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      <text>1.4.2.1 Multiplication and division of algebraic fractions</text>
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      <text><![CDATA[<h3><span style="color: inherit; font-family: inherit;">Multiplication and division of algebraic fractions</span><br></h3>
<p>To multiply together two fractions (numerical or algebraic) we multiply their numerators together and then multiply their denominators together. That is</p>

<div class="HELM_keypoint">
    <h4>Key Point</h4>
    <p style="text-align:center"><strong>Multiplication</strong></p>
    <p>\[ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd} \]</p>
</div>
<p><br></p>

<p>Any factors common to both numerator and denominator can be cancelled. This cancellation can be performed before or after the multiplication.</p>
<p>To divide one fraction by another (numerical or algebraic) we invert the second fraction and then multiply.</p>

<div class="HELM_keypoint">
    <h4>Key Point</h4>
    <p style="text-align:center"><strong>Division of fractions</strong></p>
    <p>\[ \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc} \quad b\neq 0, \, c\neq0, \, d\neq 0 \]</p>
</div>
<p><br></p>


<hr>
<h4 class="HELM_example">Example</h4>
<p>Simplify:</p>
<ol class="HELM_parts_inline">
    <li>\(\dfrac{2a}{c} \times \dfrac{4}{c}\)</li>
    <li>\(\dfrac{2a}{c} \times \dfrac{c}{4}\)</li>
    <li>\(\dfrac{2a}{c} \div \dfrac{4}{c}\)</li>
</ol>

<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
    <li>\(\dfrac{2a}{c} \times \dfrac{4}{c} = \dfrac{8a}{c^2}\)</li>
    <li>\(\dfrac{2a}{c} \times \dfrac{c}{4} = \dfrac{2ac}{4c} = \dfrac{2a}{4}= \dfrac{a}{2}\)</li>
    <li>
        <p>Division is performed by inverting the second fraction and then multiplying.</p>
        <p>\(\dfrac{2a}{c} \div \dfrac{4}{c} = \dfrac{2a}{c} \times \dfrac{c}{4}= \dfrac{a}{2}\)</p>
    </li>
</ol>

<hr>
<h4 class="HELM_example">Example</h4>
<p>Simplify:</p>
<ol class="HELM_parts_inline">
    <li>\(\dfrac{1}{5x} \times 3x\)</li>
    <li>\(\dfrac{1}{x} \times x\)</li>
</ol>

<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
  <li><p>Note that \(3x = \dfrac{3x}{1}.\)</p>
      <p>Then \(\dfrac{1}{5x} \times 3x = \dfrac{1}{5x} \times \dfrac{3x}{1} = \dfrac{3x}{5x}= \dfrac{3}{5}.\)</p></li>
  
  <li><p>\(x\) can be written as \( \dfrac{x}{1}.\)</p>
      <p>Then \(\dfrac{1}{x} \times x = \dfrac{1}{x} \times \dfrac{x}{1}= \dfrac{x}{x} = 1.\)</p>
    </li>
</ol>]]></text>
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