.html">Algebra Index</a>
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