Understanding Ratios – Comparing Two Quantities

A ratio compares two (or more) quantities of the same kind, showing how much of one there is compared to another. Written with a colon, like 3:4, a ratio says “for every 3 of the first thing, there are 4 of the second.” Ratios are everywhere: recipes, paint mixes, maps, and exchange rates are all built on this one simple idea – and having just spent this whole section mastering fractions and percentages, you already have most of the tools needed to understand them.

A rigorous theory of ratio and proportion was developed by the ancient Greek mathematician Eudoxus of Cnidus in the 4th century BCE, preserved for us today in Book V of Euclid's Elements. Eudoxus's work solved a genuine crisis in Greek mathematics: once the Pythagoreans discovered that some quantities (like the diagonal of a square, involving √2) could not be expressed as a ratio of whole numbers, mathematicians needed a more general theory of ratio that worked for these “incommensurable” quantities too. Eudoxus's elegant solution stood as the mathematical standard for comparing quantities for well over two thousand years.

What Is a Ratio?

Real-Life Application

• Recipes: flour to sugar in a specific ratio for the right texture.
• Mixing: paint colours, concrete, or squash-to-water ratios.
• Sports teams: the ratio of wins to losses.

Key Takeaways

Practice: Understanding Ratios