Why is division used to represent ratios like 1 dog to 2 cats?

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Mr Davis 97
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Why is division used for ratios? For example, if we have 1 dog for every 2 cats, then why can this relation be modeled by the division? Since the ratio can essentially have the value of 1/2 or 2, then what is the significance of these two distinct numbers in relation to the ratio? How does the value of the fraction 1/2, which is 0.50, relate to the ratio 1 dog to 2 cats? If we said we had a 0.50 ratio of dogs to cats, what does that mean?
 
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There are different reasons. First of all the ratio is a much easier number to understand, depending if it is greater or smaller than 1 (and how much grater/smaller it is) you can have an immediate idea of what's going on. If you are using small numbers (as in your example) it is clear that the ratio doesn't really add any information. However if the numbers you are dealing with are high it is much more complicated to evaluate the relationship between the two quantities just "by eye".

Moreover, the ratio gives you the percentage. For example, as you said, if I know that the ratio dogs/cats is 0.37 then this means that the number of dogs is the 37% of the number of cats.
 
Mr Davis 97 said:
Why is division used for ratios?
Because it works.

Mr Davis 97 said:
For example, if we have 1 dog for every 2 cats, then why can this relation be modeled by the division?
Because 1 dog for every 2 cats is the same as ## \frac12 ## a dog for every 1 cat.

Mr Davis 97 said:
Since the ratio can essentially have the value of 1/2 or 2, then what is the significance of these two distinct numbers in relation to the ratio?
The significance of ## \frac12 ## is that we have ## \frac12 ## a dog for every 1 cat, and the siginficance of 2 is that we have 2 cats for every 1 dog.

Mr Davis 97 said:
If we said we had a 0.50 ratio of dogs to cats, what does that mean?
That we have 0.5 dogs for every cat (which is the same as 1 dog for every 2 cats, or 2 cats for every 1 dog etc...)
 
Mr Davis 97 said:
Why is division used for ratios?

What would you prefer to use? Do you want everyone to spell it out in words every time?
 
A "ratio" is defined as a fraction which is an implied division.