2.1 Definition

Figure: A ratio splits a whole into equal-sized parts.

A ratio is a comparison of two quantities of the same kind (same units) by division. The ratio of a to b is written as a : b or a/b, where a is called the antecedent (first term) and b is called the consequent (second term).

Key properties: - A ratio has no unit — it is a pure number. - Both quantities must be expressed in the same unit before comparing (e.g., convert hours to minutes, kg to grams). - A ratio remains unchanged if both terms are multiplied or divided by the same non-zero number: a : b = ka : kb (k ≠ 0). - A ratio is normally expressed in its lowest (simplest) form, i.e., HCF of the two terms is 1. - Order matters: a : b is not the same as b : a unless a = b. - A ratio compares only two quantities at a time in its basic form; comparisons among three or more quantities together are written as a continued ratio (a : b : c), covered in Section 2.5. - If a quantity is split in a ratio, the actual values depend on a common “multiplying factor.” For a ratio a : b, the real quantities can always be written as ax and bx for some positive number x — this is the single most useful algebraic trick in this entire chapter (see Shortcut 9). - Ratios can involve fractions or decimals in their raw form (e.g., 2/3 : 5/6, or 1.2 : 0.8), but they must always be converted to whole-number form before being called “simplest.”