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<!--replace-end-7--><!--replace-end-4--><!--replace-end-1--></head><body><div class="ui fluid container universe"><!--replace-start-2--><!--replace-start-3--><!--replace-start-6--><div class="ui text container" id="zettel-container" style="position: relative"><div class="zettel-view"><article class="ui raised attached segment zettel-content"><div class="pandoc"><h1 id="title-h1">Dividing fractions</h1><p>Suppose you have the following shape:</p><p><img alt="draw.io-Page-9.drawio 1.png" src="/static/draw.io-Page-9.drawio.png" /></p><p>One part is shaded. This represents one-eighth of the original shape.</p><p><img alt="one-eighth-a.png" src="/static/one-eighth-a.png" /></p><p>Now imagine there are four instances of the shape and one-eighth remains shaded. How man one-eighths are there in four?</p><p><img alt="draw.io-Page-9.drawio 2.png" src="/static/draw.io-Page-9.drawio.png" /></p><p>The shaded proportion represents <span class="math inline">\(\frac{1}{8}\)</span> of the shape. Imagine four of these shapes, how many eighths are there?</p><p>This is a division statement: to find how many one-eighths there are we would calculate:</p><p><span class="math display">$$
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4 \div \frac{1}{8}
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$$</span></p><p>But actually it makes more sense to think of this as a multiplication. There are four shapes of eight parts meaning there are <span class="math inline">\(4 \cdot 8\)</span> parts in total, 32. One of these parts is shaded making it equal to <span class="math inline">\(\frac{1}{32}\)</span>.</p><p>From this we realise that when we divide fractions by an amount, we can express the calculation in terms of multiplication and arrive at the correct answer:</p><p><span class="math display">$$
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4 \div \frac{1}{8} = 4 \cdot 8 = 32
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$$</span></p><p>Note that we omit the numerator but that technically the answer would be <span class="math inline">\(\frac{1}{32}\)</span>.</p><h3 id="formal-specification-of-how-to-divide-fractions">Formal specification of how to divide fractions</h3><p>We combine the foregoing (that it is easier to divide by fractional amounts using multiplication) with the concept of a <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Recipricols"><a href="Reciprocals.html">reciprocol</a></span></span> to arrive at a definitive method for dividing two fractions. It boils down to: <em>invert and multiply</em>:</p><p>If <span class="math inline">\(\frac{a}{b}\)</span> and <span class="math inline">\(\frac{c}{d}\)</span> are fractions then: <span class="math display">$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$</span></p><p>We invert the divisor (the second factor) and change the operator from division to multiplication.</p><h4 id="demonstration">Demonstration</h4><p>Divide <span class="math inline">\(\frac{1}{2}\)</span> by <span class="math inline">\(\frac{3}{5}\)</span></p><p><span class="math display">$$
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\begin{split}
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\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \cdot \frac{5}{3} \\
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= \frac{5}{5}
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\end{split}
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$$</span></p><p>Divide <span class="math inline">\(\frac{-6}{x}\)</span> by <span class="math inline">\(\frac{-12}{x^2}\)</span></p><p>$$ \begin{split} \frac{-6}{x} \div \frac{12}{x^2} = \frac{-6}{x} \cdot \frac{x^2}{-12} \ = \frac{(\cancel{3} \cdot \cancel{2} )}{\cancel{x}} \cdot \frac{(\cancel{x} \cdot \cancel{x} )}{\cancel{3} \cdot \cancel{2} \cdot 2} \ = \frac{x}{2}</p><p>\end{split}</p><p>$$</p></div></article><nav class="ui attached segment deemphasized bottomPane" id="neuron-tags-pane"><div><span class="ui basic label zettel-tag" title="Tag">fractions</span><span class="ui basic label zettel-tag" title="Tag">prealgebra</span></div></nav><nav class="ui bottom attached icon compact inverted menu blue" id="neuron-nav-bar"><!--replace-start-9--><!--replace-end-9--><a class="right item" href="impulse.html" title="Open Impulse"><i class="wave square icon"></i></a></nav></div></div><!--replace-end-6--><!--replace-end-3--><!--replace-end-2--><div class="ui center aligned container footer-version"><div class="ui tiny image"><a href="https://neuron.zettel.page"><img alt="logo" src="https://raw.githubusercontent.com/srid/neuron/master/assets/neuron.svg" title="Generated by Neuron 1.9.35.3" /></a></div></div></div></body></html> |