+ Numbers on a line can lie to the right of zero (positive) or to the
+ left (negative). Temperatures below freezing, bank overdrafts, and positions
+ below sea level are everyday examples of directed numbers.
+
+
+
The number line
+
+ On a horizontal number line, moving right means the value increases; moving left means it
+ decreases. The distance from zero is the absolute value
+ \(|a|\). For example, \(|-5| = 5\) and \(|3| = 3\).
+
+
+
+
Comparing
+
+ \(a > b\) means \(a\) lies to the right of \(b\). Any negative number is less than any
+ positive number.
+
+
+
+
Adding and subtracting
+
+ Think in terms of movement on the line:
+
+
+
\(a + b\) — start at \(a\), move \(b\) units (right if \(b > 0\), left if \(b < 0\)).
+
\(a - b\) — start at \(a\), move \(-b\) units (equivalently, add the opposite of \(b\)).
+
+
+ Subtracting a negative is adding a positive: \(5 - (-3) = 5 + 3 = 8\).
+
+
+
Multiplying and dividing
+
+
+
+
+
Signs
+
Result
+
+
+
+
+
positive × positive, negative × negative
+
positive
+
+
+
positive × negative (either order)
+
negative
+
+
+
+
+
+
Ratios
+
+ A ratio 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 \(24\).
+
+ A rate compares different units. Speed is distance per time:
+ \[ \text{speed} = \frac{\text{distance}}{\text{time}}. \]
+ A unit rate tells “how much per one” — for example \(60\) km in \(1.5\) h gives
+ \(60 \div 1.5 = 40\) km/h.
+
+
+
Try it yourself
+
+ 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.
+
+
+
+
+
+
+
History
+
+ 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.”
+
+
+
+
+
+ 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.
+
+
+
+
+ In India around the 7th century, Brahmagupta 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.”
+
+
+ Ratios and proportions appear in Babylonian and Egyptian problems (sharing grain, scaling
+ recipes). The modern notation \(a:b\) and systematic treatment of unit rates
+ became standard as commerce and science needed comparable measures — speed, price per item, and
+ exchange rates between currencies.
+
+
+
Derivation
+
+
Why \((-)\times(-) = (+)\)?
+
+ 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.
+
+
+
Sharing in the ratio \(m:n\)
+
+ Suppose a total \(T\) is split into two parts in the ratio \(m:n\). Think of \(m+n\) equal
+ parts. 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.
+
+
+
Unit rate
+
+ 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.
+
+
+
Checkpoints
+
+ Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
+
+
+
+
Checkpoint
+
+ Without a calculator: is \((-4) + (-7)\) positive or negative? Where do you land if you start
+ at \(-4\) and move 7 units left?
+
+
+
+
+
Checkpoint
+
+ Simplify \(6 - (-2)\). Explain using “subtracting a negative is adding a positive.”
+
+
+
+
+
Checkpoint
+
+ Share \(72\) in the ratio \(5:4\). How many “parts” are there in total?
+
+
+
+
+
Checkpoint
+
+ A car travels \(150\) km in \(2.5\) hours. What is the average speed? Which formula did you use?
+
+
+
+
+
Checkpoint
+
+ 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?
+
+
+
+
Applications
+
+
Temperature and altitude
+
+ 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.
+
+
+
Sharing and mixtures
+
+ 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.
+
+
+
+
+
Speed and exchange rates
+
+ 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.
+
+
+
Question bank
+
+ Work each question, then open the solution to check your reasoning. Difficulty increases
+ through the sets.
+
+
+
Easy
+
+
+ Which is larger, \(-3\) or \(-7\)?
+
+
Answer: \(-3\) is larger (lies to the right on the number line).
+
Explanation: For negative numbers, the one closer to zero is greater.
Steps: \(7\) parts; each part \(84 \div 7 = 12\); shares \(24\) and \(60\).
+
+
+
+
+ A cyclist goes \(45\) km in \(3\) h. What is the average speed?
+
+
Answer: \(15\) km/h.
+
Calculation: \(45 \div 3 = 15\).
+
+
+
+
Difficult
+
+
+ The temperature was \(-2\,^{\circ}\mathrm{C}\) and fell by \(9\,^{\circ}\mathrm{C}\). Find the new temperature.
+
+
Answer: \(-11\,^{\circ}\mathrm{C}\).
+
Steps: \(-2 + (-9) = -11\).
+
+
+
+
+ Put in order from smallest to largest: \(-1\), \(0\), \(-\frac{1}{2}\), \(-2\).
+
+
Answer: \(-2 < -1 < -\frac{1}{2} < 0\).
+
Explanation: On the number line, further left means smaller.
+
+
+
+
Hardcore
+
+
+ \(A:B = 3:4\) and \(B:C = 2:5\). Find \(A:B:C\) in simplest form.
+
+
Answer: \(A:B:C = 3:4:10\) after matching \(B\) (multiply first ratio by 2: \(6:8\), second by 4: \(8:20\) → \(6:8:20 = 3:4:10\)).
+
Steps: Align \(B\) to LCM 8: \(A:B = 6:8\), \(B:C = 8:20\), so \(A:B:C = 6:8:20 = 3:4:10\).
+
+
+
+
+ Which situation matches a negative directed number?
+
+
Answer: A bank balance of \(500\) in debt (owe \(500\)) — often modelled as \(-500\).
+
Explanation: Debt below zero balance is a standard directed-number model.
+
+
+
+
References
+
+
Brahmagupta, Brāhmasphuṭasiddhānta (628 CE) — early rules for signed quantities.
+
HKDSE Compulsory Part curriculum — Number and Algebra strand (directed numbers, ratios, rates).
+
+
diff --git a/content/math/directed-numbers-rates/assets/directed-numbers-history.png b/content/math/directed-numbers-rates/assets/directed-numbers-history.png
new file mode 100644
index 0000000..94ee61b
Binary files /dev/null and b/content/math/directed-numbers-rates/assets/directed-numbers-history.png differ
diff --git a/content/math/directed-numbers-rates/core.html b/content/math/directed-numbers-rates/core.html
new file mode 100644
index 0000000..a942c92
--- /dev/null
+++ b/content/math/directed-numbers-rates/core.html
@@ -0,0 +1,88 @@
+
+
+ Numbers on a line can lie to the right of zero (positive) or to the
+ left (negative). Temperatures below freezing, bank overdrafts, and positions
+ below sea level are everyday examples of directed numbers.
+
+
+
The number line
+
+ On a horizontal number line, moving right means the value increases; moving left means it
+ decreases. The distance from zero is the absolute value
+ \(|a|\). For example, \(|-5| = 5\) and \(|3| = 3\).
+
+
+
+
Comparing
+
+ \(a > b\) means \(a\) lies to the right of \(b\). Any negative number is less than any
+ positive number.
+
+
+
+
Adding and subtracting
+
+ Think in terms of movement on the line:
+
+
+
\(a + b\) — start at \(a\), move \(b\) units (right if \(b > 0\), left if \(b < 0\)).
+
\(a - b\) — start at \(a\), move \(-b\) units (equivalently, add the opposite of \(b\)).
+
+
+ Subtracting a negative is adding a positive: \(5 - (-3) = 5 + 3 = 8\).
+
+
+
Multiplying and dividing
+
+
+
+
+
Signs
+
Result
+
+
+
+
+
positive × positive, negative × negative
+
positive
+
+
+
positive × negative (either order)
+
negative
+
+
+
+
+
+
Ratios
+
+ A ratio 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 \(24\).
+
+ A rate compares different units. Speed is distance per time:
+ \[ \text{speed} = \frac{\text{distance}}{\text{time}}. \]
+ A unit rate tells “how much per one” — for example \(60\) km in \(1.5\) h gives
+ \(60 \div 1.5 = 40\) km/h.
+
+
+
Try it yourself
+
+ 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.
+
+ 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.”
+
+
+
+
+
+ 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.
+
+
+
+
+ In India around the 7th century, Brahmagupta 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.”
+
+
+ Ratios and proportions appear in Babylonian and Egyptian problems (sharing grain, scaling
+ recipes). The modern notation \(a:b\) and systematic treatment of unit rates
+ became standard as commerce and science needed comparable measures — speed, price per item, and
+ exchange rates between currencies.
+
+
+
Derivation
+
+
Why \((-)\times(-) = (+)\)?
+
+ 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.
+
+
+
Sharing in the ratio \(m:n\)
+
+ Suppose a total \(T\) is split into two parts in the ratio \(m:n\). Think of \(m+n\) equal
+ parts. 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.
+
+
+
Unit rate
+
+ 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.
+
+
+
Checkpoints
+
+ Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
+
+
+
+
Checkpoint
+
+ Without a calculator: is \((-4) + (-7)\) positive or negative? Where do you land if you start
+ at \(-4\) and move 7 units left?
+
+
+
+
+
Checkpoint
+
+ Simplify \(6 - (-2)\). Explain using “subtracting a negative is adding a positive.”
+
+
+
+
+
Checkpoint
+
+ Share \(72\) in the ratio \(5:4\). How many “parts” are there in total?
+
+
+
+
+
Checkpoint
+
+ A car travels \(150\) km in \(2.5\) hours. What is the average speed? Which formula did you use?
+
+
+
+
+
Checkpoint
+
+ 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?
+
+
+
+
Applications
+
+
Temperature and altitude
+
+ 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.
+
+
+
Sharing and mixtures
+
+ 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.
+
+
+
+
+
Speed and exchange rates
+
+ 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.
+
+
+
Question bank
+
+ Work each question, then open the solution to check your reasoning. Difficulty increases
+ through the sets.
+
+
+
Easy
+
+
+ Which is larger, \(-3\) or \(-7\)?
+
+
Answer: \(-3\) is larger (lies to the right on the number line).
+
Explanation: For negative numbers, the one closer to zero is greater.
+ Split {formatNum(total())} in the ratio{" "}
+
+ {partA()}:{partB()}
+
+ . Each of the {sumParts()} equal parts has size{" "}
+ {formatNum(total() / sumParts())}.
+
+