4.07:_Dividing_Fractions-_Meaning

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<p class="lt-math-10579">Dividing fractions is one of the hardest ideas in elementary school mathematics.  By now, you are used to the rule: to divide by a fraction, multiply by its reciprocal. (&ldquo;invert and multiply&rdquo;). But ask yourself: Why does this rule work? Does it really make sense to you? Can you explain why it makes sense to a third grader?</p>

<p class="lt-math-10579">We are going to build up to the &ldquo;invert and multiply&rdquo; rule, but along the way, we&rsquo;ll find some more meaningful ways to understand division of fractions. So please play along: <strong>pretend that you don&rsquo;t already know the &ldquo;invert and multiply&rdquo; rule,</strong> and solve the problems in this chapter with other methods.</p>

<span id="Groups_of_Equal_Size"></span><h3 class="lt-math-10579">Groups of Equal Size</h3>

<p class="lt-math-10579">Remember the quotative model for division: $18 \div 3$ means:</p>

<blockquote>
<p class="lt-math-10579">How many groups of 3 can I find in 18?</p>
</blockquote>

<p class="lt-math-10579">We start with 18 dots (or candy bars or molecules), and we make groups of 3 dots (or 3 whatevers). We ask: how many groups can we make?</p>

<p class="mt-align-center lt-math-10579"><em>18 dots, split into groups of 3 dots.  Since there are 6 groups, we have 18 : 3 = 6.</em></p>

<p class="lt-math-10579">This same idea applies when we divide fractions.  For example, $6 \div 23$ means:</p>

<blockquote>
<p class="lt-math-10579">How many groups of $\frac{2}{3}$ can I find in 6?</p>
</blockquote>

<section class="box-example"><span id="Example:_6_.C3.B7_2.2F3"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Example: 6 &divide; 2/3</span></h5>

<p class="lt-math-10579">Let&rsquo;s draw a picture of 6 pies, and see how many groups of $\frac{2}{3}$ we can find:</p>

<p class="lt-math-10579">We found nine equal groups of size $\frac{2}{3}$, so we conclude that \[6 \div \frac{2}{3} = 9 \ldotp \nonumber \]</p>
</section>

<p class="lt-math-10579">Unfortunately, it&rsquo;s not always quite so straightforward to find the equal groups.  For example, $\frac{3}{4} \div \frac{1}{3}$ asks the question:</p>

<blockquote>
<p class="lt-math-10579">How many groups of $\frac{1}{3}$ can I find in $\frac{3}{4}$?</p>
</blockquote>

<section class="box-example"><span id="Example:_3.2F4_.C3.B7_1.2F3"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Example: 3/4 &divide; 1/3</span></h5>

<p class="lt-math-10579">Let&rsquo;s draw a picture of $\frac{3}{4}$ of a pie, and see how many groups of $\frac{1}{3}$ we can find:</p>

<p class="lt-math-10579">The first pictures shows $\frac{3}{4}$ of a pie.  The second picture shows two equal groups of $\frac{1}{3}$ inside of $\frac{3}{4}$, but there&rsquo;s a little bit left over.  We conclude $$\frac{3}{4} \div \frac{1}{3} = 2 + \text{a tiny bit more} \ldotp$$But how much more? Can we figure it out exactly?</p>
</section>

<p class="lt-math-10579">Here&rsquo;s a method that will let you do the computation exactly.  We&rsquo;ll use rectangular pies, and divide them up into rows and columns based on the denominators of the numbers we&rsquo;re dividing.</p>

<section class="box-example"><span id="Example:_3.2F4_.C3.B7_1.2F3_2"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Example: 3/4 &divide; 1/3</span></h5>

<p class="lt-math-10579">Start by drawing two identical rectangles, each with 4 rows (from the denominator of $\frac{3}{4}$ and 3 columns (from the denominator of $\frac{1}{3}$).</p>

<p class="lt-math-10579">Shade $\frac{3}{4}$ of the first rectangle (this is exactly three rows), and shade $\frac{1}{3}$ of the second rectangle (so that&rsquo;s one column).</p>

<p class="lt-math-10579">Now ask: how many copies of $\frac{1}{3}$ can I find in $\frac{3}{4}$?  Well, $\frac{1}{3}$ is equal to four of the smaller squares.  So we find groups equal to that:</p>

<p class="lt-math-10579">In the picture of $\frac{3}{4}$, we can find:</p>

<ul>
    <li class="lt-math-10579">two groups of four squares (two groups of $\frac{1}{3}$), and</li>
    <li class="lt-math-10579">one square left over, which is $\frac{1}{4}$ of the group we&rsquo;re looking for.</li>
</ul>

<p class="lt-math-10579">We conclude: \[\frac{3}{4} \div \frac{1}{3} = 2 \frac{1}{4} \ldotp \nonumber \]</p>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-10579">Use either method above to find the following quotients.  Remember, pretend that you don&rsquo;t know any method to divide fractions except finding equal-sized groups.</p>

<p class="lt-math-10579">\[\frac{3}{4} \div \frac{1}{2} \qquad \frac{1}{3} \div \frac{1}{2} \qquad \frac{4}{9} \div \frac{1}{3} \qquad \frac{4}{5} \div \frac{1}{3} \qquad \frac{3}{5} \div \frac{3}{4} \qquad \frac{3}{2} \div \frac{1}{2} \qquad \frac{2}{3} \div \frac{1}{2} \nonumber \]</p>
</section>

<span id="Common_Denominator_Method"></span><h3 class="lt-math-10579">Common Denominator Method</h3>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-10579">Solve each of the following fraction division problems using the &ldquo;groups of equal size&rdquo; method: $$\frac{6}{4} \div \frac{3}{4} \qquad \frac{6}{10} \div \frac{3}{10} \qquad \frac{8}{9} \div \frac{4}{9} \qquad \frac{6}{33} \div \frac{2}{33} \qquad \frac{5}{4} \div \frac{2}{4} \qquad \frac{5}{2} \div \frac{2}{2} \qquad \frac{5}{10} \div \frac{2}{10}$$What do you notice?</p>
</section>

<p class="lt-math-10579">This leads to our first fraction division method:</p>

<section class="box-definition"><span id="Common_denominator_method"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Common denominator method</span></h5>

<p class="lt-math-10579">If two fractions have the same denominator, then when you divide them, you can just divide the numerators. In symbols, \[\frac{a}{d} \div \frac{b}{d} = \frac{a}{b} \ldotp \nonumber \]</p>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_3"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ul>
    <li class="lt-math-10579">Use the common denominator method to find these quotients: $$\frac{1}{3} \div \frac{2}{3}, \qquad \frac{5}{8} \div \frac{3}{8}, \qquad \frac{3}{8} \div \frac{5}{8}, \qquad \frac{15}{33} \div \frac{1}{33},$$</li>
    <li class="lt-math-10579">What if the fractions do not have a common denominator? Is the method useless, or can you find a way to make it work? Can you solve these problems?$$\frac{3}{5} \div \frac{3}{4}, \qquad \frac{3}{4} \div \frac{8}{7}, \qquad \frac{2}{3} \div \frac{1}{2}, \qquad \frac{5}{8} \div \frac{1}{4} \ldotp$$</li>
</ul>
</section>

<span id="Missing_Factor_Method"></span><h2 class="lt-math-10579">Missing Factor Method</h2>

<p class="lt-math-10579">We know that we can always turn a division problem into a &ldquo;missing factor&rdquo; multiplication problem. Can that help us compute fraction division? Sometimes!</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_4"></span><h5 class="box-legend lt-math-10579"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-10579">For each division problem, rewrite it as a missing factor multiplication question. Then find the quotient using what you know about multiplying fractions.</p>

<p class="lt-math-10579">\[\frac{9}{10} \div \frac{3}{5}, \qquad \frac{7}{8} \div \frac{1}{4}, \qquad \frac{6}{7} \div \frac{3}{7}, \qquad \frac{10}{9} \div \frac{2}{3}, \qquad \frac{25}{12} \div \frac{5}{6} \ldotp \nonumber \]</p>
</section>

<p class="lt-math-10579">Unfortunately, the missing factor method doesn&rsquo;t always work out so nicely.  For example,</p>

<p class="lt-math-10579">\[\frac{3}{4} \div \frac{1}{3} = \_\_ \nonumber \]</p>

<p class="lt-math-10579">can be rewritten as</p>

<p class="lt-math-10579">\[\frac{1}{3} \cdot \_\_ = \frac{3}{4} \ldotp \nonumber \]</p>

<p class="lt-math-10579">There isn&rsquo;t a nice ratio of whole numbers that obviously fills in the blank, but we&rsquo;ll come back to this idea and resolve it soon.</p>

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