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236 lines
10 KiB
HTML
236 lines
10 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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For much of history, mathematicians treated only positive quantities as numbers. Negative
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solutions to equations were often rejected as meaningless. Practical needs — debts, losses,
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and temperatures below zero — still pushed people to work with “numbers less than nothing.”
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</p>
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<figure class="book-figure">
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<img
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class="book-figure__img"
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src="/chapters/math/directed-numbers-rates/directed-numbers-history.png"
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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"
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width="960"
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height="640"
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loading="lazy"
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decoding="async"
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/>
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<figcaption class="book-figure__caption">
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A sketch of how directed numbers and ratios were understood — from Greek reluctance to use
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negative roots, through Indian and Chinese models of fortune and debt, to modern rates such
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as speed and exchange.
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</figcaption>
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</figure>
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<p>
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In India around the 7th century, <strong>Brahmagupta</strong> gave rules for operating with
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positive and negative quantities (fortunes and debts). Chinese scholars used counting rods with
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two colours to represent opposite signs. European acceptance grew slowly; even in the 18th century
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some writers called negative roots “false.”
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</p>
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<p>
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Ratios and proportions appear in Babylonian and Egyptian problems (sharing grain, scaling
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recipes). The modern notation \(a:b\) and systematic treatment of <strong>unit rates</strong>
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became standard as commerce and science needed comparable measures — speed, price per item, and
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exchange rates between currencies.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation</h2>
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<h3 id="why-minus-times-minus" class="book-prose__heading">Why \((-)\times(-) = (+)\)?</h3>
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<p>
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One useful pattern: multiplication should keep the rules of arithmetic consistent. For any
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real \(a\), you want \(a \times 0 = 0\). Then
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\[ a \times 0 = a \times (1 + (-1)) = a\cdot 1 + a\cdot(-1) = a + a\cdot(-1). \]
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So \(a\cdot(-1) = -a\). Now apply that twice:
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\[ (-a)\cdot(-1) = -((-a)) = a, \]
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which shows that multiplying two negatives gives a positive.
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</p>
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<h3 id="sharing-ratio" class="book-prose__heading">Sharing in the ratio \(m:n\)</h3>
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<p>
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Suppose a total \(T\) is split into two parts in the ratio \(m:n\). Think of \(m+n\) equal
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<em>parts</em>. Each part has size
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\[ \frac{T}{m+n}. \]
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The first quantity gets \(m\) of those parts:
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\[ \frac{m}{m+n}\,T, \]
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and the second gets \(\frac{n}{m+n}\,T\). The parts add to \(T\), as required.
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</p>
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<h3 id="unit-rate" class="book-prose__heading">Unit rate</h3>
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<p>
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If \(d\) units of quantity are produced in \(t\) units of time, the average rate is
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\(d/t\). “Per one hour” means divide by \(t\) when \(t \neq 0\):
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\[ \text{rate per unit time} = \frac{d}{t}. \]
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Re-arranging gives \(d = \text{rate} \times t\), the formula used for uniform motion.
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</p>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<p>
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Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
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</p>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Without a calculator: is \((-4) + (-7)\) positive or negative? Where do you land if you start
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at \(-4\) and move 7 units left?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Simplify \(6 - (-2)\). Explain using “subtracting a negative is adding a positive.”
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Share \(72\) in the ratio \(5:4\). How many “parts” are there in total?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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A car travels \(150\) km in \(2.5\) hours. What is the average speed? Which formula did you use?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Currency A costs \(1.2\) units of Currency B per unit of A. You exchange \(50\) units of A.
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How many units of B do you receive?
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</p>
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</div>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<h3 id="temperature-altitude" class="book-prose__heading">Temperature and altitude</h3>
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<p>
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A weather station records \(3\,^{\circ}\mathrm{C}\) at noon and the temperature drops
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\(8\,^{\circ}\mathrm{C}\) by midnight. The change is \(-8\), so the midnight temperature is
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\[ 3 + (-8) = -5\,^{\circ}\mathrm{C}. \]
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Altitude above sea level is positive; below sea level (e.g. Dead Sea shore) is negative
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elevation relative to sea level.
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</p>
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<h3 id="sharing-mixture" class="book-prose__heading">Sharing and mixtures</h3>
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<p>
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Paint is mixed in the ratio blue : white \(= 2:5\). For \(21\) litres total, there are
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\(7\) equal parts, each \(3\) litres, giving \(6\) L blue and \(15\) L white.
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</p>
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<div class="math-widget-mount" data-widget="ratio-demo" data-pagefind-ignore></div>
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<h3 id="speed-exchange" class="book-prose__heading">Speed and exchange rates</h3>
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<p>
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Walking \(6\) km in \(1.5\) h gives speed \(\frac{6}{1.5} = 4\) km/h. An exchange rate of
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\(1\) USD \(=\) \(7.8\) HKD means \(100\) USD converts to \(780\) HKD by multiplying by the
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unit rate.
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</p>
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<h2 id="question-bank" class="book-prose__heading">Question bank</h2>
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<p>
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Work each question, then open the solution to check your reasoning. Difficulty increases
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through the sets.
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</p>
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<h3 id="questions-easy" class="book-prose__heading">Easy</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Which is larger, \(-3\) or \(-7\)?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(-3\) is larger (lies to the right on the number line).</p>
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<p><strong>Explanation:</strong> For negative numbers, the one closer to zero is greater.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Evaluate \((-2) + 9\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(7\).</p>
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<p><strong>Steps:</strong> Start at \(-2\), move 9 units right.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">\((-4) \times 6 = \) ______</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(-24\).</p>
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<p><strong>Explanation:</strong> Different signs → negative product.</p>
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</div>
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</details>
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<h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Evaluate \(5 - (-3) \times 2\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(11\).</p>
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<p><strong>Steps:</strong> \((-3)\times 2 = -6\); then \(5 - (-6) = 5 + 6 = 11\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Divide \(84\) in the ratio \(2:5\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(24\) and \(60\).</p>
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<p><strong>Steps:</strong> \(7\) parts; each part \(84 \div 7 = 12\); shares \(24\) and \(60\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A cyclist goes \(45\) km in \(3\) h. What is the average speed?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(15\) km/h.</p>
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<p><strong>Calculation:</strong> \(45 \div 3 = 15\).</p>
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</div>
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</details>
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<h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>
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<details class="book-question">
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<summary class="book-question__prompt">The temperature was \(-2\,^{\circ}\mathrm{C}\) and fell by \(9\,^{\circ}\mathrm{C}\). Find the new temperature.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(-11\,^{\circ}\mathrm{C}\).</p>
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<p><strong>Steps:</strong> \(-2 + (-9) = -11\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Put in order from smallest to largest: \(-1\), \(0\), \(-\frac{1}{2}\), \(-2\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(-2 < -1 < -\frac{1}{2} < 0\).</p>
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<p><strong>Explanation:</strong> On the number line, further left means smaller.</p>
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</div>
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</details>
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<h3 id="questions-hardcore" class="book-prose__heading">Hardcore</h3>
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<details class="book-question">
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<summary class="book-question__prompt">\(A:B = 3:4\) and \(B:C = 2:5\). Find \(A:B:C\) in simplest form.</summary>
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<div class="book-question__solution">
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<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>
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<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>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Which situation matches a <em>negative</em> directed number?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> A bank balance of \(500\) in debt (owe \(500\)) — often modelled as \(-500\).</p>
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<p><strong>Explanation:</strong> Debt below zero balance is a standard directed-number model.</p>
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</div>
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</details>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li>Brahmagupta, <em>Brāhmasphuṭasiddhānta</em> (628 CE) — early rules for signed quantities.</li>
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<li>HKDSE Compulsory Part curriculum — Number and Algebra strand (directed numbers, ratios, rates).</li>
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</ul>
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</div>
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