Understanding the relationship between ratios and fractions

[logicgate]

TL;DR Summary

Got two questions. First, why ratios are considered fractions ?. Second, why multiplying any ratio no matter how many numbers are involved by a constant it stays the same?.

As I understand, a ratio is a comparison between two or more quantities. Ratios involve two or more numbers. Whereas a fraction is a single real number. Why are ratios and fractions the same when ratios involve two or more different numbers whereas fractions represent only ONE real number like for example the ratio 4 : 10 can be expressed as a single number 4/10 which is 0.4 . Also, how do you express ratios such as 2 : 3 : 5 as a single fraction ? Does it become 2/3/5 ? It doesn't make sense to me. Last question is why when we multiply every number in a ratio by a constant the ratio stays the same? For example the ratio a : b : c : d is the same as ax : bx : cx : dx.

 

[Steve4Physics]

As I understand, a ratio is a comparison between two or more quantities. Ratios involve two or more numbers. Whereas a fraction is a single real number. Why are ratios and fractions the same when ratios involve two or more different numbers whereas fractions represent only ONE real number like for example the ratio 4 : 10 can be expressed as a single number 4/10 which is 0.4 .

If you want a comparison of the relative sizes – which is what a ratio is - of two quantities, A and B, you can express it in two ways:

- use a single value – the number by which A (say) needs to bee multiplied to get B; this single number could be written as a fraction, e.g. 1/8;

- use a pair of values which have the correct relative sizes of A and B, e.g. 1 : 8.

A fraction and a ratio look very simila similar but are not quite identical. For example you can add 'normal' fractions but it usually doesn’t make sense to add ratios.

Also, how do you express ratios such as 2 : 3 : 5 as a single fraction ? Does it become 2/3/5 ?

No. We don't use fractional form. It doesn't work. There isn't single value which relates 3 (oe more) independent values.

...why when we multiply every number in a ratio by a constant the ratio stays the same? For example the ratio a : b : c : d is the same as ax : bx : cx : dx.

Yes, assuming x is not zero.

E.g. a cake recipe needs 200g butter (B), 300g sugar (S) and 400g flour (F). (Please don’t try this!) We would express the ratio as B : S : F = 200 : 300 : 400 = 2 : 3 : 4.

If we wanted to make 3 cakes the ratio can be expressed as 3*200 : 3*300 : 3*400 = 600 : 900 : 1200 so we know how much of each ingredient is needed to keep the proportions (ratios) correct.

 

[PeroK]

If quantities $x$ and $y$ are in the ratio $a:b$, then$$\frac x y = \frac a b$$Everything about ratios follows from that observation.

 

[PeroK]

PS if quantities $x,y,z$ are in the ratio $a:b:c$, then this means quantities $x,y$ are in the ratio $a:b$; and, quantities $y,z$ are in the ratio $b:c$.

 

[Joseph Austin]

I like the way you think! In my opinion, a ratio and a fraction are in fact different things, even though they have some aspects in common.
In physics, numbers usually represent a quantity of "something", e.g. meters, kilograms, seconds, coulombs, etc. So a "ratio" is a relationship between quantities of two different things. But curiously, often the ratio, or it's inverse the product, are constant. For example: F = m a, or a = F / m. So to achieve the same acceleration, if you double the mass, you must also double the force (same ratio). Alternatively, given a certain force, if you double the mass, you will halve the acceleration (same product).

If the two "things" in a ratio are the same, e.g. length and width are both distances, the "units" cancel out and we are left with a "pure" number.
Whether a number is a "fraction" or a "whole number" depends on the measurement scale, or the base of the number system used. So, for example, 3 (whole) inches is 1/4 (or 0.25) feet. So "fraction", understood as "part of a whole," is actually more a property of the number "system" than of the quantity itself!
So, you can think of a "ratio" as a "pure number" coefficient of a relationship between "units" of kinds (or sizes) of quantities, e.g. 30 "miles per hour" = 1/2 "hours per mile", that is, 1 hour per 2 miles, or 1/2 hour per 1 mile
Representing the "ratio" as a "fraction" is just a short-hand notation giving the same "pure number" coefficient result.
BTW, the same "puzzle" your fraction question represents also applies to products.
Here's a puzzle I would give my students:
What is 2 apples times 3 oranges?
Answer: 6 "pears" (pairs)!

As for expressing 2:3:5 as a single "fraction", of course we cannot do that, as a "fraction" has only two terms and your example has three. We could, of course, invent three-term entities and define arithmetic for them, but for what application would that be worth the trouble?
One example that comes to mind is music, where notes in "Just Intonation" have "harmonic" ratios 1:2:3:4:5:6...
So a just major tonic chord has note frequency ratios 4:5:6.
Suppose you have a chord with notes do mi so = 240:300:360
The "dominant" chord would have frequencies 3:2 or 3/2 times that:
so ti re = (3/2)*240:(3/2)*300:(3/2)*360 = 360:450:540.
and the sub-dominant chord would have frequencies 2:3 or 2/3 times that:
fa la do = (2/3)*240:(2/3)*300:(2/3)*360 = 160:200:240.
(BTW, I think it makes more sense to say the scale is defined by the principal chords than to try to define the chords by positions on the "scale", but that's another whole subject.)

Keep asking these kinds of questions! Sometimes our "shortcuts" obscure important things.

 

[Mark44]

Got two questions. First, why ratios are considered fractions ?.

Because every ratio can be written as a fraction; e.g., the ratio 2:3 says that the first quantity is 2/3 the latter.

Second, why multiplying any ratio no matter how many numbers are involved by a constant it stays the same?

If you multiply each term of a ratio by the same number, that's equivalent to multiplying the numerator and denominator by that same number, which in turn is equivalent to multiplying the fraction by 1.

Last question is why when we multiply every number in a ratio by a constant the ratio stays the same? For example the ratio a : b : c : d is the same as ax : bx : cx : dx.

Your last question is the third of the two questions you said you were asking.
It seems to me that what you're asking about is a proportion, not a ratio. A proportion states that two ratios are equal. They are usually written in the form a : b :: c : d, and read as "a is to b in the same proportion as c is to d." This is also the same as writing $\frac a b = \frac c d$. For example 5: 8 :: 10 : 16. If you multiply each number in the proportion by the same number you're essentially multiplying the fraction that each ratio represents by 1, changing nothng.

In my opinion, a ratio and a fraction are in fact different things, even though they have some aspects in common.

OTOH, a fraction is a rational number. I didn't see anything in your post that shows that ratios and fractions are different.

 

[symbolipoint]

Got two questions. First, why ratios are considered fractions ?. Second, why multiplying any ratio no matter how many numbers are involved by a constant it stays the same?.

I did not read everything.
Most of the time, but do not need to be, ratio is comparison of two whole numbers.
If you multiply a ratio by a factor, you need to multiply the ENTIRE ratio; which means BOTH parts.

 

[Mark44]

@logicgate you posted another thread about a related topic a week or so ago -- https://www.physicsforums.com/threads/modern-vs-euclids-definition-of-equivalent-ratios.1064671/

In that thread there were some good responses, but nothing more from you other than the original post. It would be helpful for you to follow up on threads that people put effort into to provided answers to you.

 

[logicgate]

I like the way you think! In my opinion, a ratio and a fraction are in fact different things, even though they have some aspects in common.
In physics, numbers usually represent a quantity of "something", e.g. meters, kilograms, seconds, coulombs, etc. So a "ratio" is a relationship between quantities of two different things. But curiously, often the ratio, or it's inverse the product, are constant. For example: F = m a, or a = F / m. So to achieve the same acceleration, if you double the mass, you must also double the force (same ratio). Alternatively, given a certain force, if you double the mass, you will halve the acceleration (same product).

If the two "things" in a ratio are the same, e.g. length and width are both distances, the "units" cancel out and we are left with a "pure" number.
Whether a number is a "fraction" or a "whole number" depends on the measurement scale, or the base of the number system used. So, for example, 3 (whole) inches is 1/4 (or 0.25) feet. So "fraction", understood as "part of a whole," is actually more a property of the number "system" than of the quantity itself!
So, you can think of a "ratio" as a "pure number" coefficient of a relationship between "units" of kinds (or sizes) of quantities, e.g. 30 "miles per hour" = 1/2 "hours per mile", that is, 1 hour per 2 miles, or 1/2 hour per 1 mile
Representing the "ratio" as a "fraction" is just a short-hand notation giving the same "pure number" coefficient result.
BTW, the same "puzzle" your fraction question represents also applies to products.
Here's a puzzle I would give my students:
What is 2 apples times 3 oranges?
Answer: 6 "pears" (pairs)!

As for expressing 2:3:5 as a single "fraction", of course we cannot do that, as a "fraction" has only two terms and your example has three. We could, of course, invent three-term entities and define arithmetic for them, but for what application would that be worth the trouble?
One example that comes to mind is music, where notes in "Just Intonation" have "harmonic" ratios 1:2:3:4:5:6...
So a just major tonic chord has note frequency ratios 4:5:6.
Suppose you have a chord with notes do mi so = 240:300:360
The "dominant" chord would have frequencies 3:2 or 3/2 times that:
so ti re = (3/2)*240:(3/2)*300:(3/2)*360 = 360:450:540.
and the sub-dominant chord would have frequencies 2:3 or 2/3 times that:
fa la do = (2/3)*240:(2/3)*300:(2/3)*360 = 160:200:240.
(BTW, I think it makes more sense to say the scale is defined by the principal chords than to try to define the chords by positions on the "scale", but that's another whole subject.)

Keep asking these kinds of questions! Sometimes our "shortcuts" obscure important things.

Thank you for your answer.