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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">Multiplying fractions</h1><p>To find the product of two fractions <span class="math inline">\(\frac{a}{b}\)</span> and <span class="math inline">\(\frac{c}{d}\)</span> multiply their numerators and denominators and then reduce:</p><p><span class="math display">$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$</span></p><h3 id="example">Example</h3><p><span class="math display">$$
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\frac{1}{3} \cdot \frac{2}{5} = \frac{1 \cdot 2}{3 \cdot 5} = \frac{2}{15}
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$$</span></p><h2 id="prime-factorisation-in-place">Prime factorisation in place</h2><p>The example above did not require a reduction, so here is a more complex example:</p><p><span class="math display">$$
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\frac{14}{15} \cdot \frac{30}{140} = \frac{420}{2100}
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$$</span></p><p>It would be laborious to reduce such a large product using factor trees or the repeated application of divisors, as defined in <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Reducing fractions to their lowest terms"><a href="Reducing_fractions.html">reducing fractions</a></span></span>. We can use a more efficient method. This method can be applied at the point at which we conduct the multiplication rather than afterwards once we have the product. We express the the initial multiplicands as prime factors:</p><p><span class="math display">$$
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\frac{14}{15} \cdot \frac{30}{140} = \frac{(2 \cdot 7) \cdot (2 \cdot 3 \cdot 5) }{(3 \cdot 5) \cdot (2 \cdot 2 \cdot 7 \cdot 5)}
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$$</span></p><p>We now have the product in factorised form before we have applied the multiplication so we can go ahead and cancel:</p><p><span class="math display">$$
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\frac{\cancel{2}, \cancel{7}, \cancel{2}, \cancel{3}, \cancel{5}}{\cancel{3}, \cancel{5}, \cancel{2}, \cancel{2}, \cancel{7}, 5} = \frac{1}{5}
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$$</span></p><p><strong>Note that in the above case, there was only a single 5 left as a denominator and no value left as a numerator. This is equivalent to there just being “one five” so we write <span class="math inline">\(\frac{1}{5}\)</span></strong></p><h2 id="example-of-multiplying-fractions-with-negative-fractions-containing-variables">Example of multiplying fractions with negative fractions containing variables</h2><p>Calculate: <span class="math display">$$- \frac{6x}{55y} \cdot - \frac{110y^2}{105x^2}$$</span></p><p>First multiply in place:</p><p><span class="math display">$$
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\frac{(3 \cdot 2 \cdot x) \cdot (5 \cdot 2 \cdot 11 \cdot y \cdot y)}{(5 \cdot 11 \cdot y) \cdot (7 \cdot 5 \cdot 3 \cdot x \cdot x)}
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$$</span></p><p>Then cancel:</p><p><span class="math display">$$
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\frac{(\cancel{3} \cdot 2 \cdot \cancel{x}) \cdot (\cancel{5} \cdot 2 \cdot \cancel{11} \cdot \cancel{y} \cdot y)}{(\cancel{5} \cdot \cancel{11} \cdot \cancel{y}) \cdot (7 \cdot 5 \cdot \cancel{3} \cdot \cancel{x} \cdot x)} =
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\frac{2 \cdot 2 \cdot y}{7 \cdot 5 \cdot x}
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$$</span></p><p>Then reduce:</p><p><span class="math display">$$
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\frac{2 \cdot 2 \cdot y}{7 \cdot 5 \cdot x} = \frac{4y}{35x}
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$$</span></p></div></article><nav class="ui attached segment deemphasized bottomPane" id="neuron-tags-pane"><div><span class="ui basic label zettel-tag" title="Tag">arithmetic</span><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>
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