Concrete Definition for a Ratio

[watercolor]

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!

 

[joyfulerst]

A ratio compares two quantities, showing how many of one there are for each of the other. For example, 3 apples to 5 oranges is a 3 : 5 ratio.

 

[Aion]

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!

Hi! You’re thinking along the right lines. A ratio is fundamentally a relation between two quantities of the same kind, written [imath]A:B[/imath], that expresses how many times one quantity contains or relates to the other. This is the more general, conceptual definition.

When the quantities are measurable in a common unit, the ratio can be represented numerically as a fraction:
[math]A:B = \frac{A}{B}[/math]
For example, the ratio of [imath]\text{dogs} : \text{cats} = 2:3[/imath] becomes meaningful if we measure both in the same unit (e.g., number of animals). With 48 cats and 32 dogs, the idea of forming groups of 2 dogs and 3 cats works perfectly, since

[math]32 \div 2 = 16 \text{ groups of dogs}, \quad 48 \div 3 = 16 \text{ groups of cats}[/math]
This also illustrates the scaling property of ratios: multiplying both terms by the same factor does not change the ratio, so
[math]2:3 = (16\cdot 2) : (16\cdot 3)[/math]
If the ratio [imath]A:B[/imath] is a constant [imath]k[/imath], then

[math]A:B = \frac{A}{B} = k \quad \Rightarrow \quad A = kB[/math]
Showing that [imath]A[/imath] varies directly with [imath]B[/imath]. This means that if [imath]B[/imath] changes, [imath]A[/imath] changes proportionally to maintain the same ratio.

In this way, fractions measure ratios numerically, while the ratio itself expresses the relationship between quantities and is preserved under scaling.

 

[Dr.Peterson]

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!

There is some question as to what you mean by "rigorous" and "concrete"; to me, they are somewhat different. It's true that ratios are not quite the same as fractions, and that you will often see vague statements that "a ratio is a relationship" without a clear definition of what sort of thing a ratio actually is.

One thing that is necessary in considering a proper definition of a ratio is that, on one hand, ratios, unlike fractions, can have zero for either term; and on the other hand, unlike fractions, we can have ratios of more than two terms, such as 2:3:4 for length:width:height, or the like. A fraction represents a number; a ratio can't always be equated to a fraction, though often it can.

I would define a ratio as an ordered pair (or triple, or n-tuple) representing the fact that some quantities are the same multiple of those numbers, so that for example the ratio a:b:c means that three quantities are, respectively, ka, kb, and kc for some number k. Here, k can be thought of as a group size, or as a scaling factor. (None of the numbers involved need to be integers, in general.)

That fits well with "2 dogs for every 3 cats"; if there are 2k dogs, then there are 3k cats.

 

[JeffM]

A ratio is a statement about relative quantities in different sets.

“The ratio of girls to boys in that class is 2 to 1” simply means that are twice as many girls in that class as there are boys. The statement tells us nothing about how many boys actually are in the class; it might be one, or it might be fifteen. It is a useful statement if, but only if, what is relevant is the relative sizes of the set of male students compared to the set of female students.
But frequently relative size is what is relevant or comprehensible.

Now it will seldom be the case that a ratio will be relevant if none of the sets being discussed has any elements. Fractions become pertinent once we know the concrete number of elements in just one set with a positive number of elements

“The ratio of set X to set Y is 2 to 7, and the number of elements in set Y is 28” means that the number of elements in set X is equal to (2/7) times 28 = 8.”

Fractions are the way we turn the relative sizes expressed by ratios into countable quantities, and the numerator and denominator of the needed fraction are given by the ratio.