# Representing ratios with division

This might seem like a silly question, but why do we use division to compare two quantities, i.e., a ratio? I've always taken for granted that dividing two physical quantities tells how many of one quantity there is for the other quantity, but why exactly does this work? Why don't we define some new operation that represents the comparison between two quantities rather than division?

jedishrfu
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Don't we have that with the > and < operators?

Mark44
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Related to what jedishrfu said, we can also compare numbers with subtraction, as in x - y. If x is larger than y, then x - y > 0. If x is the smaller of the two, then x - y < 0. Of course, x - y = 0 if the two numbers are equal.

PeroK
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Also, some things like earthquakes are measured on a logarithmic scale. That's another way to compare two quantities.

I think what is being suggested by the question is this kind of thing...?

There are 3 oranges and no apples.

- What is the ratio of oranges to apples? "Three to zero" or 3:0 is OK, but 3/0 is not defined.
- What is the ratio of apples to oranges? "Zero to three" or 0:3 is OK, and 0/3 is defined but following through to get "0" loses information about how many oranges.

I think what is being suggested by the question is this kind of thing...?

There are 3 oranges and no apples.

- What is the ratio of oranges to apples? "Three to zero" or 3:0 is OK, but 3/0 is not defined.
- What is the ratio of apples to oranges? "Zero to three" or 0:3 is OK, and 0/3 is defined but following through to get "0" loses information about how many oranges.

That is more along the lins of what I am asking. I'm wondering why division is used as a way to compare quantities. From your example, it would seem as though using the notation such as a:b to be a better way...

Mark44
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That is more along the lins of what I am asking. I'm wondering why division is used as a way to compare quantities. From your example, it would seem as though using the notation such as a:b to be a better way...
Not really. It is very rare for someone to compare, say, 3 oranges and 0 apples.

If you want to know how many times larger one quantity is than another, you divide, and that's your ratio. As far as the a : b notation, I've never, ever seen it used when the second quantity (which is to be compared to) is zero.

Not really. It is very rare for someone to compare, say, 3 oranges and 0 apples.

If you want to know how many times larger one quantity is than another, you divide, and that's your ratio. As far as the a : b notation, I've never, ever seen it used when the second quantity (which is to be compared to) is zero.

I do see how division can be useful for expressing how two quantities compare, but my question is really why was the idea conceived in the first place? It seems that division is just the inverse operation of multiplication. So how come it turns out to be useful? This question also applies to rates to, as that is a specific type of ratio.

Mark44
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I do see how division can be useful for expressing how two quantities compare, but my question is really why was the idea conceived in the first place?
I doubt that anyone knows when division first came about. An obvious use is in allocating n things amongst m people. Any time someone needs to allocate a number of things equally, they're almost certainly doing division.
Mr Davis 97 said:
It seems that division is just the inverse operation of multiplication. So how come it turns out to be useful? This question also applies to rates to, as that is a specific type of ratio.

Not really. It is very rare for someone to compare, say, 3 oranges and 0 apples.

If you want to know how many times larger one quantity is than another, you divide, and that's your ratio. As far as the a : b notation, I've never, ever seen it used when the second quantity (which is to be compared to) is zero.

Zero apples is not a rare thing, it is the state wherever and whenever apples are not present... zero apples is the state of the entirety of my house right now and virtually all sub-volumes of the universe for all time...

What have you seen used when the second quantity is zero?

3:0 is still informative of the magnitude of the second compared to that of the first... it is not multiplicative information (because division by 0 is undefined), but still retains some additive information as to the magnitude of the difference in value.

Seems to me that X:Y is simply two values and the colon is not any intrinsic operation. Choosing an operation to use for comparing them is another step. One may compare these numbers additively with respect to zero, which allows either number to be zero, or one may impose a division by looking at how many times larger one is of the other, which is either undefined or results in zero if either one is zero.

It depends on what we want to compare. Consider a report that claims that there is a 9:2 ratio of men to women in engineering. This can immediately be visualized by our minds by rendering 9 men for every group of 2 women. Division is an operation that answers the same type of grouping question: a 9/11 fraction of men in engineering can also be visualized immediately by rendering 9 men in every group of 11 engineers.
Compare the alternative type of descriptor: there are 35,448 more men than women in engineering. Without knowing the exact amount of total engineers, this number may be alarming (there are only 36,000 engineers) or marginal (there are 35,000,000 engineers). So this method of comparison is not suited to describing this particular type of information concisely.
A description of "how many times one group fits inside the other", a ratio, is more immediately descriptive in this case. As you can see, the : operator is also used to describe ratios just as efficiently as division, but division has the added benefit of being able to be reduced to a single real number descriptor.

• Mr Davis 97
This might seem like a silly question, but why do we use division to compare two quantities, i.e., a ratio? I've always taken for granted that dividing two physical quantities tells how many of one quantity there is for the other quantity, but why exactly does this work? Why don't we define some new operation that represents the comparison between two quantities rather than division?

Ratios between two quantities give us the comparison of a quantity with a unit value of the other quantity. Now lets say a bag contains 10 oranges and 5 mangoes. the ratio is 10:2 or 10/2 = 5. That is, for every one mango we have 5 oranges in the bag or we can say 5 oranges per mango. Same applies if there are 3 oranges and 4 mangoes. For every 1 mango we would have a 0.75 of an orange.

Now why does this work? When we divide 10 by 2 we find the number that when repeatedly subtracted twice (or repeatedly added twice gives us 10) gives us 0. This means for every '1' in '2' we need to subtract a 5 from 10. Keep repeating this (twice) till we get a 0. Thus for every '1' we have a 5 just like we had 5 oranges for every mango in the above example.
I hope this was clear.

Mark44
Mentor
Zero apples is not a rare thing, it is the state wherever and whenever apples are not present... zero apples is the state of the entirety of my house right now and virtually all sub-volumes of the universe for all time...
Of course zero apples is not a rarity, but almost no one would be interested in a ratio comparison between three oranges and zero apples, and a ratio is exactly what you get with the : notation. See the wikipedia article, http://en.wikipedia.org/wiki/Ratio.
bahamagreen said:
What have you seen used when the second quantity is zero?
Subtraction, as in there are three more oranges than apples.
bahamagreen said:
3:0 is still informative of the magnitude of the second compared to that of the first... it is not multiplicative information (because division by 0 is undefined), but still retains some additive information as to the magnitude of the difference in value.

Seems to me that X:Y is simply two values and the colon is not any intrinsic operation.
The colon notation IS division, just in a different form of notation. It's a ratio, and the indicated operation is division.

bahamagreen said:
Choosing an operation to use for comparing them is another step.
Nope.
bahamagreen said:
One may compare these numbers additively with respect to zero, which allows either number to be zero, or one may impose a division by looking at how many times larger one is of the other, which is either undefined or results in zero if either one is zero.
Sorry, this is nonsense.

Mark44,

Thanks for the remarks... let me elaborate on my thinking and maybe we can get to the bottom of what's amiss.

I looked at the Wiki page you linked; I saw nothing mentioning 0 specifically as a term in a ratio, but did notice that all the Euclidean examples when converted to modern form stipulated positive integers... that suggests 0 may be improper in ratios.

I can see how all defined divisions are ratios, but no ratios where the second place is 0 are defined divisions.
Therefore it seems improper to hold that ratios and divisions are the same unless ratios must not have 0 in the second place.
To hold that all ratios are divisions one must restrict ratios to those for which the corresponding division must not include a division by zero.

However, this restriction seems to extend to natural examples that seem perfectly defined with a zero in the second place.
Mixtures of paint colors A and B can be a ratio A:B in which either A or B may range from 0 to N..
A bucket of paint that is all A and no B has the ratio A:B where B=0 and the color is fully saturated A..

How do you view the paint example? Does it represent a class of improper ratios?

jedishrfu
Mentor
I think we are all in agreement here although it may not seem to be true. There are several notions floating around the use of ratios, the use of fractions and the use of percentages all of which are commonly used to describe mixtures of items or material.

As has been noted, we can have mixtures of fruit where we have 5 apples for every 10 oranges and we would represent it as the ratio 5 : 10 with the understanding that it's apples to oranges. In this case, we would never consider dividing one by the other because the items are different and so we wouldn't use a fraction to represent it. However, we might use percentages like 33% apples to 66% oranges in the mixture of fruit.

With respect to ratios representing division, I would argue that it's more general than that in its common usage. Consider the case of apples, oranges and bananas, we might say that 5 : 10 : 15 to mean 5 apples to 10 oranges to 15 bananas. It wouldn't make sense to say it's

My take is that we can always use ratios or percentages for mixtures no matter how many items or how much material is used. However, when ratios show zero elements then it's meaningless to use fractions and by extension division.

Perhaps, this is why we learn about all three techniques in school to cover the case where we may have to deal with zero.

Having said that, I think the thread has run its course and it's time to close it now.

Mark44
Mentor
I looked at the Wiki page you linked; I saw nothing mentioning 0 specifically as a term in a ratio, but did notice that all the Euclidean examples when converted to modern form stipulated positive integers... that suggests 0 may be improper in ratios.
Not "may be improper" -- division by zero is undefined. Period.
bahamagreen said:
I can see how all defined divisions are ratios, but no ratios where the second place is 0 are defined divisions.
Therefore it seems improper to hold that ratios and divisions are the same unless ratios must not have 0 in the second place.
To hold that all ratios are divisions one must restrict ratios to those for which the corresponding division must not include a division by zero.

However, this restriction seems to extend to natural examples that seem perfectly defined with a zero in the second place.
Mixtures of paint colors A and B can be a ratio A:B in which either A or B may range from 0 to N..
A bucket of paint that is all A and no B has the ratio A:B where B=0 and the color is fully saturated A..
If something is not present in a mixture, you don't describe its absence by a ratio. If you buy a gallon of paint, they don't list all the things that are not in the mixture.
bahamagreen said:
How do you view the paint example?
Does it represent a class of improper ratios?

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