14 KiB
+++ 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" +++
You can already add, subtract, multiply, and divide whole numbers using written calculation (columns, carry, and borrow). On a number line, those values lie on or to the right of zero. Numbers can also lie to the left of zero — we call them 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 \(24\).
Sharing in ratio \(m:n\)
\[ \text{one part} = \frac{\text{total}}{m+n}, \quad \text{first share} = \frac{m}{m+n}\times\text{total}, \quad \text{second share} = \frac{n}{m+n}\times\text{total} \]
Rates
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.”
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.
Evaluate \((-2) + 9\).
Answer: \(7\).
Steps: Start at \(-2\), move 9 units right.
\((-4) \times 6 = \) ______
Answer: \(-24\).
Explanation: Different signs → negative product.
Intermediate
Evaluate \(5 - (-3) \times 2\).
Answer: \(11\).
Steps: \((-3)\times 2 = -6\); then \(5 - (-6) = 5 + 6 = 11\).
Divide \(84\) in the ratio \(2:5\).
Answer: \(24\) and \(60\).
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).