I’m having trouble understanding how fractions relate to ratios. A ratio like 3:5 isn’t directly related to the fraction 3/5, is it? I see how that ratio could be expressed in terms of the two fractions 3/8 and 5/8, but 3/5 doesn’t seem to relate (or be useful) when considering a ratio of 3:5.
Many textbooks I’ve seen, when introducing the topic of ratios, say something along the lines of “3:5 can be expressed in many ways, it can be expressed directly in words as ‘3 parts to 5 parts’, or it can be expressed as a fraction 3/5, or it can be…” and so on. Some textbooks will clarify that 3/5, when used this way, isn’t “really” a fraction, its just representing a ratio. This makes absolutely no sense to me. Why express 3:5 as 3/5 at all?
Best Answer
The notation $a:b$ emphasizes a relative relationship between $a$ and $b$, and the notation $\frac{a}{b}$ emphasizes an operation on two elements $a$ and $b$.
But ultimately the two symbols represent the same thing (at least, when using integers): a relative size between $a$ and $b$. If you think carefully about what it means for two ratios to be equivalent, you'll find that the definition of equality of $\frac{a}{b}$ and $\frac{c}{d}$ is just that the ratios $a:b$ and $c:d$ are equal.
Actually, I do believe that I've seen posters from certain countries actually use "$:$" for division, to the confusion of the rest of us.
One difference in these two notations is that you can link a lot of ratios together at once like this: $1:2:4:7$. This expresses a bunch of ratios at once: $1:2$, $2:4$, $4:7$, $1:4$, $2:7$ etc. If these were ratios of ingredients in some mixture recipe, then you could rather handily increase and decrease the size of your recipe as you desired using this notation.
But this does not translate over to the slash notation, which becomes problematic if you're thinking of the slash as an operation.
This is a bit of a reach, but one way to think of it is that $a:b$ is kind of like "a division operation you are postponing." This is why you can stack them together because no operation is intended. (If you used slashes, the urge would be to carry out the operations until you have a single fraction, but this would require parentheses to make the expression unambiguous.)