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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">Adding and subtracting fractions</h1><h2 id="adding-subracting-fractions-with-common-denominators">Adding/ subracting fractions with common denominators</h2><p>For two fractions <span class="math inline">\(\frac{a}{c}\)</span> and <span class="math inline">\(\frac{b}{c}\)</span> with a common denominator, their sum is defined as:</p><p><span class="math display">$$
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\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}
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$$</span></p><p>For example:</p><p><span class="math display">$$
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\frac{2}{8} + \frac{3}{8} = \frac{5}{8}
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$$</span></p><p>The same applies to subtraction:</p><p><span class="math display">$$
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\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}
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$$</span></p><h2 id="adding-subracting-fractions-without-common-denominators">Adding/ subracting fractions without common denominators</h2><ul><li>Find the lowest common denominator for the two fractions</li><li>Use this to create two equivalent fractions</li><li>Add/subtract</li><li>Reduce</li></ul><h3 id="lowest-common-denominator-and-lowest-common-multiple">Lowest common denominator and lowest common multiple</h3><p>Given the symmetry between <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Factors and divisors"><a href="Factors_and_divisors.html">factors and divisors</a></span></span> these properties are related. Note however that the LCM is more generic: it applies to any set of numbers not just fractions. Whereas the LCD is explicitly to do with fractions (hence ‘denominator’).</p><ul><li><p>For two fractions <span class="math inline">\(a, b\)</span> (or a set), the LCD is the smallest number divisble by both the denominator of <span class="math inline">\(a\)</span> and the denominator of <span class="math inline">\(b\)</span> (or each member of the set).</p></li><li><p>For two fractions <span class="math inline">\(a, b\)</span> (or a set), the LCM is the smallest number that is a multiple of the denominator of <span class="math inline">\(a\)</span> and the denominator of <span class="math inline">\(b\)</span> (or each member of the set).</p></li></ul><p>In order to find the LCM of the set <span class="math inline">\(\{12, 16\}\)</span> we list the multiples of both:</p><p><span class="math display">$$
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12, 24, 36, 48 \\
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16, 32, 48
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$$</span></p><p>Until we identify the smallest number common to both lists. In this case it is 48. Thus the LCM of 12 and 16 is 48.</p><p>The relationship between LCM and LCD is that <em>the least common denominator is the least common multiple of the fractions’ denomintors</em>.</p><h3 id="demonstration-addition">Demonstration: addition</h3><p>We can now use this to calculate the addition of two fractions without common denominators: <span class="math inline">\(\frac{4}{9} + \frac{1}{6}\)</span>.</p><p>First identify the common multiples of 9 and 6:</p><p><span class="math display">$$
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9, 18, ... \\
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6, 12, 18, ...
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$$</span></p><p>The least common multiple is 18. We then think: what do we need to multiply each denominator by to get 18?</p><p>In the case of the first fraction (<span class="math inline">\(\frac{4}{9}\)</span>) it is 2:</p><p><span class="math display">$$
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\frac{4}{9 \cdot 2} = \frac{4}{18}
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$$</span></p><p>But what we do to the denominator, we must also do to the numerator, hence:</p><p><span class="math display">$$
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\frac{4 \cdot 2}{9 \cdot 2} = \frac{8}{18}
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$$</span></p><p>We then do the same to the second fraction (<span class="math inline">\(\frac{1}{6}\)</span>). We need to multiply its denominator by 3 to get 18 and we apply this also to the numerator.</p><p><span class="math display">$$
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\frac{1 \cdot 3}{6 \cdot 3} = \frac{3}{18}
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$$</span></p><p>We now have two fractions that share a common denominator so we can sum:</p><p><span class="math display">$$
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\frac{8}{18} + \frac{3}{18} = \frac{11}{18}
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$$</span></p><h3 id="demonstration-subtraction">Demonstration: subtraction</h3><p>Calculate:</p><p><span class="math display">$$
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\frac{3}{5} - \frac{2}{3}
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$$</span></p><p>Once again we need to find the least common denominator for the two fractions. We start by listing the common multiples for the two denominators 5 and 3:</p><p><span class="math display">$$
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5, 10, 15, ... \\
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3, 6, 9, 12, 15,...
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$$</span></p><p>The lowest common multiple is 15. From the first fraction we get 15 by multiplying by 3. With the second fraction we get 15 by multiplying by 5. Thus:</p><p><span class="math display">$$
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\frac{3 \cdot 3}{5 \cdot 3} = \frac{9}{15}
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$$</span></p><p><span class="math display">$$
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\frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}
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$$</span></p><p>We can now carry out the subtraction:</p><p><span class="math display">$$
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\frac{9}{15} - \frac{10}{15} = -\frac{1}{15}
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$$</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">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> |