Ratios

Ratios are usually used when you want to specify the amount of substance A relative to substance B. For example, whether there should be twice as much of A, half as much and so on. Of course, a ratio doesn’t have to be only between two substances; more of them can be involved.

Relative ratios

Ratios are always relative; they are not an absolute quantity. If a recipe says “add half a kilo of sugar and a quarter of a kilo of meat”, that is something different from “add a bit of meat and twice as much sugar”. In the first case we have no choice but to buy half a kilo of sugar, whereas in the second case it is enough to have twice as much sugar as meat. Either way, we’ll cook up a revolting sludge.

Ratios are usually written with a colon. For example 1 : 2 or 5 : 4. What does this notation mean? The first one means that the first substance should be present in half the amount of the second substance. The second one says that substance A should be in the bowl 5 times, while the same amount of substance B only four times.

Suppose we want to mix an alcoholic drink called “Bound-to-Be-Sick”, which has a ratio of vodka to rum of 5 : 2. That means we pour 5 decilitres of vodka and 2 decilitres of rum into the glass. Since it is a ratio, we also meet the requirement if we pour 2.5 decilitres of vodka and 1 decilitre of rum. In short, it is enough that there is 2.5 times more vodka than rum. We got to the number 2.5 by dividing 5 / 2. So if the ratio of substances A to B is 5 : 1, it means there should be five times more of A (5 / 1). For a ratio of 7 : 3 it would be about 2.3 times more (7 / 3). For a ratio of 1 : 1 there would be equal amounts of both, and finally, for a ratio of 2 : 3 there would be 1.5 times more of B (3 / 2), or A would be only about 0.67 times the amount of B (2 / 3).

Worked examples

We have £100 in our wallet, all in small change. We also have two workers in the garden who dug a grave all night for the neighbour who keeps peering through your window into your living room. Because Oliver worked faster than George, he gets more money, in the ratio 7 : 3. So how much money will they get for their morbid work, assuming we aren’t stingy and give them the whole £100?

It’s easy to work out. We have the ratio 7 : 3. So Oliver gets seven parts, whereas George gets only three parts. We add up the parts of the ratio, 7 + 3 = 10, and divide the £100 we want to give them by this number: 100 / 10 = 10. That tells us how many pounds one part is worth. Oliver is to get seven parts: 10 · 7 = 70 and George three parts: 3 · 10 = 30. And that’s solved – Oliver gets £70 and George £30.

This is the simplest way to solve problems with ratios. A ratio really just expresses the parts of some whole. If the ratio is 1 : 2, it means that substance A makes up 1/3 of the whole and substance B 2/3. They are just fractions written differently. We find the denominator by adding up all the parts of the ratio, and for the numerators we just substitute the given parts. If we add up all the parts, we must get the whole, that is, one. 1/3 + 2/3 = 3/3 = 1. We can also express it as percentages: 33.3% and 66.7% (approximately). Another simple example:

Lily and Jack have been friends since childhood, but unfortunately each of them went to a different university, which separated them for a long time. Lily studied history and Jack studied maths and physics. Years later they met in their favourite pub, “The Soggy Kelpie”, and got talking. Lily told Jack that since they last saw each other she had had sixteen partners (Lily was a very beautiful girl and a bit flighty). But Jack was a sly one and didn’t tell Lily the exact number of his girlfriends, only that the ratio of her partners to his partners was 8 : 3. A typical maths-and-physics student… But Lily didn’t just have perfect 89-61-88 measurements; she was clever too, so she started to work it out: Right, my sixteen partners correspond to the eight in the ratio Jack told me. So if I want to find out how many blokes correspond to one part of the ratio, I have to divide 16 / 8 = 2. So one part corresponds to two partners. Jack has three parts, so the number of his partners is two times three, which is six. Jack agreed and ordered another drink. After that evening both their scores went up by exactly one, but that’s another story.

Amelia worked as a barmaid in a place called “Sk8 is Gr8”. One evening a slightly tipsy bloke came up to the bar and wanted a special mixed drink. He gave the barmaid his instructions: “Love, I want it in a half-litre glass. I want banana juice, cola and coffee in it. All in the ratio 5 : 2 : 3.” Amelia was puzzled, but the customer is always right. She got to work. First she added up all the parts of the ratio: 5 + 2 + 3 = 10. A half-litre glass holds 500 ml of liquid. So she had to find out how many millilitres correspond to one part of the ratio. She did that easily: 500 / 10 = 50. One part corresponds to 50 millilitres. So she poured 5 · 50 = 250 ml of banana juice, 2 · 50 = 100 ml of cola and 3 · 50 = 150 ml of coffee into the glass. When the bloke showed up a second time, he asked for the keys to the loo.

Map scale

A classic use of ratios is also the map scale.