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254 lines
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<title>How to Safely Convert From One Unit to Another</title>
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<article id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">How to Safely Convert Between Units</h1>
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<p>We can convert from <b>km/h</b> (kilometers per hour) to <b>m/s</b> (meters per second) like this:</p>
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<div class="example">
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<p>A kilometer has 1000 meters, and an hour has 3600 seconds, so a kilometer per hour is:</p>
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<ul>
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<li><span class="intbl"><em>1000</em><strong>3600</strong></span> = 0.2777... m/s</li>
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</ul>
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</div>
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<p>How did I know to make it <span class="intbl"><em>1000</em><strong>3600</strong></span> and not <span class="intbl"><em>3600</em><strong>1000</strong></span> (the other way around)?</p>
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<p>The trick is to do the conversions as <a href="../fractions-menu.html">fractions</a>!</p>
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<div class="example">
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<h3>Example 1:</h3>
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<p>Let's start with a simple example: <b>convert 3 km to m</b> (3 kilometers to meters). There are 1000 m in 1 km, so the conversion is easy, but let's follow a system.</p>
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<p>The system is:</p>
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<ul>
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<li>Write the conversion as a fraction that equals 1</li>
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<li>Multiply it out (leaving all units in the answer)</li>
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<li>Cancel any units that are both top and bottom</li>
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</ul>
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<p>We can write the conversion as a <b>fraction that equals 1</b>:</p>
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<p class="center larger"><span class="intbl"><em>1000 m</em><strong>1 km</strong></span> = 1</p>
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<p>And it is safe to multiply by 1 (does not affect the answer):</p>
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<p class="center larger">3 km × <span class="hilite">1</span> = 3 km</p>
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<!-- 3 km * 1 = 3 km -->
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<p> so we can do this:</p>
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<p class="center larger">3 km × <span class="intbl hilite">
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<em>1000 m</em>
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<strong>1 km</strong></span> = <span class="intbl"><em>3000 km · m</em>
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<strong> 1 km</strong></span></p>
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<p>The answer looks strange! But we aren't finished yet ... we can "cancel" any units that are both top and bottom:</p>
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<p class="center larger"><span class="intbl">
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<em>3000 <span style="text-decoration:line-through">km</span> · m</em>
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<strong>1 <span style="text-decoration:line-through">km</span></strong></span> = 3000 m</p>
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<p>So, 3 km equals 3000 m. Well we knew that, but we want to follow a <b>system</b>, so that when things get harder we know what to do!</p>
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<p>And when we do it correctly we get to cancel units that are both top and bottom, and get a neat answer.</p>
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<div style="background-color: #ffdddd; padding: 5px;">
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<p>Note: if we do it <b>wrong</b> (with the conversion upside down) we get this:</p>
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<p class="center larger">3 km × <span class="intbl" style="background-color:#FCC">
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<em>1 km</em>
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<strong>1000 m</strong></span> = <span class="intbl">
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<em>3 km · km</em>
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<strong>1000 m</strong></span></p>
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<p>And that doesn't let us do any cancelling!</p>
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</div>
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</div>
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<div class="example">
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<h3>Example 2:</h3>
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<p>Let's use this method to solve the <b>km/h</b> to <b>m/s</b> conversion from the top of the page.</p>
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<p>We will do it in two stages:</p>
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<ol>
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<li>from km/h (kilometers per hour) to m/h (meters per hour), then</li>
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<li>from m/h (meters per hour) to m/s (meters per second).</li>
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</ol>
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<h4>1. From km/h (kilometers per hour) to m/h (meters per hour)</h4>
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<p class="center larger"><span class="intbl">
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<em>1 km</em>
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<strong> h</strong></span> × <span class="intbl hilite"><em>1000 m</em><strong>1 km</strong></span> = <span class="intbl">
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<em>1000 km · m</em>
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<strong>1 h · km</strong></span></p>
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<p>Now "cancel out" any units that are both top and bottom:</p>
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<p class="center larger"><span class="intbl">
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<em>1000 <span style="text-decoration:line-through">km</span> · m</em>
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<strong>1 h · <span style="text-decoration:line-through">km</span></strong>
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</span> = <span class="intbl">
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<em>1000 m</em>
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<strong> 1 h</strong>
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</span></p>
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<p>
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It is now in meters per hour.
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</p>
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<h4>2. From m/h (meters per hour) to m/s (meters per second)</h4>
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<p>To go from m/h (meters per hour) to m/s (meters per second) we put the "3600 seconds in an hour" conversion "upside down" because we want an "h" on top (so they will cancel later) :</p>
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<p class="center larger"><span class="intbl">
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<em>1000 m</em>
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<strong> 1 h</strong>
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</span> × <span class="intbl hilite">
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<em>1 h</em>
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<strong>3600 s</strong>
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</span> = <span class="intbl">
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<em>1000 m · h</em>
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<strong>3600 h · s</strong>
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</span></p>
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<p>Then "cancel out" any units that are both top and bottom:</p>
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<p class="center larger"><span class="intbl">
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<em>1000 m · <span style="text-decoration:line-through">h</span></em>
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<strong>3600 <span style="text-decoration:line-through">h</span> · s </strong>
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</span> = <span class="intbl">
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<em>1000 m</em>
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<strong> 3600 s</strong>
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</span></p>
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<p>And so our anwer is:</p>
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<p class="center larger"><span class="intbl">
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<em>1000 m</em>
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<strong>3600 s</strong>
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</span> = <b>0.2777... m/s</b></p>
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<p> </p>
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<div style="background-color: #ffdddd; padding: 5px;">
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<p>Doing it <b>wrong</b> (with the the 3600 seconds/hour the other way around) gets this:</p>
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<p class="center larger"><span class="intbl">
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<em>1000 m</em>
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<strong>1 h</strong></span> × <span class="intbl" style="background-color:#FCC">
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<em>3600 s</em>
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<strong>1 h</strong></span> = <span class="intbl">
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<em>1000 × 3600 m · s</em>
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<strong>1 h · h</strong></span> </p>
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<p>And there is nothing to cancel! </p>
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<p>So we know we made a mistake, and can correct it.</p>
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</div>
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<h4>All In One Go</h4>
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<p>With experience we can do it in one line like this:</p>
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<p class="center larger"><span class="intbl">
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<em>1 km</em>
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<strong>1 h</strong></span> x <span class="intbl hilite">
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<em>1000 m</em>
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<strong>1 km</strong>
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</span> × <span class="intbl hilite">
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<em>1 h</em>
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<strong>3600 s</strong></span> </p><p class="center larger">= <span class="intbl"><em>1000 <span style="text-decoration:line-through">km</span> · m · <span style="text-decoration:line-through">h</span></em>
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<strong>3600 <span style="text-decoration:line-through">h</span> · <span style="text-decoration:line-through">km</span> · s</strong>
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</span> </p><p class="center larger">= <span class="intbl">
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<em>1000 m</em>
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<strong>3600 s</strong></span></p>
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<p>Or even "all in one go" (crossing out as we go) like this:</p>
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<p class="center larger"><span class="intbl">
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<em>1 <span style="text-decoration:line-through">km</span></em>
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<strong>1 <span style="text-decoration:line-through">h</span></strong>
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</span> x <span class="intbl hilite">
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<em>1000 m</em>
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<strong>1 <span style="text-decoration:line-through">km</span></strong>
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</span> × <span class="intbl hilite">
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<em>1 <span style="text-decoration:line-through">h</span></em>
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<strong>3600 s</strong>
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</span> = <span class="intbl">
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<em>1000 m</em>
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<strong>3600 s</strong>
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</span></p>
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</div>
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<div class="example">
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<h3>Example 3</h3>
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<p>Now let's use this method to do a real-world conversion. </p>
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<p>What is 60 mph (miles per hour) in m/s (meters per second) ?</p>
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<p class="center larger"><span class="intbl">
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<em>60 mile</em>
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<strong>h</strong></span> × <span class="intbl hilite">
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<em>1609 m</em>
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<strong>mile</strong></span> × <span class="intbl hilite">
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<em>1 h</em>
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<strong>3600 s</strong></span> </p><p class="center larger">= <span class="intbl">
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<em>60 × 1609 <span style="text-decoration:line-through">mile</span> · m · <span style="text-decoration:line-through">h</span></em>
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<strong>3600 <span style="text-decoration:line-through">h</span> · <span style="text-decoration:line-through">mile</span> · s</strong></span> </p><p class="center larger">= 26.82 m/s</p>
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</div>
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<h2>Summary</h2>
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<p>The important points are:</p>
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<ul>
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<li>Write the conversion as a fraction (that equals one)</li>
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<li>Multiply it out (leaving all units in the answer)</li>
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<li>Cancel any units that are both top and bottom</li>
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</ul>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(1915, 1916, 1039, 1040, 1913, 1914, 3720, 3721, 3722, 3723);</script>
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</div>
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<div class="related"><a href="../unit-conversion-tool.html">Unit Conversion Tool </a><a href="index.html">Measurement Index</a></div>
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Copyright © 2019 MathsIsFun.com
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