4.03:_The_Key_Fraction_Rule

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<p class="lt-math-9844">We know that $\frac{a}{b}$ is the answer to a division problem: $$\frac{a}{b}$$represents the amount of pie an individual child receives when <img alt="a" height="8" title="Rendered by QuickLaTeX.com" width="9" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/16379/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.svg?revision=1" /> pies are shared equally by <img alt="b" height="13" title="Rendered by QuickLaTeX.com" width="8" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/16386/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.svg?revision=1" /> children.</p>

<p class="lt-math-9844">What happens if we double the number of pie and double the number of kids? Nothing! The amount of pie per child is still the same:</p>

<p class="lt-math-9844">\[\frac{2a}{2b} = \frac{a}{b} \ldotp \nonumber \]</p>

<p class="lt-math-9844">For example, as the picture shows, $\frac{6}{3}$ and $\frac{12}{6}$ both give two pies for each child.</p>

<p class="lt-math-9844">And tripling the number of pies and the number of children also does not change the final amount of pies per child, nor does quadrupling each number, or one trillion-billion-tupling the numbers!</p>

<p class="lt-math-9844">\[\frac{6}{3} = \frac{12}{6} = \frac{18}{9} = \ldots = \text{two pies per child} \ldotp \nonumber \]</p>

<p class="lt-math-9844">This leads us to want to believe:</p>

<section class="box-definition"><span id="Key_Fraction_Rule"></span><h5 class="box-legend lt-math-9844"><span class="lt-icon-default">Key Fraction Rule</span></h5>

<p class="lt-math-9844">\[\frac{xa}{xb} = \frac{a}{b}, \nonumber \]</p>

<p class="lt-math-9844">(at least for positive whole numbers <img alt="x" height="8" title="Rendered by QuickLaTeX.com" width="10" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/16504/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.svg?revision=1" />).</p>

<p class="lt-math-9844">We say that the fractions $\frac{xa}{xb}$ and $\frac{a}{b}$ are <strong>equivalent</strong>.</p>
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<section class="box-example"><span id="Example:_Fractions_equivalent_to_3.2F5"></span><h5 class="box-legend lt-math-9844"><span class="lt-icon-default">Example: Fractions equivalent to 3/5</span></h5>

<p class="lt-math-9844">For example, </p>

<p class="lt-math-9844">\[\frac{3}{5}\; \text{(sharing three pies among five kids)} \nonumber \]</p>

<p class="lt-math-9844">yields the same result as</p>

<p class="lt-math-9844">\[\frac{3 \cdot 2}{5 \cdot 2} = \frac{6}{10}\; \text{(sharing six pies among ten kids)} \nonumber \]</p>

<p class="lt-math-9844">\[\frac{3 \cdot 100}{5 \cdot 100} = \frac{300}{500}\; \text{(sharing 300 pies among 500 kids)} \ldotp \nonumber \]</p>
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<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h5 class="box-legend lt-math-9844"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-9844">Write down a lot of equivalent fractions for $\frac{1}{2}$, for $\frac{10}{3}$, and for 1.</p>
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<section class="box-example"><span id="Example:_Going_Backwards"></span><h5 class="box-legend lt-math-9844"><span class="lt-icon-default">Example: Going Backwards</span></h5>

<p class="lt-math-9844">\[\frac{20}{32}\; \text{(sharing 20 pies among 32 kids)} \nonumber \]</p>

<p class="lt-math-9844">is the same problem as:</p>

<p class="lt-math-9844">\[\frac{5 \cdot 4}{8 \cdot 4} = \frac{5}{8}\; \text{(sharing five pies among eight kids)} \ldotp \nonumber \]</p>

<p class="lt-math-9844">Most people say we have <i>cancelled</i> or taken a common factor 4 from the numerator and denominator.</p>

<p class="lt-math-9844">Mathematicians call this process <i>reducing</i> the fraction to lowest terms. (We have made the numerator and denominator smaller, in fact as small as we can make them!)</p>

<p class="lt-math-9844">Teachers tend to say that we are <i>simplifying</i> the fraction. (You have to admit that $\frac{5}{8}$ does look simpler than $\frac{20}{32}$.)</p>
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<section class="box-example"><span id="Example:_How_Low_Can_You_Go.3F"></span><h5 class="box-legend lt-math-9844"><span class="lt-icon-default">Example: How Low Can You Go?</span></h5>

<p class="lt-math-9844">As another example, $\frac{280}{350}$ can certainly be simplified by noticing that there is a common factor of 10 in both the numerator and the denominator:</p>

<p class="lt-math-9844">\[\frac{280}{350} = \frac{28 \cdot 10}{35 \cdot 10} = \frac{28}{35} \ldotp \nonumber \]</p>

<p class="lt-math-9844">We can go further as 28 and 35 are both multiples of 7:</p>

<p class="lt-math-9844">\[\frac{28}{35} = \frac{4 \cdot 7}{5 \cdot 7} = \frac{4}{5} \ldotp \nonumber \]</p>

<p class="lt-math-9844">Thus, sharing 280 pies among 350 children gives the same result as sharing 4 pies among 5 children!</p>

<p class="lt-math-9844">\[\frac{280}{350} = \frac{4}{5} \ldotp \nonumber \]</p>

<p class="lt-math-9844">Since 4 and 5 share no common factors, this is as far as we can go with this example (while staying with whole numbers).</p>
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<p class="lt-math-9844">Mix and Match: On the top are some fractions that have not been simplified. On the bottom are the simplified answers, but in random order. Which simplified answer goes with which fraction? (Notice that there are fewer answers than questions!)</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h5 class="box-legend lt-math-9844"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-9844">Use the &ldquo;Pies Per Child Model&rdquo; to explain <strong>why</strong> the key fraction rule holds. That is, explain why each individual child gets the same amount of pie in these two situations:</p>

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