Concrete Definition for a Ratio

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Hi there. I had a question about ratios. I couldn't seem to find a rigorous definition on them. I understand the concept of fractions being numbers of the form a/b where a is divided by b (or a multiplied by 1/b) and this represents a number for ex. 2/3. Ratios on the other hand compare relative quantities so 2/3 represents the relationship between 2 and 3 where 2/3 is the number multiplied by 3 to get 2. But I don't understand how 2:3 plays a role in this. I don't know if this makes sense but saying 2 dogs for every 3 cats doesn't fit with this definition that I'm comfortable with. I see 2/3 as a number and 2 dogs : 3 cats seems more like a statement. However, I do see that if I have 48 cats and 32 dogs then the idea that there are 2 dogs to 3 cats fits because there are 16 such groups. I'm just having trouble connecting all of these ideas in my head and the definitions for what a ratio or a fraction are don't seem concrete. I like the definition of a fraction being a number in the form a divided by b as this seems very unambiguous. Thank you again!