openfreebooks-bookgenerator/content/math/directed-numbers-rates/_index.md

+++
title = "Directed numbers, rates & ratios"
description = "Operate with negative numbers and solve ratio, rate, and proportion problems in context."
template = "chapter.html"
weight = 2
[extra]
subject = "math"
chapter_id = "directed-numbers-rates"
strand = "Number & Algebra"
+++

<div class="book-prose">
  <p>
  You can already add, subtract, multiply, and divide whole numbers using <strong>written
  calculation</strong> (columns, carry, and borrow). On a number line, those values lie on or to the
  <strong>right</strong> of zero.
  Numbers can also lie to the <strong>left</strong> of zero — we call them <strong>negative</strong>.
  Temperatures below freezing, bank overdrafts, and positions below sea level are everyday examples
  of <strong>directed numbers</strong>.
  </p>

  <h2 id="number-line" class="book-prose__heading">The number line</h2>
  <p>
  On a horizontal number line, moving right means the value <em>increases</em>; moving left means it
  <em>decreases</em>. The distance from zero is the <strong>absolute value</strong>
  \(|a|\). For example, \(|-5| = 5\) and \(|3| = 3\).
  </p>

  <div class="book-callout book-callout--formula" role="note">
    <p class="book-callout__label">Comparing</p>
    <p class="book-formula">
      \(a &gt; b\) means \(a\) lies to the right of \(b\). Any negative number is less than any
      positive number.
    </p>
  </div>

  <h2 id="adding-subtracting" class="book-prose__heading">Adding and subtracting</h2>
  <p>
  Think in terms of movement on the line:
  </p>
  <ul>
    <li><strong>\(a + b\)</strong> — start at \(a\), move \(b\) units (right if \(b &gt; 0\), left if \(b &lt; 0\)).</li>
    <li><strong>\(a - b\)</strong> — start at \(a\), move \(-b\) units (equivalently, add the opposite of \(b\)).</li>
  </ul>
  <p>
  Subtracting a negative is adding a positive: \(5 - (-3) = 5 + 3 = 8\).
  </p>

  <h2 id="multiplying-signs" class="book-prose__heading">Multiplying and dividing</h2>
  <div class="book-table-wrap">
    <table class="book-table">
      <thead>
        <tr>
          <th scope="col">Signs</th>
          <th scope="col">Result</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td>positive × positive, negative × negative</td>
          <td>positive</td>
        </tr>
        <tr>
          <td>positive × negative (either order)</td>
          <td>negative</td>
        </tr>
      </tbody>
    </table>
  </div>

  <h2 id="ratios" class="book-prose__heading">Ratios</h2>
  <p>
  A <strong>ratio</strong> compares quantities of the same kind. The ratio \(2:3\) means “2 parts
  to 3 parts” — five equal parts in total. To share \(40\) in the ratio \(2:3\), one part is
  \(40 \div 5 = 8\), so the shares are \(16\) and \(24\).
  </p>

  <div class="book-callout book-callout--formula" role="note">
    <p class="book-callout__label">Sharing in ratio \(m:n\)</p>
    <p class="book-formula">
      \[ \text{one part} = \frac{\text{total}}{m+n}, \quad
      \text{first share} = \frac{m}{m+n}\times\text{total}, \quad
      \text{second share} = \frac{n}{m+n}\times\text{total} \]
    </p>
  </div>

  <h2 id="rates" class="book-prose__heading">Rates</h2>
  <p>
  A <strong>rate</strong> compares different units. <strong>Speed</strong> is distance per time:
  \[ \text{speed} = \frac{\text{distance}}{\text{time}}. \]
  A <strong>unit rate</strong> tells “how much per one” — for example \(60\) km in \(1.5\) h gives
  \(60 \div 1.5 = 40\) km/h.
  </p>

  <h2 id="try-it-yourself" class="book-prose__heading">Try it yourself</h2>
  <p>
  Use the number-line explorer to pick two directed numbers and see their sum on the line. Then try
  the ratio sharer in the applications section below.
  </p>
</div>

<div class="math-widget-mount" data-widget="number-line" data-pagefind-ignore></div>

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    For much of history, mathematicians treated only positive quantities as numbers. Negative
    solutions to equations were often rejected as meaningless. Practical needs — debts, losses,
    and temperatures below zero — still pushed people to work with “numbers less than nothing.”
  </p>

  <figure class="book-figure">
    <img
      class="book-figure__img"
      src="/chapters/math/directed-numbers-rates/directed-numbers-history.png"
      alt="Illustrated timeline: ancient rejection of negative roots, Indian and Chinese rules for signed quantities, European skepticism, and the development of ratios from barter to modern rates"
      width="960"
      height="640"
      loading="lazy"
      decoding="async"
    />
    <figcaption class="book-figure__caption">
      A sketch of how directed numbers and ratios were understood — from Greek reluctance to use
      negative roots, through Indian and Chinese models of fortune and debt, to modern rates such
      as speed and exchange.
    </figcaption>
  </figure>

  <p>
    In India around the 7th century, <strong>Brahmagupta</strong> gave rules for operating with
    positive and negative quantities (fortunes and debts). Chinese scholars used counting rods with
    two colours to represent opposite signs. European acceptance grew slowly; even in the 18th century
    some writers called negative roots “false.”
  </p>
  <p>
    Ratios and proportions appear in Babylonian and Egyptian problems (sharing grain, scaling
    recipes). The modern notation \(a:b\) and systematic treatment of <strong>unit rates</strong>
    became standard as commerce and science needed comparable measures — speed, price per item, and
    exchange rates between currencies.
  </p>

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

  <h3 id="why-minus-times-minus" class="book-prose__heading">Why \((-)\times(-) = (+)\)?</h3>
  <p>
    One useful pattern: multiplication should keep the rules of arithmetic consistent. For any
    real \(a\), you want \(a \times 0 = 0\). Then
    \[ a \times 0 = a \times (1 + (-1)) = a\cdot 1 + a\cdot(-1) = a + a\cdot(-1). \]
    So \(a\cdot(-1) = -a\). Now apply that twice:
    \[ (-a)\cdot(-1) = -((-a)) = a, \]
    which shows that multiplying two negatives gives a positive.
  </p>

  <h3 id="sharing-ratio" class="book-prose__heading">Sharing in the ratio \(m:n\)</h3>
  <p>
    Suppose a total \(T\) is split into two parts in the ratio \(m:n\). Think of \(m+n\) equal
    <em>parts</em>. Each part has size
    \[ \frac{T}{m+n}. \]
    The first quantity gets \(m\) of those parts:
    \[ \frac{m}{m+n}\,T, \]
    and the second gets \(\frac{n}{m+n}\,T\). The parts add to \(T\), as required.
  </p>

  <h3 id="unit-rate" class="book-prose__heading">Unit rate</h3>
  <p>
    If \(d\) units of quantity are produced in \(t\) units of time, the average rate is
    \(d/t\). “Per one hour” means divide by \(t\) when \(t \neq 0\):
    \[ \text{rate per unit time} = \frac{d}{t}. \]
    Re-arranging gives \(d = \text{rate} \times t\), the formula used for uniform motion.
  </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>
      Without a calculator: is \((-4) + (-7)\) positive or negative? Where do you land if you start
      at \(-4\) and move 7 units left?
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Simplify \(6 - (-2)\). Explain using “subtracting a negative is adding a positive.”
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Share \(72\) in the ratio \(5:4\). How many “parts” are there in total?
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      A car travels \(150\) km in \(2.5\) hours. What is the average speed? Which formula did you use?
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Currency A costs \(1.2\) units of Currency B per unit of A. You exchange \(50\) units of A.
      How many units of B do you receive?
    </p>
  </div>

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

  <h3 id="temperature-altitude" class="book-prose__heading">Temperature and altitude</h3>
  <p>
    A weather station records \(3\,^{\circ}\mathrm{C}\) at noon and the temperature drops
    \(8\,^{\circ}\mathrm{C}\) by midnight. The change is \(-8\), so the midnight temperature is
    \[ 3 + (-8) = -5\,^{\circ}\mathrm{C}. \]
    Altitude above sea level is positive; below sea level (e.g. Dead Sea shore) is negative
    elevation relative to sea level.
  </p>

  <h3 id="sharing-mixture" class="book-prose__heading">Sharing and mixtures</h3>
  <p>
    Paint is mixed in the ratio blue : white \(= 2:5\). For \(21\) litres total, there are
    \(7\) equal parts, each \(3\) litres, giving \(6\) L blue and \(15\) L white.
  </p>

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

  <h3 id="speed-exchange" class="book-prose__heading">Speed and exchange rates</h3>
  <p>
    Walking \(6\) km in \(1.5\) h gives speed \(\frac{6}{1.5} = 4\) km/h. An exchange rate of
    \(1\) USD \(=\) \(7.8\) HKD means \(100\) USD converts to \(780\) HKD by multiplying by the
    unit rate.
  </p>

  <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">Which is larger, \(-3\) or \(-7\)?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(-3\) is larger (lies to the right on the number line).</p>
      <p><strong>Explanation:</strong> For negative numbers, the one closer to zero is greater.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Evaluate \((-2) + 9\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(7\).</p>
      <p><strong>Steps:</strong> Start at \(-2\), move 9 units right.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">\((-4) \times 6 = \) ______</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(-24\).</p>
      <p><strong>Explanation:</strong> Different signs → negative product.</p>
    </div>
  </details>

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

  <details class="book-question">
    <summary class="book-question__prompt">Evaluate \(5 - (-3) \times 2\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(11\).</p>
      <p><strong>Steps:</strong> \((-3)\times 2 = -6\); then \(5 - (-6) = 5 + 6 = 11\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Divide \(84\) in the ratio \(2:5\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(24\) and \(60\).</p>
      <p><strong>Steps:</strong> \(7\) parts; each part \(84 \div 7 = 12\); shares \(24\) and \(60\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">A cyclist goes \(45\) km in \(3\) h. What is the average speed?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(15\) km/h.</p>
      <p><strong>Calculation:</strong> \(45 \div 3 = 15\).</p>
    </div>
  </details>

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

  <details class="book-question">
    <summary class="book-question__prompt">The temperature was \(-2\,^{\circ}\mathrm{C}\) and fell by \(9\,^{\circ}\mathrm{C}\). Find the new temperature.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(-11\,^{\circ}\mathrm{C}\).</p>
      <p><strong>Steps:</strong> \(-2 + (-9) = -11\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Put in order from smallest to largest: \(-1\), \(0\), \(-\frac{1}{2}\), \(-2\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(-2 &lt; -1 &lt; -\frac{1}{2} &lt; 0\).</p>
      <p><strong>Explanation:</strong> On the number line, further left means smaller.</p>
    </div>
  </details>

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

  <details class="book-question">
    <summary class="book-question__prompt">\(A:B = 3:4\) and \(B:C = 2:5\). Find \(A:B:C\) in simplest form.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(A:B:C = 3:4:10\) after matching \(B\) (multiply first ratio by 2: \(6:8\), second by 4: \(8:20\) → \(6:8:20 = 3:4:10\)).</p>
      <p><strong>Steps:</strong> Align \(B\) to LCM 8: \(A:B = 6:8\), \(B:C = 8:20\), so \(A:B:C = 6:8:20 = 3:4:10\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Which situation matches a <em>negative</em> directed number?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> A bank balance of \(500\) in debt (owe \(500\)) — often modelled as \(-500\).</p>
      <p><strong>Explanation:</strong> Debt below zero balance is a standard directed-number model.</p>
    </div>
  </details>

  <h2 id="references" class="book-prose__heading">References</h2>
  <ul>
    <li>Brahmagupta, <em>Brāhmasphuṭasiddhānta</em> (628 CE) — early rules for signed quantities.</li>
    <li>HKDSE Compulsory Part curriculum — Number and Algebra strand (directed numbers, ratios, rates).</li>
  </ul>
</div>