Ratios and their simplest form

To talk about this properly we need some notation. The ratio of width to height of image A is written 60 : 40 (read '60 to 40'). The numbers 60 and 40 are called the terms of the ratio. A ratio compares two quantities of the same kind, in the same unit, and the order matters: 60 : 40 is not 40 : 60.

So image C is 30 : 20 and image D is 90 : 60.

What does a : b mean? It says that for every a units of the first quantity there are b units of the second. In image A: for every 60 mm of width there are 40 mm of height. (A ratio of 3 : 2 doesn't mean exactly 3 and 2; it means 3 × n and 2 × n, for the same n.)

Images A, C and D are proportional because their terms change by the same factor. Multiply both terms of 60 : 40 by ½ and you get 30 : 20, which is image C. (What factor takes 60 : 40 to 90 : 60? Work it out.)

Hunting for the factor each time is slow. A more systematic way is to reduce every ratio to its simplest form and compare those. To reduce a ratio, divide both terms by their HCF:

A and D both come to 3 : 2, so they're proportional, with no searching needed.

And the odd ones: B comes to 2 : 1 and E to 1 : 1. Neither is 3 : 2, which is exactly why they look wrong.

When two ratios have the same simplest form, we say they are in proportion (or proportional), and write them with a double colon: a : b :: c : d. So 60 : 40 :: 30 : 20, and 60 : 40 :: 90 : 60.