Some Notes on Ratios and Sets

When given a ratio, it’s important of course to look at if the ratio is the ratio of a subset to a total larger set, or the ratio of a subset to another subset. For example, given the ratio 1/4, this could be the ratio of cats to total animals, or a ratio of cats to dogs. If a ratio of cats to total animals, that means the ratio will always be 1 cat per every 4 animals. So if you had 20 animals, you’d have 5 cats.

On the other hand, if 1/4 were a ratio of cats to dogs, that’s quite different from the first example. Whereas 4 was the total of animals in the first example, here the total amount of animals would be 5; 1 cat plus 4 dogs So the ratio of cats to the total amount of animals here would be a/(a+b), where a is cats and b is dogs, so it would be 1/5. The ratio of dogs to the total amount of animals here would be b/(a+b), in other words 4/5.

One thing I never thought about before is if the ratio of cats to dogs is 1/4, you can get that by putting the ratio of cats to total over the ratio of dogs to total, so (1/5)/(4/5)=1/4. And all of the subsets if they don’t overlap add up to 1, like 1/5 + 4/5.

Internal tag: math

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