Ratios & scale

The juice bottle says: 1 part concentrate to 3 parts water. Whether you mix one glass or a whole jug, the juice tastes the same as long as you keep that ratio. That is exactly what this is about: ratios don't describe how much of something you have, but how the parts relate to each other.

What a ratio tells you

A ratio like 1 : 3 (say: one to three) means: for every 1 part of one kind, there are 3 parts of the other. For the juice that means: 1 cup of concentrate, 3 cups of water, 4 cups of drink in total. If you double both sides, the taste stays the same: 2 : 6 is the same ratio as 1 : 3. So you may simplify and expand a ratio just like a fraction, as long as you multiply both sides by the same number or divide both by the same number. And just as with a fraction you can ask what share of the whole one part is: with 1 : 3 it is 1 cup out of 4, so the concentrate makes up one quarter of the mixture.

Comparing ratios

Which juice is stronger: 1 : 3 or 2 : 5? Turn both into fractions of the whole. With 1 : 3, the concentrate is 1 of 4 parts, so one quarter. With 2 : 5, it is 2 of 7 parts, so two sevenths. To compare them, put both over the same denominator: one quarter equals 7 of 28 parts, two sevenths equal 8 of 28 parts. So two sevenths is a little more, which means the 2 : 5 mixture tastes stronger. If you can compare fractions, you can compare ratios.

Scale: the world in pocket size

Maps work with ratios too. The scale 1 : 25,000 means: 1 cm on the map corresponds to 25,000 cm in reality, so 250 m. If you measure 4 cm on the map, that is 4 times 250 m = 1000 m outside, one kilometre. The same idea powers scaling recipes: if a cake for 4 people needs 200 g of flour, then for 8 people you need double, because all ingredients grow in the same ratio.