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174 lines
7.7 KiB
HTML
174 lines
7.7 KiB
HTML
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<title>Cross Multiply</title>
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<h1 align="center">Cross Multiply</h1>
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<p>
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To cross multiply is to go
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</p>
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<div class="tbl">
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<div class="row"><span class="left">from this:</span><span class="right">
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<p class="center"><span class="intbl">
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<em>8</em>
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<strong>12</strong>
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</span> = <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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</span></div>
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<div class="row"><span class="left">to this:</span><span class="right">
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<p class="center"><span class="large">8 × 3 = 12 × 2</span></p></span></div>
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</div>
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<h2>How Does it Work?</h2>
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<p>Multiplying the top <i>and</i> bottom of a fraction by the same amount doesn't change its value. </p>
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<div class="example">
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<p>Step 1: Multiply the top and bottom of the <i><b>first</b></i> fraction by the bottom number of the <i><b>second</b></i> fraction.</p>
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<p class="center large"><span class="intbl">
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<em>8 <span style="background-color:yellow;">× 3</span></em>
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<strong>12 <span style="background-color:yellow;">× 3</span></strong>
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</span> = <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>Step 2: Multiply the top and bottom of the <i><b>second</b></i> fraction by the bottom number that the <i><b>first</b></i> fraction had.</p>
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<p class="center large"><span class="intbl">
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<em>8 × 3</em>
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<strong>12 × 3</strong>
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</span> = <span class="intbl">
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<em>2 <span style="background-color:yellow;">× 12</span></em>
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<strong>3 <span style="background-color:yellow;">× 12</span></strong>
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</span></p>
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<p class="center">And Magic! The bottom of <b>both</b> fractions is now <span class="large">12 × 3</span></p>
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<p>Step 3: We can get rid of the <span class="large">12 × 3</span> (as we are dividing both sides by the same amount) and the equation is still true:</p>
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<p align="center"><span class="large">8 × 3 = 12 × 2</span></p>
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<p><b>Job Done!</b></p>
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</div>
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<p>In practice, though, it is easier to skip the steps and go straight to the "cross-multiplied" form.</p>
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<h2>Using Variables</h2>
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<p>The general case, using variables instead of numbers, is:</p>
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<div class="tbl">
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<div class="row"><span class="left">To cross multiply is to go from this:</span><span class="right"> <span class="intbl">
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<em>a</em>
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<strong>b</strong>
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</span> = <span class="intbl">
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<em>c</em>
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<strong>d</strong>
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</span> </span></div>
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<div class="row"><span class="left">To this:</span><span class="right">ad = bc</span></div>
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</div>
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<div class="def">
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<p class="center"><span class="large">To remember think <b>cross</b> (x) multiply:<br>
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<br>
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</span><img src="images/cross-multiply.svg" alt="cross multiply illustration" /><br />
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</p>
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</div>
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<p> </p>
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<p>Cross multiplication can help speed up a solution. Like in this example: </p>
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<div class="example">
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<h3>Example:
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Find "x" in <span class="intbl">
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<em>x</em>
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<strong>8</strong>
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</span> = <span class="intbl">
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<em>2</em>
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<strong>x</strong>
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</span></h3>
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<div class="tbl">
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<div class="row"><span class="left">Start with: </span><span class="right"><span class="intbl">
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<em>x</em>
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<strong>8</strong>
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</span> = <span class="intbl">
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<em>2</em>
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<strong>x</strong>
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</span></span></div>
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<div class="row"><span class="left">Cross multiply:</span><span class="right">x<sup>2</sup> = 8 × 2</span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">x<sup>2</sup> = 16</span></div>
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<div class="row"><span class="left">And solve:</span><span class="right">x = 4 or −4</span></div>
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</div>
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<p>Check: Does <span class="intbl">
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<em>4</em>
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<strong>8</strong>
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</span> = <span class="intbl">
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<em>2</em>
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<strong>4</strong>
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</span> and <span class="intbl">
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<em>−4</em>
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<strong>8</strong>
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</span> = <span class="intbl">
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<em>2</em>
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<strong>−4</strong>
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</span> ? </p>
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</div>
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<p> </p>
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<h2>Terminology</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../numbers/images/fraction.svg" alt="fraction numerator denominator" /></p>
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<p>I said "top" and "bottom" of the fractions ... but the correct words are <b>numerator</b> and <b>denominator</b>, OK? (I just wanted to keep it simple.) </p>
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<h2>Caution: Zero</h2>
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<p>Be careful, though! </p>
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<p>We cannot use it when a denominator ("b" and "d" above) is zero, as <a href="../numbers/dividing-by-zero.html">dividing by zero</a> is "illegal". </p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(334, 335, 2045, 2046, 3174, 3175, 3176, 3876, 3877, 3878);</script> </div>
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<div class="related"> <a href="index.html">Algebra Index</a> <a href="../fractions-menu.html">Fractions Index</a> <a href="../geometry/mean-proportional.html">Mean Proportional</a></div>
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