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.
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.
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.
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.
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?
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.
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.
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.
Work each question, then open the solution to check your reasoning. Difficulty increases through the sets.
Answer: \(-3\) is larger (lies to the right on the number line).
Explanation: For negative numbers, the one closer to zero is greater.
Answer: \(7\).
Steps: Start at \(-2\), move 9 units right.
Answer: \(-24\).
Explanation: Different signs → negative product.
Answer: \(11\).
Steps: \((-3)\times 2 = -6\); then \(5 - (-6) = 5 + 6 = 11\).
Answer: \(24\) and \(60\).
Steps: \(7\) parts; each part \(84 \div 7 = 12\); shares \(24\) and \(60\).
Answer: \(15\) km/h.
Calculation: \(45 \div 3 = 15\).
Answer: \(-11\,^{\circ}\mathrm{C}\).
Steps: \(-2 + (-9) = -11\).
Answer: \(-2 < -1 < -\frac{1}{2} < 0\).
Explanation: On the number line, further left means smaller.
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 \(A:B:C = 6:8:20 = 3:4:10\).
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.