Ratio and Proportion: O-Level / IGCSE Maths (0580)

A ratio compares two or more quantities of the same kind, written in the form $a : b$ (read ”$a$ to $b$”). This topic begins with ratio notation and the idea that a ratio can be simplified by dividing every part by a common factor, just like simplifying a fraction. Equivalent ratios such as $2 : 3$ and $6 : 9$ describe the same comparison, and order matters, so $a : b$ is generally different from $b : a$.

A central skill is dividing a quantity in a given ratio. To share an amount in the ratio $a : b$, you find the total number of parts, $a + b$, work out the value of one part by dividing the quantity by that total, then multiply to get each share. Closely linked to this is the ratio–fraction relationship: in the ratio $a : b$, the first quantity is $\frac{a}{a+b}$ of the whole, which lets you move freely between ratio language and fraction or percentage language. The topic also covers proportion. Two quantities are in direct proportion when one is a constant multiple of the other, so $y = kx$ and the ratio $\frac{y}{x}$ stays fixed; they are in inverse proportion when their product is constant, $xy = k$, so as one increases the other decreases. The unitary method (finding the value of one unit first) is a reliable way to solve both kinds of problem.

These ideas appear throughout the course in scale drawings, currency conversion, best-buy comparisons, mixing and recipe questions, and speed–distance–time work. The worked examples below are original, written from the syllabus objective to show ratio notation, sharing in a ratio, the ratio–fraction link, and direct and inverse proportion in clear, exam-style steps with full worked solutions.