openfreebooks-bookgenerator/content/math/algebraic-fractions-formulas/supplement.html

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    For centuries, algebra was written in words. In the 16th and 17th centuries, mathematicians such
    as <strong>François Viète</strong> began to use letters systematically for both unknowns and
    known parameters. That made <strong>formulas</strong> — general relationships such as
    \(v = d/t\) — easy to rearrange without repeating long verbal rules.
  </p>
  <p>
    Rational expressions (fractions built from polynomials) appear whenever you divide one polynomial
    by another. Scientists from <strong>Galileo</strong> onward used ratios and proportionalities in
    motion; later, <strong>Newton’s</strong> laws and electrical laws (e.g. \(V = IR\)) relied on
    changing the subject of a formula to solve for the quantity you can measure.
  </p>
  <p>
    Today, algebraic fractions show up in simplifying models, in calculus limits, and whenever a
    relationship is written as one quantity divided by another.
  </p>

  <h2 id="derivation" class="book-prose__heading">Derivation</h2>

  <h3 id="why-cancel" class="book-prose__heading">Why can we cancel a common factor?</h3>
  <p>
    If \(c \neq 0\), then \(\dfrac{ac}{bc} = \dfrac{a}{b}\) because multiplying numerator and
    denominator by the same non-zero number does not change the value of a numeric fraction; the
    same idea holds when \(c\) is a polynomial factor shared by top and bottom.
  </p>
  <p>
    You <em>cannot</em> cancel in \(\dfrac{a+c}{b+c}\) in general — that would mean, for example,
    \(\dfrac{1+2}{3+2} = \dfrac{1}{3}\), which is false. Only <strong>factors</strong>, not
    <strong>terms</strong>, may be cancelled.
  </p>

  <h3 id="why-reverse-ops" class="book-prose__heading">Why reverse the order of operations?</h3>
  <p>
    A formula like \(v = d/t\) builds \(v\) by first dividing \(d\) by \(t\). To recover \(t\),
    undo division by multiplying: \(vt = d\), then undo multiplication by \(v\) (when \(v \neq 0\))
    to get \(t = d/v\). Each step reverses one operation that was applied to the subject you want.
  </p>

  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
  <p>
    Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
  </p>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Simplify \(\dfrac{x^2 - 4}{x + 2}\). What value of \(x\) must be excluded?
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Can you cancel the \(x\) in \(\dfrac{x + 1}{x}\)? Explain briefly.
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Write \(\dfrac{3}{x} - \dfrac{1}{2x}\) as a single fraction in simplest form.
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Make \(d\) the subject of \(v = \dfrac{d}{t}\). State any conditions on \(t\) and \(v\).
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      A wire obeys \(V = IR\). If voltage and resistance are known, which formula gives current?
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Simplify \(\dfrac{1}{x} + \dfrac{1}{x+1}\). What is the common denominator?
    </p>
  </div>

  <h2 id="applications" class="book-prose__heading">Applications</h2>

  <h3 id="uniform-motion" class="book-prose__heading">Uniform motion</h3>
  <p>
    For constant speed, distance \(d\), speed \(v\), and time \(t\) are related by \(v = d/t\). If a
    train travels \(240\) km in \(3\) h, then:
  </p>
  <p class="book-formula">\[
    \begin{aligned}
      v &= \frac{240}{3} \\
        &= 80\ \text{km/h}
    \end{aligned}
  \]</p>
  <p>
    If you know \(v\) and \(t\) but need distance, rearrange to \(d = vt\).
  </p>

  <div class="math-widget-mount" data-widget="formula-rearrange-demo" data-pagefind-ignore></div>

  <h3 id="density" class="book-prose__heading">Density</h3>
  <p>
    Density \(\rho\) (rho) is mass per unit volume: \(\rho = m/V\). A block of mass \(2.7\) kg with
    volume \(0.001\) m³ has:
  </p>
  <p class="book-formula">\[
    \begin{aligned}
      \rho &= \frac{2.7}{0.001} \\
           &= 2700\ \text{kg/m}^3
    \end{aligned}
  \]</p>
  <p>
    To find mass from \(\rho\) and \(V\), use \(m = \rho V\).
  </p>

  <h3 id="pressure-circuits" class="book-prose__heading">Pressure and electric circuits</h3>
  <p>
    Pressure \(P = F/A\) links force \(F\) and area \(A\). Hydraulic and weather models often solve
    for \(F = PA\). In a resistor, \(V = IR\) gives voltage from current and resistance; rearrange
    to \(I = V/R\) when you know voltage and resistance.
  </p>

  <div class="math-widget-mount" data-widget="fraction-simplify-demo" data-pagefind-ignore></div>

  <h2 id="question-bank" class="book-prose__heading">Question bank</h2>
  <p>
    Work each question, then open the solution to check your reasoning. Difficulty increases
    through the sets.
  </p>

  <h3 id="questions-easy" class="book-prose__heading">Easy</h3>

  <details class="book-question">
    <summary class="book-question__prompt">Simplify \(\dfrac{6x}{3x}\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(2\), with \(x \neq 0\).</p>
      <p><strong>Steps:</strong> Cancel the common factor \(3x\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Simplify \(\dfrac{x^2}{x}\) for \(x \neq 0\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(x\).</p>
      <p><strong>Explanation:</strong> Cancel one factor \(x\) from numerator and denominator.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Make \(d\) the subject of \(v = d/t\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(d = vt\) (\(t \neq 0\) when rearranging from the original).</p>
      <p><strong>Steps:</strong> Multiply both sides by \(t\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Make \(R\) the subject of \(V = IR\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(R = V/I\) (\(I \neq 0\)).</p>
      <p><strong>Steps:</strong> Divide both sides by \(I\).</p>
    </div>
  </details>

  <h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>

  <details class="book-question">
    <summary class="book-question__prompt">Simplify \(\dfrac{x^2 - 9}{x + 3}\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(x - 3\), with \(x \neq -3\).</p>
      <p><strong>Steps:</strong> \(x^2 - 9 = (x-3)(x+3)\); cancel \((x+3)\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">\(\dfrac{2}{x} + \dfrac{1}{x} = \) ______</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\dfrac{3}{x}\), \(x \neq 0\).</p>
      <p><strong>Explanation:</strong> Same denominator already.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Simplify \(\dfrac{x^2 - 1}{x^2 + x}\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\dfrac{x-1}{x}\), \(x \neq 0,\; x \neq -1\).</p>
      <p><strong>Steps:</strong> Factorise; cancel \((x+1)\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Make \(m\) the subject of \(\rho = m/V\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(m = \rho V\) (\(V \neq 0\) in the original).</p>
      <p><strong>Steps:</strong> Multiply both sides by \(V\).</p>
    </div>
  </details>

  <h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>

  <details class="book-question">
    <summary class="book-question__prompt">\(\dfrac{1}{x} + \dfrac{2}{x+1} = \) ______</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\dfrac{3x+1}{x(x+1)}\), \(x \neq 0,\; x \neq -1\).</p>
      <p><strong>Steps:</strong> LCD \(x(x+1)\); numerator \( (x+1) + 2x = 3x+1\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">\(\dfrac{x}{2} \times \dfrac{4}{x^2} = \) ______</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\dfrac{2}{x}\), \(x \neq 0\).</p>
      <p><strong>Steps:</strong> Multiply; cancel a factor \(x\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Simplify \(\dfrac{1}{x-2} - \dfrac{3}{x^2 - 4}\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\dfrac{x-1}{(x-2)(x+2)}\), \(x \neq 2,\; x \neq -2\).</p>
      <p><strong>Steps:</strong> Write \(x^2-4 = (x-2)(x+2)\); common denominator \((x-2)(x+2)\).</p>
    </div>
  </details>

  <h3 id="questions-hardcore" class="book-prose__heading">Hardcore</h3>

  <details class="book-question">
    <summary class="book-question__prompt">\(\dfrac{x^2 - 1}{x+1} \div \dfrac{x-1}{2} = \) ______</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(2\), with \(x \neq -1,\; x \neq 1\).</p>
      <p><strong>Steps:</strong> Factorise \(x^2-1\); multiply by reciprocal; cancel \((x-1)\) and \((x+1)\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Make \(t\) the subject of \(v = \dfrac{d}{t}\) and then find \(t\) when \(d = 150\) km and \(v = 75\) km/h.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong></p>
      <p class="book-formula">\[
        \begin{aligned}
          t &= \frac{d}{v} \\
            &= \frac{150}{75} \\
            &= 2\ \text{h}
        \end{aligned}
      \]</p>
      <p><strong>Steps:</strong> \(vt = d \Rightarrow t = d/v\); substitute values.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Simplify fully: \(\dfrac{2x^2 - 8}{x^2 - 4}\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(2\), \(x \neq 2,\; x \neq -2\).</p>
      <p><strong>Steps:</strong> Factorise numerator \(2(x^2-4)\) and denominator \((x-2)(x+2)\); cancel \((x-2)(x+2)\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">A formula is \(P = \dfrac{F}{A}\). The force is \(120\) N and the area is \(0.05\) m². Find \(P\), then make \(F\) the subject and verify your force value.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong></p>
      <p class="book-formula">\[
        \begin{aligned}
          P &= \frac{120}{0.05} \\
            &= 2400\ \text{Pa}
        \end{aligned}
      \]</p>
      <p class="book-formula">\[
        \begin{aligned}
          F &= PA \\
            &= 2400 \times 0.05 \\
            &= 120\ \text{N}
        \end{aligned}
      \]</p>
      <p><strong>Explanation:</strong> Rearranging checks that the formula is consistent.</p>
    </div>
  </details>

  <h2 id="references" class="book-prose__heading">References</h2>
  <ul>
    <li>Viète, <em>In artem analyticem isagoge</em> (1591) — early systematic use of letters in algebra.</li>
    <li>HKDSE Compulsory Part — Number and Algebra (algebraic fractions, formulas and identities).</li>
  </ul>
</div>