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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">Reducing fractions to their lowest terms</h1><blockquote><p>A fraction is said to be <em>reduced to its lowest terms</em> if the <a href="Factors%20and%20divisors.md#greatest-common-divisor">greatest common divisor</a> of the numerator and the denominator is $1$.</p></blockquote><blockquote><p>Whenever we reduce a fraction, the resultant fraction will always be <a href="Equivalent%20fractions.md">equivalent</a> to the fraction we started with.</p></blockquote><p>Thus the fraction <span class="math inline">\(\frac{2}{3}\)</span> is reduced to its lowest terms because the greatest common divisor is 1. Neither the numerator or the denominator can be reduced to any lower terms. In contrast, the fraction <span class="math inline">\(\frac{4}{6}\)</span> is not reduced to its lowest terms because the greatest common divisor of both 4 and 6 is 2, not 1.</p><h3 id="1-reducing-with-repeated-application-of-divisors">1. Reducing with repeated application of divisors</h3><p>The following demonstrates the process of reducing a fraction to its lowest terms in a series of steps:</p><p><span class="math display">$$
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\\frac{18}{24} = \frac{18/2}{24/2} = \frac{9}{12} = \frac{9/3}{12/3} = \frac{3}{4}
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$$</span></p><p>_Once we get to <span class="math inline">\(\frac{3}{4}\)</span> the greatest common divisor is 1, therefore <span class="math inline">\(\frac{18}{24}\)</span> has been reduced to its lowest terms _.</p><h3 id="2-reducing-in-one-step-with-the-highest-common-divisor">2. Reducing in one step with the highest common divisor</h3><p>In the previous example the reduction took two steps: first we divided by two and then we divided by three. There is a more efficient way: find the <a href="Factors%20and%20divisors.md#greatest-common-divisor">highest common divisor</a> of the numerator and denominator and then use this as the basis for the reduction. With this method, the reduction can be completed in a single step.</p><p>The greatest common divisor of 18 and 24 is 6, thus:</p><p><span class="math display">$$
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\frac{18}{24} = \frac{18/6}{24/6} = \frac{3}{4}
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$$</span></p><p>Note how our earlier two divisors 2 and 3 are <a href="Factors%20and%20divisors.md#factors">factors</a> of 6, showing the consistency between the two methods.</p><h3 id="3-reducing-with-factors-and-cancellation">3. Reducing with factors and cancellation</h3><p>The two methods above are not very systematic and are quite heuristic. The third approach is more systematic and relies on the <a href="Factors%20and%20divisors.md">interchangeability of factors and divisors</a>.</p><p>Instead of thinking asking what is the greatest common divisor of 18 and 24 we could ask: which single number can we multiply by to get 18 and 24? Obviously both numbers are in the six times table. This is therefore to say that 6 is a <a href="Factors%20and%20divisors.md#factors">factor</a> of both: we can multiply some number by 6 to arrive at both 18 and 24. The numbers are 3 and 4 respectively:</p><p><span class="math display">$$
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\\begin{split}
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3 \cdot 6 = 18 \\
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4 \cdot 6 = 24
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\\end{split}
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$$</span></p><p>Here, 3 and 4 are the multiplicands of the factor 6. As <span class="math inline">\(\frac{3}{4}\)</span> doesn’t have a lower common factor, it is therefore defined in its lowest terms.</p><p>Once we have reached this point, we no longer need the common factor 6, we can therefore cancel it out, leaving the multiplicands as the reduced fraction:</p><p><span class="math display">$$
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\\begin{split}
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3 \cancel{\cdot6= 18}\\
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4 \cancel{\cdot6= 24}
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\\end{split}
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$$</span></p><h3 id="4-reducing-with-prime-factorisation">4. Reducing with prime factorisation</h3><p>This is still a bit long-winded however particularly when finding the factors of larger numbers because we have to go through the factors of both numbers to find the largest held in common.</p><p>A better method is to utilise <a href="Prime%20factorization.md">prime factorization</a> combined with the canceling technique.</p><p>First we find the prime factors of both the numerator and denominator: <img alt="drawio-Page-7.drawio.png" src="/static/drawio-Page-7.drawio.png" /></p><p>This gives us:</p><p><span class="math display">$$
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\\frac{18}{24} = \frac{2 \cdot 3 \cdot 3}{2 \cdot 2 \cdot 2 \cdot 3}
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$$</span></p><p>We then cancel out the factors held in common between the numerator and denominator:</p><p><span class="math display">$$
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\\frac{\cancel{2} \cdot \cancel{3} \cdot 3}{\cancel{2} \cdot 2 \cdot 2 \cdot \cancel{3}}
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$$</span></p><p>This gives us:</p><p><span class="math display">$$
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\\frac{3}{2 \cdot 2}
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$$</span></p><p>We then simplify the fraction as normal to its lowest term (conducting any multiplications required by what is left from the prime factorization):</p><p><span class="math display">$$
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\\frac{3}{4}
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$$</span></p><h2 id="reducing-fractions-that-contain-variables">Reducing fractions that contain variables</h2><p>Superficially this looks to be more difficult but in fact we can apply the same prime factorization method to get the result.</p><h3 id="demonstration">Demonstration</h3><p><em>Reduce the following fraction to its lowest terms: <span class="math display">$$\frac{25a^3b}{40a^2b^3}$$</span></em></p><p>The prime factors of the numerator and denominator:</p><p><span class="math display">$$
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\\begin{split}
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25 = {5, 5} \\
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40 = {2,2,2,5}
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\\end{split}
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$$</span></p><p>Now we apply canceling but we include the variable parts, treating them exactly the same as the coefficients. We break them out of their exponents however.</p><p><span class="math display">$$\frac{25a^3b}{40a^2b^3} =\frac{5 \cdot 5 \cdot a \cdot a \cdot a \cdot b}{2 \cdot 2 \cdot 2 \cdot 5 \cdot a \cdot a \cdot b \cdot b \cdot b }$$</span></p><p>Canceled:</p><p><span class="math display">$$\frac{\cancel{5} \cdot 5 \cdot \cancel{a} \cdot \cancel{a} \cdot a \cdot \cancel{b}}{2 \cdot 2 \cdot 2 \cdot \cancel{5} \cdot \cancel{a} \cdot \cancel{a} \cdot \cancel{b} \cdot b \cdot b }$$</span></p><p>Which gives us:</p><p><span class="math display">$$
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\\frac{5 \cdot a}{2 \cdot 2 \cdot 2 \cdot b \cdot b} = \frac{5a}{8b^2}
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$$</span></p><h2 id="reducing-fractions-that-contain-negative-values">Reducing fractions that contain negative values</h2><p><em>Reduce the following fraction to its lowest terms: <span class="math display">$$\frac{14y^5}{-35y^3}$$</span></em></p><ul><li><p>This fraction is an instance of a <a href="Handling%20negative%20fractions.md#fractions-with-unlike-terms">fraction with unlike terms</a>.</p></li><li><p>Apply <a href="Prime%20factorization.md">Prime factorization</a>:</p><p><img alt="draw.io-Page-8.drawio.png" src="/static/draw.io-Page-8.drawio.png" /></p></li><li><p>Cancel the coefficients and variable parts</p><p><span class="math display">$$
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\\frac{14y^5}{-35y^3}=\frac{5 \cdot 7 \cdot 2 \cdot y \cdot y \cdot y \cdot y \cdot y}{-5 \cdot 7 \cdot y \cdot y \cdot y} = - \frac{2y^2}{5}
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$$</span></p></li></ul><p><em>Reduce the following fraction to its lowest terms: <span class="math display">$$\frac{- 12xy^2}{ - 18xy^2}$$</span></em></p><ul><li><p>This fraction is an instance of a <a href="Handling%20negative%20fractions.md#fractions-with-like-terms">fraction with like terms</a>.</p></li><li><p>Apply <a href="Prime%20factorization.md">Prime factorization</a>:</p></li></ul><p><img alt="draw.io-Page-8.drawio 1.png" src="/static/draw.io-Page-8.drawio%201.png" /></p><ul><li><p>Cancel the coefficients and variable parts</p><p>$$</p><ul><li>\frac{12xy^2}{18xy^2}=\frac{3 \cdot 2 \cdot 2 \cdot x \cdot y \cdot y}{3 \cdot 7 \cdot 3 \cdot 2 \cdot x \cdot x \cdot y} = - \frac{2y}{3x} $$</li></ul></li></ul></div></article><nav class="ui attached segment deemphasized backlinksPane" id="neuron-backlinks-pane"><h3 class="ui header">Backlinks</h3><ul class="backlinks"><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Multiplying_fractions.html">Multiplying fractions</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><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></div></li></ul></li></ul></nav><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> |