Talk:Ratio

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Ratios between more than two quantities

I've written an example under the section on ratios and fractions that addresses this issue, I hope.--Dwetherow 05:18, 23 February 2007 (UTC)

Hmmm. I thought you could have ratios between more than two quantities. E.g. If my fruit bowl has apples, pears and bananas in the ratio 1:3:4 and there are 2 apples in there then there are 6 pears and 8 bananas.

So, why does the article limit rations to being between only two quantities? —The preceding unsigned comment was added by 217.22.155.67 (talk • contribs) .

Why wouldn't you just put 2:6:8 ? —The preceding unsigned comment was added by 81.4.160.194 (talk • contribs) .

Usually you try to express ratios in lowest terms. - dcljr (talk) 08:28, 13 April 2006 (UTC)

What you are defining are relative proportions, not a ratio. —The preceding unsigned comment was added by 192.124.26.250 (talk • contribs) .

really —The preceding unsigned comment was added by 71.96.145.159 (talk • contribs) .

Isn't a proportion the same as a ratio? --116.14.34.220 (talk) 13:13, 16 June 2009 (UTC)

Well... it is true that if the fruits are in the "ratio" of 1:3:4 (I have seen this wording in textbooks before), as described above, then the ratio of apples to pears is 1:3, pears to bananas 3:4, and apples to bananas 1:4, so there's nothing wrong with applying the concept of "ratio" to this situation, you just have to think about it two things at a time. Strictly speaking, the word "ratio" refers to a relationship between two quantities only, but proportions (or "proportional" things) can involve any number of quantities (for example, the corresponding sides of any two similar figures are proportional, regardless of how many sides they have). Finally, any ratio can be explained in terms of proportions, as well: if the ratio of pears to bananas is 3:4, then the proportions of pears and bananas, respectively, are 3/7 and 4/7 of the total number of fruits. (And in the previous example, the proportions of apples, pears and bananas are 1/8, 2/8 = 1/4, and 4/8 = 1/2 of the total.) - dcljr (talk) 08:28, 13 April 2006 (UTC)

Therefore a ratio between more than two quantities is a shorthand for expressing several ratios? --72.140.146.246 13:35, 3 June 2006 (UTC)

Yes. --116.14.34.220 (talk) 13:08, 16 June 2009 (UTC)

You could think of either ratio or proportion as a recipe with a number of terms. For one example in the Greek orders of architecture fenestration's (door and window openings)moldings and other details are often considered to be proportionate to column diameters and elements in different ratios depending on which order they are. — Preceding unsigned comment added by 142.0.102.93 (talk) 18:38, 23 May 2014 (UTC)

Ratios as percentages

Another question. So a ratio can never be expressed as a percentage? —The preceding unsigned comment was added by 202.4.4.48 (talk • contribs) .

If the ratio of apples to oranges is 2 to 1, then the number of apples is 200% (twice) the number of oranges, and the number of oranges is 50% (half) the number of apples. - dcljr (talk) 08:28, 13 April 2006 (UTC)

I think that a ratio is always 100% of everything you are talking about. For example, if you have a 2:1 ratio of apples to oranges then two thirds or approximately 67% of your fruit are apples and one third or 33% are oranges, for a total of 100% or three thirds.--Dwetherow 05:18, 23 February 2007 (UTC)

I think I'd go with User:Dcljr's definition. The whole thing doesn't have to be 100%. Rather, you can fix one to be 100% and see what the other one is. Isn't that what you do with fractions, which are essentially the same as ratios? --116.14.34.220 (talk) 13:10, 16 June 2009 (UTC)

Maps present another interesting example. A map ratio of 1:100 sensibly presented as a percentage would be 1%. Meaning a distance on the map is 1% of the distance in the real world. It makes no sense to add the distance on the map to the distance in the real world, and use this combined 'real' and 'virtual' distance in any calculation. —Preceding unsigned comment added by 89.242.145.251 (talk) 17:43, 14 February 2010 (UTC)

a ratio can be both a proportion and a percentage. Suppose for example we were to compare the ratio of pennies (100:1) in a dollar to nickles (20:1) or quarters (4:1)in a dollar. You might count a random dollars worth of change and determine the proportion of nickles to pennies, dimes, quarters or half dollars then compare the percentage of each.142.0.102.93 (talk) 18:47, 23 May 2014 (UTC)

Comparing ratios

So if you have two ratios, 1:2000 and 1:4000, which one is "higher"? —The preceding unsigned comment was added by 69.157.57.16 (talk • contribs) .

Well, you divide 1 by 2000, get a number. Then, divide 1 by 4000, get another number. See which one of the two obtained numbers is bigger. Oleg Alexandrov 20:36, 27 September 2005 (UTC)

Can you actually compare ratios? --116.14.34.220 (talk) 13:10, 16 June 2009 (UTC)

Scale map

If you were drawing a map and were using the ratio 1cm:20km how many cm would 22km be? 1.1? —The preceding unsigned comment was added by 81.178.228.183 (talk • contribs) .

Yes. 1 is to 20 as 1.1 is to 22. Or: (1/20)=(x/22) → x=1.1. - dcljr (talk) 06:36, 13 April 2006 (UTC)

Are ratios necessarily linear?

The article begins by declaring a ratio to be a linear relationship. What about, say, the ratio of a square's perimeter to its area? That's nonlinear; is it a ratio? --VP 38.113.17.3 21:41, 17 April 2006 (UTC)

Yes, I'd like to hear the reasoning for stating that ratios are linear relationships. This implies to me a relationship across the range of magnitudes of a quantity, which is not a (necessary) feature of a ratio. A ratio between two continuous quantities of the same kind is a real number. Ratios between different quantities are not numbers at all. For example 1cm/1g is not a number. However, it is possible to form a ratio between the numbers arising from measurements. You might say that there is a 1:1 correspondence between the volume, say in $cm^{3}$, and mass, in grams, required to store some material. Then, the ratio is between numbers which are measurements, not between quantities. Holon 01:08, 12 May 2006 (UTC)

1 cm / 1 g is a number, but it is not dimensionless (ie it has units of cm/g). A ratio between, say, the area of two different shapes would give a dimensionless real number (ie no units). --72.140.146.246 19:10, 3 June 2006 (UTC)

1cm/1g is not a number. Take a simpler example; 1g. One gram is not a number: it is a quantity; an amount of mass. The 1 in 1 g is a measure of quantity of mass, and to know its dimension is to know its unit. Quantities themselves are not numbers. Measures of quantities are numbers. Holon 02:16, 4 June 2006 (UTC)

you can easily compare ratios of measures of length, area, or volume by making a linear measure the side of a square or a cube.

in which case they become non linear since they are exponential. Further you can change the rate of increase at an increasing rate over time introducing a fourth dimension. The ancient problems of squaring a circle, doubling a cube and trisecting an angle were solvable so long as you didn't add constraints such as limiting their construction to ruler and compass.142.0.102.93 (talk) 19:04, 23 May 2014 (UTC)

Comensurability allows you to take two different scales and put them in harmony. For example a Mesopotamian foot was three hands of 100 mm each, an ordinary cubit was 5 hands or 1/2 meter and a great cubit was 6 hands, 2 feet or 600 mm. To make that sexigesimal division commensurable with a septenary Egyptian ruler which used palms of 75 mm rather than hands of 100 mm you used a ration of 4 palms to 3 hands to make one foot. The Egyptian ruler would have an ordinary cubit of 5 palms or 450 mm and a royal cubit of 525 mm. Its ell of 8 palms would be two feet (1 nibw) which was the same as a Mesopotamian Great cubit. The use of different scales used to be a feature of slide rules which could be used for logarithmic computation.142.0.102.93 (talk) 19:04, 23 May 2014 (UTC)

Can Ratios Be Negative?

Lets say I'm dealing in apples and oranges, and I am in debt apples but have a surplus of oranges.

I may have a ratio of (-2 apples / 3 oranges), and a ratio of (-3 apples / 2 oranges). Which is the larger ratio of apples to oranges?

If thought about as a fraction then -2/3 = -.666, and -3/2 = -1.5.

Therefore, -2/3 is a bigger ratio of apples to oranges because it is 'less negative' compared with -1.5.

However, if thought about in absolute terms, there are more apples to oranges in the -3/2 ratio.

Can ratios work with one (or more) parts of the ratio being negative? Or are ratios strictly absolute?

71.142.81.237 08:11, 22 February 2007 (UTC)Dave A

If you think of a negative number as being directional, then that is what you are doing to get a negative ratio. I'm not really comfortable with that, and haven't seen any definition that would accommodate it. To me, what you really have is a ratio of 2:3, where the units are apples owed to surplus oranges. Trishm 05:02, 18 June 2007 (UTC)

Classically, the theory of ratios was developed in antiquity, long before the Western philosophers and mathematicians had learned about negative numbers. (There were some sorts of theory of negative numbers very early in China and India, but these ideas did not reach Europe until much later.) Therefore, of course the ratios considered by the ancients were all positive.

In modern usage, a negative ratio sometimes makes sense; it depends on the kind entities you compare. JoergenB (talk) 18:54, 11 January 2010 (UTC)

Agreed, but the meaning can vary with context, so there is no general theory of negative ratio. Dbfirs 21:43, 25 January 2010 (UTC)

Correctly stating a ratio

I have some confusion over ratios and fractions - In the opening statement, the example 2:3 is used and is described as a whole consisting of 5 parts. In the first example, the ratio 1:4 refers to four parts in the whole. Which is the correct description of a ratio? This has always confused me. Stating the question in other terms - if I have a solution consisting of 1 part X and 3 parts Y, do I describe the ratio as 1:3 or 1:4? —Preceding unsigned comment added by 67.161.203.22 (talk • contribs)

The example was incorrect. To answer your question, the ratio of X to Y would be 1:3 (1 to 3), not 1:4. The proportion of the whole that is X would be 1/4 (one out of 4). I've completely rewritten the article to try to clarify the situation. - dcljr (talk) 19:05, 14 August 2006 (UTC)

I've changed the edits because the concept of ratio is fundamental to the very definition of measurement throughout the physical sciences, and arguably all science. I'm open to debate on how to present the two subtly different usages of the term. Let me know if you have qualms, I'm always open to suggestions Holon 11:07, 16 August 2006 (UTC)

I'm a HS teacher, and have always taught that fractions are synonymous with ratios; that the ratio 2:3, for example, is the quotient of 2 divided by 3 or 2/3. Under this interpretation, a ratio would be "a fraction turned on its side." This is in many textbooks, for example Dolciani "Algebra-Structure and Method." I'd be interested in seeing a reference with an alternative interpretation, making 1/2 not equal to 1:2. While on the subject: Dolciani defines a proportion as "an equation that states that two ratios are equal." So, 2:3 = 4:6 would be a proportion. Splendiff 23:42, 6 June 2007 (UTC)

Dude! You're freaking me out! I'm in my first year of teaching - Grade 6 - and don't understand a ratio to be a fraction-on-its-side at all. I think you're right in that 2:3 is equivalent to 4:6, but I don't think 2:3 is two thirds. Take this set for example: [xxyyy] - ²/₅ are x's, three fifths are y's, and the ratio of x to y is 2:3 (no '5' present in the ratio). A fraction represents a part of a whole - a ratio compares parts of a whole (usually all the parts). Another example [1 apple and 2 bananas] - one third is apple, two thirds are banana and the ratio of apple to banana is 1:2 (for every apple there are two bananas; for every banana there's half an apple). Add an orange and you have a ratio of 1:2:1, which shows the proportion of parts of the whole rather than representing one part of the whole (such as 1/4 being oranges). Neither specify how big the whole is. Arcrawfo (talk) 11:47, 14 June 2008 (UTC)

OK, Arcrawfo again. I've been thinking about the fraction thing. Splendiff; I think I may get what you might mean... Back to the fruit bowl example: 8 apples, 12 oranges; the apples to oranges ratio is 3:4. This means that for every 3 apples there are four oranges. It also means that however many oranges there are, there will be ³/₄ the amount of apples. For 6 apples and 9 oranges - 2:3 - however many oranges there are, there will be two thirds the apples. And this is how the ratio is used as a fraction. Si?Arcrawfo (talk) 01:17, 15 June 2008 (UTC)

Try "The VNR Concise Encyclopedia of Mathematics", by W. Gellert and H. Kuestner and M. Hellwich, and H. Kaestner, published in 1977, ISBN 9780442205904.

The ratio of two numbers of is value of one number in terms of the other, and is expressed as the quotient of their measures. A ratio is a general means of comparing any two numbers in a multiplicative sense.

Rather than thinking of a ratio as a special case of a fraction, think of the ratio as the way numbers were compared in a multiplicative sense, before fractions assumed their full power.Ratios are more flexible than fractions, because they can be used to compare part to part, such as often used in chemistry or maths (like a 3:4:5 triangle), or part to whole, as in a fraction. The techniques for manipulating fractions are much more powerful though (I don't think I've ever tried to add dissimilar ratios), so we use fractions most of the time now. Trishm 04:53, 18 June 2007 (UTC)

The example given under concentration not only makes no sense, it's incorrect. 1:5 means 1 part TO 5 parts (i.e. 1 part of X added to 5 parts Y) and should not be confused with 1 in 5 (which means 1 part in 5 parts TOTAL). The same is true for smaller ratios such as 1:100 which is not the same as 1% but actually, in terms of concentration, refers to 1 part added to 100 parts which equates to 0.91%. An easy way to understand it is to look at a 1:1 mixture. Using Splendiff's explanation above, this would be equivalent to 1/1 or 100%. When in fact it represents a 50% mixture of one component in the other or 1/2. I'm not sure how they teach these concepts in high school, but I teach it this way to our pharmacy students - and we're pretty picky with life-threatening concentration calculations. PharmaG (talk) 14:03, 15 July 2010 (UTC)

The problem is in the imprecise way in which we state ratios. We should always specify clearly whether we are stating the ratio of one component to another or the ratio of one component to the total. Both are used in real science. In high school, questions are usually set with precision and clarity, as with your examples to pharmacy students. Unfortunately, industrial chemists and biologists seem not always to specify which ratio they are stating. I've moved this confusing example nearer to the end, and mentioned that dilution factors are not normally used in pharmacy (though I suspect that they are in homeopathy?) Dbfirs 20:07, 15 July 2010 (UTC)

In ancient times people used to use unit fractions computationally. 2/3 and 3/4 were special cases, but lets suppose I'm an architect and I want to instruct a carpenter or a mason how to build a circular arch using unit fractions. This problem actually was solved at Saqarra c 3000 BC. http://en.wikipedia.org/wiki/File:Egyptian_circle.png The architect created a grid of vertical lines 1 ordinary cubit apart and then marked a height on each vertical through which the workmen bent a stick to get a fair line. This method is still used today in lofting boats where each frame of the boat has a different curve. 142.0.102.93 (talk) 19:12, 23 May 2014 (UTC)

no chickens!

someone added the word 'chicken' to the start of the page. it wasn't in context and i assume it was a mistake so i removed it. if you're terribly fond of chickens and find this edit to be offensive, i apologize.

gba 05:11, 12 February 2007 (UTC)

I'm sorry. I don't edit and don't know what the standards are involved. I just wanted to point out something I think needs correction:

Under: Ratios and fractions

"a) If you have three apples for every four oranges then you have a 3:4 ratio b) If you want to determine what fraction of the total fruit will be apples or oranges then you add the parts of the ratio to determine the total fruit, in this case: 3+4=7 c) The total fruit becomes the common denominator and the parts of the ratio become the numerators, in this case: 3/7 of the fruit are apples and 1/7 are oranges d) The fractions implied in a ratio will always total one whole (or 100% of the fruit), in this case: 3/7 + 4/7 = 7/7 = 1"

Specifically "c) The total fruit becomes the common denominator and the parts of the ratio become the numerators, in this case: 3/7 of the fruit are apples and 1/7 are oranges" I believe the "1/7 are oranges" should read "4/7 are oranges"

24.61.93.51 15:30, 17 March 2007 (UTC)peter

Poorly defined in the lead section

Current definition is very bad. It mentions the word proportional which leads to circular definition. Somebody provided a better one above:

The ratio of two numbers of is value of one number in terms of the other, and is expressed as the quotient of their measures. A ratio is a general means of comparing any two numbers in a multiplicative sense.

"The VNR Concise Encyclopedia of Mathematics", by W. Gellert and H. Kuestner and M. Hellwich, and H. Kaestner, published in 1977, ISBN 9780442205904.

I believe this one also has an issue, as it defines "ratio of numbers" and isn't it supposed to be a "ratio of quantities" per discussion above? By the way Rate also seems to have a broken lead section, it contradicts this article. Could someone comment or correct? I'm not feeling "bold" enough because of my poor English. --Kubanczyk 15:25, 11 October 2007 (UTC)

I think the confusion in the article reflects widespread confusion in the use of the terms. As a result, I see the definitions themselves keep going around in circles over time! I agree the current definition is not good, but also agree that the proposed definition above is problematic. Another concern is to make the article encyclopedic rather than just definitions with examples.

Ratio is used in three related but distinct senses. There are ratios of two quantities of the same kind (e.g. 2 cm/1 cm), which are real numbers. There are ratios of numbers, which are themselves also real numbers. Rates are written using the same notation as ratios, but they are clearly not real numbers. For example, 3 cm/ sec is a rate and 3 cm/ 1 cm is a ratio. Most uses of ratio imply that a ratio is a number, whereas a rate clearly is not a number. I think rate is more clearly defined as the change in the magnitude of one quantity with respect to change in another kind of quantity. Nevertheless, it is not unusual for definitions of rate to refer to ratio so it clearly needs to be noted.

My preference based on these considerations is to clarify the different usages of the terms with respect to the key concepts, with examples, rather than to have rather confusing (to me) and potentially contradictory (depending on definition) usages. This should be done, preferably, with reference to some good sources. Sources should include, but most certainly not be limited to mathematical ones such as the dictionary cited above; for example physics sources and sources relating to historical usage are very much relevant. I will try to find some time to do this, but it probably won't be soon.

On other points, unitless is a term I don't find particularly helpful: if there are no units, it is a number pure and simple. Again, it's commonly used so it is reasonable to note the usage but this does not mean the term should feature prominently in the opening lines as is currently the case. An acid test for me is to ask: what does it add to the reader's understanding? In addition, I find the use of quantity in the article potentially quite confusing. Quantity in the definition refers to numbers, whereas quantity more often refers to those such as 1 cm, 2 ohm, 2 sec etc., which are clearly not numbers. Nevertheless, quantity is used very often in mathematics to refer to (pure) numbers, particularly when relationships between more than one variable are described, so this needs to be noted.

Thoughts? Holon 10:39, 13 October 2007 (UTC) (updated to clarify the main points Holon 12:22, 15 October 2007 (UTC))

Ratios often do have units and sometimes those differ. Consider the ratio of boys to girls in a student body. This may be 7 boys to 13 girls == 7 boys : 13 girls == 7 boys / 13 girls. That ratio is not a rate as we have no rule to establish an influential relationship between the two measurements. A change in number of girls in the group would not imply or necessitate a change in the number of boys. To stretch the example a bit if there were a requirement for at least 1 teaching assistant per 10 students, then we have a rate. Adding one more students and another teaching assistant is added as well to maintain the relationship.

Rate is a special case of ratio that establishes a relationship. The contradiction on the ratio and rate entries comes from assertion on ratio entry that rate is per unit of time. Per example above of "3 cm/ sec", that is a rate, but also a ratio. (Technically just stating "3 cm/ sec" is not enough information to be certain it is a rate. But we humans have trouble seperating anything with time as non-relational. What if "3 cm" is length of wood block and "1 sec" is amount of time a second hand was observed to move? There is a comparison of two unrelated measurements and therefore is not a rate, though it still qualifies as a ratio.)

-Geek Jason (talk) 16:21, 12 May 2009 (UTC)

Ratio, definitions and examples

RATIO

A ratio is an ordered finite set of quantities provided two sets are equivalent when their correspondent elements are proportional. For example, <a, b c, d,…> = <A, B, C, D,…> if A=ka, B=kb, C=kc, … for some k≠0. The equality sign is usually used in this case to designate equivalence of the two sets.

The quantities may or may not have units of measurement. Colon is usually used as separator, for example, a:b:c:d:… or A:B:C:D…

Main property of ratios: For any two pairs of corresponding elements, say b, d and B, D, the following equality is held: bD=dB.

Example: An ordered set <5 dogs, 2 houses, 3 apples, 4 oranges> is a ratio if it represents a structure, so that <5 dogs, 2 houses, 3 apples, 4 oranges> = <10 dogs, 4 houses, 6 apples, 8 oranges> = <15 dogs, 6 houses, 9 apples, 12 oranges> = …, where k=2, 3,…, respectively. Using a colon notation, we get 5 dogs : 2 houses : 3 apples : 4 oranges = 10 dogs : 4 houses : 6 apples : 8 oranges = 15 dogs : 6 houses : 9 apples : 12 oranges = … The chain of equalities may be continued further using arbitrary values of k≠0. In this example the main property of ratios leads to the following equalities: 2 houses • 6 apples = 4 houses • 3 apples = 12 apple-houses, 6 apples • 12 oranges = 8 oranges • 12 apples = 72 apple-oranges etc.

Recipes provide good examples of ratios. Thus, a recipe that reads "For serving two persons take 2 pound rabbit, ½ cup flour, 1 tablespoon butter, and 1cup red wine", may be written as the ratio 2 persons : 2 lb : ½ cup : 1 tablespoon : 1 cup. This ratio tells us that depending on the number of persons served, the amounts of each product should be increased or decreased proportionally.

Two-element ratio related to direct variation of two quantities is called a rate. For example, <126 miles, 3 gallons> =126 miles : 3 gallons is a rate of gasoline consumption. A rate with the second element equals to one is called a unit rate. Thus, this example may be written as 126 miles : 3 gallons = 42 miles : 1 gallon, with the latter rate being the unit rate.

Rates and other two-element ratios without units of measurement possess the following property of fractions: they do not change their meaning if their elements are multiplied or divided by any non-zero number. In particular, they may be expressed in lower terms. For example, the following ratios are equivalent: 126:3 = 42:1. Nevertheless, rates or two-element ratios are not fractions, because they do not possess all of the fractions' properties. For instance, ratios cannot be added or subtracted; if we formally add or subtract them as fractions, the result may or may not make sense as a ratio. —Preceding unsigned comment added by Hostosv (talk • contribs) 01:01, 5 November 2007 (UTC)

ų —Preceding unsigned comment added by 41.242.171.21 (talk) 11:38, 17 August 2008 (UTC)

Adding references

The article had gotten quite long considering there were no references cited. I've added some older sources on the assumption that there haven't been a lot of changes to the subject in the last hundred years. It will take some time to go through the article section by section to see what needs to be further referenced, what should be deleted as OR, what should be expanded based on the material in the sources, and what is OK as is. I'm leaving the references needed tag until this is all sorted out.--RDBury (talk) 21:45, 4 November 2009 (UTC)

The problem being, of course, that referenced material can be rubbish. Now, I cannot speak for the quality or otherwise of the Penny Encylopledia in general, but I would like to in this particular. The PE (of which I do not have a copy, so I am taking this on trust) is referenced and, perhaps, *interpreted* to say that the ratio n:m is the fraction n/m. No it is not. The ratio n:m is the fraction n/(n+m). The Penny Encyclopedia is wrong, you get what you pay for! Paul Beardsell (talk) 10:23, 26 December 2009 (UTC)

An example to illustrate. The ratio 1:1 is not 1/1 but 1/2. If an equal amount of butter is added to sugar by weight what is the weight of the butter in ratio to that of the sugar? 1:1. What is the weight of the butter as a proportion of the total? 1/2 Paul Beardsell (talk) 10:23, 26 December 2009 (UTC)

So, I intend to repair the article, but I have been reverted once. Anything to say? Paul Beardsell (talk) 10:23, 26 December 2009 (UTC)

Firstly, you don't need a copy of the Penny Cylopledia, there is a link to it online at the bottom of the page. Second, the Penny Cyclopedia isn't the only source on this; Fundamentals of practical mathematics (also available online through a link at the bottom of the page) uses the same definition. I could probably find more but the statement is already referenced. If you don't agree with the statement then the burden is on you to find a reference supporting your claim, not on me to keep finding more and more references until you're satisfied.

In your butter and sugar example, the ratio of butter to sugar is still 1 to 1, however the ratio of butter to the total is 1 to 2. This is normally called the proportion of butter, to the proportion of butter in your example is 1/2; maybe that's what you're thinking of.--RDBury (talk) 19:32, 26 December 2009 (UTC)

Let's see, you agree that often with ratios that n:m corresponds to n/(n+m)? The article doesn't say that. Paul Beardsell (talk) 17:19, 27 December 2009 (UTC)

The article is right, if you think it's wrong then give a reference. Engaging in pointless arguments on a talk page is a waste of time for both of us.--RDBury (talk) 00:39, 28 December 2009 (UTC)

Stating that proportions often is taken with respect to the total is not the same as stating that "n:m corresponds to n/(n+m)". Actually, n/m often is defined as an equivalence class of ratios, including n:m. (Two ratios a:b and c:d then are considered as equivalent, if there are non-zero or otherwise restricted elements e and f, such that ae = cf and be = df; cf. localization of a ring, where the ratios are identified with certain pairs (r,s).) JoergenB (talk) 19:17, 11 January 2010 (UTC)

Paul Beardsell, what statement do you wish to add (or remove) to the article to explain your understanding of the word "ratio"? --Robin (talk) 00:56, 28 December 2009 (UTC)

Ratios and commensurability

Right now (19:27, 11 January 2010 (UTC)), the article completely seems to lack creferences to commensurability issues. This is a pity; at least, the historical relation between ratios of commensurable entities and rational numbers ought to be explained. IMHO, the explanation of the term "rational" is clearly of encyclopedial interest. JoergenB (talk) 19:26, 11 January 2010 (UTC)

Yes, it is of interest, but mathematicians often use irrational and even transcendental ratios! Dbfirs 21:55, 25 January 2010 (UTC)

Yes, already Eudoxus developed a theory for more general ratios or "proportions". However, these are not in general commensurable. Indeed, the main reason for the Eudoxus's rather abstract treatment of proportions is the trouble with noncommensurability - or "irrationality", to use the modern term. JoergenB (talk) 17:24, 6 February 2010 (UTC)

Perhaps we should have a separate article on ratios of commensurable entities? I don't think a long discussion on the history of commensurability and rationality belongs here. Dbfirs 20:57, 15 July 2010 (UTC)

Rewrite

This article needs much work. I've fiddled around with it a bit whilst watching TV, but it needs restructuring. I suggest a brief lead, followed by the simple examples for those who come here wondering what the word means, or how to use the concept, with the technical definitions and philosophy of rationality at the end. What does anyone else think? Thanks to 24.51.192.180 for removing the bit I didn't like but wasn't brave enough to remove, and to Anonymous Dissident for improving my wording. Dbfirs 23:16, 25 January 2010 (UTC)

Thanks to RDBury for improving the article, including removing my heading that turned out to be OR (though I thought when I used it that I could find someone else who called it "Normal Form"). I've restored the content of the paragraph because it can be found in most elementary texts on ratio. Dbfirs 23:24, 23 February 2010 (UTC)

Clarity in opening and basic definition?

I find the opening sentences to be somewhat confusing, particularly to a person who is looking for a basic understanding of ratio. The first sentence definition makes sense and serves as an adequate definition of ratio: "In mathematics, a ratio expresses the magnitude of quantities relative to each other." That's clear.

But, the opening goes on: "Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second..." What does that mean? Right now, the graphic example given next to the opening paragraph is a "4:3" ratio. A reader would presumably look over at that example and try to puzzle out the opening. Using the second sentence definition, that means "the ratio of 4 to 3 indicates how many times 4 is contained in 3." What?? The Penny Encyclopedia is cited for this, but I looked at the entry linked at the bottom of the page, and I don't see this idea mentioned prominently. It's a long article, so it might be buried in there somewhere, but in general the source seems to go more with the definition of the first sentence, i.e., a ratio compares the relative magnitude of two things, which means 4:3 is the same as 3:4, it's just the order of the things expressed (width:height vs. height:width) has changed -- so why the focus on which quantity is "contained" in which other quantity?

The opening continues: "and may be expressed algebraically as their quotient." (I don't disagree with this, but it seems to focus on only two-term ratios that represent fractional relationships, which is only one class of ratio -- is that appropriate for a general opening?) And then "Example: For every Spoon of sugar, you need 2 spoons of flour ( 1:2 )" This isn't a very good example for what was just stated (quotient relationships), since expressing this ratio as 1/2 is potentially misleading. In terms of "spoons," there is 1/2 as much sugar as flour, but -- as is clear from some confusion expressed in earlier Talk Page discussion -- this is not the only possible fractional relationship to be derived from such a ratio. If the ratio clearly represented a part-to-whole relationship (as many ratios do, such as "3 students compared to the entire class of 12 students," or 3:12), it would make sense to represent it as a quotient (3/12, or 1/4). In the case of the example provided, however, the application of a quotient as mentioned in the previous sentence is ambiguous.140.247.240.127 (talk) 22:08, 19 July 2010 (UTC)

By the way, here's a possible draft suggestion to replace the opening. I'm not going to edit the article myself (I long ago stopped participating in edit wars on Wikipedia), but this might serve to push the discussion forward for improvement:

"In mathematics, a ratio expresses the magnitude of quantities relative to each other. Although ratios can represent the relative size of any number of quantities, ratios are most commonly used to compare two quantities. Such a relationship can compare relative amounts or sizes of two things. For example, if a lake is 3 miles long and 2 miles wide, the length and width create a 3:2 ratio with each other. Ratios can be expressed in many ways; a recipe that states "for every spoon of sugar, you need 2 spoons of flour" implies a 1:2 ratio of sugar to flour. In some circumstances, a ratio represents a part-to-whole relationship (similar to a fraction), and in that case the ratio may be expressed algebraically as a quotient. For example, if there are 5 blond students in a class of 17 students, the ratio of blond students to the entire class would be 5:17, or, equivalently, 5/17." 140.247.240.127 (talk) 22:40, 19 July 2010 (UTC)

Edit request from 66.245.6.251, 1 January 2011

Please clarify the first sentence under Number of Terms to say something like,

"In general, when comparing the quantities of a two-quantity ratio, this can be expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, the amount/size/volume/number of the first quantity will be 2/3 that of the second quantity. This pattern also holds in ratios with more than two terms; however, a ratio with more than two terms cannot be converted into a single fraction, where a single fraction can represent only one part of the ratio."

I'm requesting this clarification because I often find that my students think that ratios are the same thing as fractions. While this is clearly demonstrated in the Fraction and Proportion subheadings, the Number of Terms text muddies and confuses this important distinction.

Thanks

66.245.6.251 (talk) 02:54, 1 January 2011 (UTC)

Ok done. Thank you for this contribution. I made a bit of a change to the final clause of your edit request, as I found it a bit unclear. Feel free to let me know if I've ruined everything by doing this. Also, why don't you make an account and stay for a while. That way you'll be able to make edits like these by yourself without needing to submit a request first. Zachlipton (talk) 03:01, 1 January 2011 (UTC)

What of ratios of quantities of different measures?

Is an expression like "2mg substance per 50 ml water" a ratio? If so, some changes are needed. If not, what is it? 118.107.149.10 (talk) 01:51, 23 June 2011 (UTC)

This "w/v" expression is often used in science, and is not what mathematicians would call a pure ratio, but it is a ratio of mg of substance to ml of water. Have you any suggestions on how to improve the article? Should we have a separate paragraph on w.v ratios? Dbfirs 08:07, 23 June 2011 (UTC)

... (later) I've added a couple of sentences to the last paragraph. Does this help? Dbfirs 08:40, 23 June 2011 (UTC)

Remen as a Proportion or ratio to other units.

In antiquity writers such as Vitruvus went back as far as the historian Herodotus checking for ratios between the unit fractions used up until medieval times for calculation and unit measures and discovered that the body measures which were the basis for Egyptian inscription grids were expanded to the agricultural measures used for Egyptian and Mesopotamian fields, and further expanded into architectural proportions based on structural loads and spans and distance measures such as the surveyed length of a days march or sail. The proportion of foot to remen can be either 4:5 making it the hypotenuse or 3:4 making it the side of a right triangle. If the remen is the hypotenuse of a 3:4:5 triangle then the foot is one side and the quarter another so the proportions are 3:4 quarter to foot, 4:5 foot to remen and 3:5 quarter to Remen. The quarter is 1/4 yard. The foot is 1/3 yard. The remen is

The remen may also be the side of a square whose diagonal is a cubit The proportion of remen to cubit is 4:5

The proportion of palm to remen is 1:5

The proportion of hand to remen is 1:4

The proportion of palm to foot is 1:4

The proportion of hand to foot is 1:3

The table below demonstrates a harmonious system of proportion much like the musical scales, with fourths and fifths, and other scales based on geometric divisions, diameters, circumferences, diagonals, powers, and series coordinated with the canons of architectural proportion, Pi, phi and other constants..

In Mesopotamia and Egypt the Remen could be divided into different proportions as a similar triangle with sides as fingers, palms, or hands. The Egyptians thought of the Remen as proportionate to the cubit or mh foot and palm.

They used it as the diagonal of a unit rise or run like a modern framing square. Their relatedseked gives a slope. Its convenient to think of remen as intermediate to both large and small scale elements.

Even before the Greeks like Solon, Herodotus, Pythagorus, Plato, Ptolomy, Aristotle, Eratosthenes, and the Romans like Vitruvius, there seems to be a concept that all things should be related to one another proportionally.

Its not certain whether the ideas of proportionality begin with studies of the elements of the body as they relate to scaling architecture to the needs of humans, or the divisions of urban planning laying out cities and fields to the needs of surveyors.

In all cultures the canons of proportion are proportional to reproducable standards.

In ancient cultures the standards are divisions of a degree of the earths circumference into mia chillioi, mille passus, and stadia.

Stadia, are used to lay out city blocks, roads, large public buildings and fields

Fields are divided into acres using as their sides, furlongs, perches, cords, rods, fathoms, paces, yards, cubits, and remen which are proportional to miles and stadia

Buildings are divided into feet, hands, palms and fingers, which are also systematized to the sides of agricultural units.

Inside buildings the elements of the architectural design follow the canons of proportion of the the inscription grids based on body measures and the orders of architectural components.

In manufacturing the same unit fraction proportions are systematized to the length and width of boards, cloth and manufactured goods.

The unit fractions used are generally the best sexigesimal factors, three quarters, halves, 3rds, fourths, fifths, sixths, sevenths, eighths, tenths, unidecimals, sixteenths and their inverses used as a doubling system

Greek Remen generally have long, median and short forms with their sides related geometrically as arithmetric or geometric series based on hands and feet.

• The Egyptian bd is 300 mm and its remen is 375 mm. the proportion is 1:1.25
• The Ionian pous and Roman pes are a short foot measuring 296 mm their remen is 370 mm
• the Old English foot is 3 hands (15 digits of 20.32 mm) = 304.8 mm and its remen is 381 mm
• The Modern English foot is 12 inches of 25.4 mm = 304.8 mm and its remen is 381 mm (15")
• The Attic pous measures 308.4 mm its remen is 385.5
• The Athenian pous measures 316 mm and is considered of median length its remen is 395 mm
• Long pous are actually Remen (4 hands) and pygons
• See cubit for the discussion of the choice of division into hands or palms
• See the table below for proportions relative to other ancient Mediterranean units

Roman Remen generally have long, and short forms with their sides related geometrically as arithmetric or geometric series based on fingers palms and feet.

By Roman times the Remen is standardized as the diagonal of a 3:4:5 triangle with one side a palmus and another a pes. The Remen and similar forms of sacred geometry formed the basis of the later system of Roman architectural proportions as described by Vitruvius.

Generally the sexagesimal (base-six) or decimal (base-ten) multiples have Mesopotamian origins while the septenary (base-seven) multiples have Egyptian origins.

Unit Proportions to Greek Remen

Unit	Finger	Culture	Metric	Palm	Hand	Foot	Remen	Pace	Fathom
(1 ŝuŝi	1 (little finger)	Mesop	14.49 mm		.2	0.067	0.05
1 ŝushi	1 (ring finger)	Mesop	16.67 mm		.2	0.67	0.05
1 shushi	1 (ring finger)	Mesop	17 mm		.2	0.67	0.05
1 digitus	1 (long finger)	Roman	18.5 mm	.25		0.0625	0.04
1 dj	1 (long finger)	Egyptian	18.75 mm	.25		0.0625	0.04
1 daktylos	1 (index finger)	Greek	19.275 mm		.2	0.067	0.04
1 uban	1 (index finger)	Mesop	.2		.2	0.067	0.04
1 finger	1 (index finger)	Old English	20.32 mm		.2	0.067	0.045
1 inch	(thumb)	English	25.4 mm			0.083	.067
1 uncia	(thumb or inch)	Roman	24.7 mm	.25		0.083	.067
1 condylos	2 (daktylos)	Greek	38.55 mm	.5		2	.1
1 palaiste, palm	4 (daktylos)	Greek	77.1 mm	1		0.25	.2
1 palaistos, hand	5 (daktylos)	Greek	96.375 mm		1	0.333	.25
1 hand	5 (fingers)	English	101.6mm		1	0.333	.25
1 dichas,	8 (daktylos)	Greek	154.2 mm	2		0.5	.4
1 spithame	12 (daktylos)	Greek	231.3 mm	3		.75	.6
1 pous, foot of 4 palms	16 (daktylos)	Ionian Greek	296 mm	4		1	.8
1 pes, foot	16 (digitus)	Roman	296.4 mm	4		1	.8
1 uban, foot	15 (uban)	Mesop	300 mm		3	1	.75
1 bd, foot	16 (dj)	Egyptian	300 mm	4		1	.8
1 foote(3 hands)	15 (fingers)	Old English	304.8 mm		3	1	.75
1 foot, (12 inches)	16 (inches)	English	308.4 mm		3	1	.75
1 pous, foot of 4 palms	16 (daktylos)	Attic Greek	308.4 mm	4		1	.8
1 pous, foot of 3 hands	15 (daktylos)	Athenian Greek	316 mm	4		1	.8
1 pygon, remen	20 (daktylos)	Greek	385.5 mm	5	1.25	1.25	1
1 pechya, cubit	24 (daktylos)	Greek	462.6 mm	6		1.5	1.1
1 cubit of 17.6" 6 palms	25 (fingers)	Egyptian	450 mm	6		1.5	1.3
1 cubit of 19.2" 5 hands	25 (fingers)	English	480 mm		5	1.62	1.3
1 mh royal cubit	28 (dj)	Egyptian	525 mm	7		2.33	1.4
1 bema	40 (daktylos)	Greek	771 mm	10		2.5	2
1 yard	48 (finger)	English	975.36 mm	12		3	2.4
1 xylon	72 (daktylos)	Greek	1.3878 m	18		4.55	3.64
1 passus pace	80 (digitus)	Roman	1.542 m	20		5	4	1
1 orguia	96 (daktylos)	Greek	1.8504 m	24		6	5		1
1 akaina	160 (daktylos)	Greek	3.084 m	40		10	8	2
1 English rod	264 (fingers)	English	5.365 m	66		16.5	13.2		1
1 hayt	280 (dj)	Egyptian	5.397 m	70		17.5	14		3
1 perch	1,056 (fingers)	English	20.3544 m	264		66	53.4		11
1 plethron	1,600 (daktylos)	Greek	30.84 m	400		100	80	20
1 actus	1,920 (digitus)	Roman	37.008 m	480		120	96	24	20
khet side of 100 royal cubits	2,800 (dj)	Egyptian	53.97 m	700		175	140	35
iku side	3,600 (ŝushi)	Mesop	60m		720	240	180	48	40
acre side	3,333 (daktylos)	English	64.359 m	835		208.71	168.9
1 stade of Eratosthenes	8,400 (dj)	Egyptian	157.5 m	2100		525	420	84	70
1 stade	8,100 (shushi)	Persian	162 m		2700	900	525	85
1 minute	9,600 (daktylos)	Egyptian	180 m	2400		600	480	96	80
1 stadion 600 pous	9,600 (daktylos)	Greek	185 m	2400		600	480	96	80
1 stadium625 pes	9,600 (daktylos)	Roman	185 m	2400		625	500	100
1 furlong 625 pes	10,000 (digitus)	Roman	185.0 m	2640		660	528	132	88
1 furlong 600 pous	9900 (daktylos)	English	185.0 m	1980		660	528	132	88
1 Olympic Stadion 600 pous	10,000 (daktylos)	Greek	192.8 m	2500		625	500	100
1 furlong 625 fote	10,000(fingers)	Old English	203.2 m	2500		635	500	100
1 stade	11,520 (daktylos)	Persian	222 m	2880		720	576	144	120
1 cable	11,520 (daktylos)	English	222 m	2880		720	576	144	120
1 furlong 660 feet	10,560 (inches)	English	268.2 m	2640		660	528	132	110
1 diaulos	19,200 (daktylos)	Greek	370 m	4800		1,200	960	192	160
1 English myle	75,000(fingers)	Old English	1.524 km		15000	5,000	4000	800
1 mia chilioi	80,000 (daktylos)	Greek	1.628352 km	20,000		5,000	1000
1 mile	84,480 (fingers)	English	1.628352 km	21,120		5,280	4224	1056	880
1 dolichos	115,200 (daktylos)	Greek	2.22 km	28,800	7,200	5760	4800
1 stadia of Xenophon	280,000 (daktylos)	Greek	5.397 km	70,000		17,500	1400	3500
1/10 degree	560,000 (daktylos)	Greek	10.797 km	140,000		35,000	2800	7000
1 schϓnus	576,000 (daktylos)Z	Greek	11.1 km	144,000		36,000	288000	28800	24000
1 stathmos	1,280,000 (daktylos)	Greek	24.672 km	320,000		80,000	64000	16000
1 degree	5,760,000 (digitus)	Roman	111 km	1,440,000		360,000	288000	72000	60000

1 daktulos (pl. daktuloi), digit

= 1/16 pous

1 condulos

= 1/8 pous

1 palaiste, palm

= ¼ pous

1 dikhas

= ½ pous

1 spithame, span

= ¾ pous

1 pous (pl. podes), foot

≈ 316 mm, said to be 3/5 Egyptian royal cubit. There are variations, from 296 mm (Ionic) to 326 mm (Doric)

1 pugon, Homeric cubit

= 1¼ podes

1 pechua, cubit

= 1½ podes ≈ 47.4 cm

1 bema, pace

= 2½ podes

1 khulon

= 4½ podes

1 orguia, fathom

= 6 podes

1 akaina

= 10 podes

1 plethron (pl. plethra)

= 100 podes, a cord measure

1 stadion (pl. stadia)

= 6 plethra = 600 podes ≈ 185.4 m

1 diaulos (pl. diauloi)

= 2 stadia, only used for the Olympic footrace introduced in 724 BC

1 dolikhos

= 6 or 12 diauloi. Only used for the Olympic foot race introduced in 720 BC

1 parasanges

= 30 stadia ≈ 5.5 km. Persian measure used by Xenophon, for instance

1 skhoinos (pl. skhoinoi, lit. "reefs")

= 60 stadia ≈ 11.1 km (usually), based on Egyptian river measure iter or atur, for variants see there

1 stathmos

≈ 25 km, one day's journey. May have been variable, dependent on terrain

For variant, the stadion at Olympia measures 192.3 m. With a widespread use throughout antiquity, there were many variants of a stadion, from as short as 157.5 m up to 222 m, but it is usually stated as 185 m.

The Greek root stadios means 'to have standing'. Stadions are used to measure the sides of fields.

In the time of Herodotus, the standard Attic stadion used for distance measure is 600 pous of 308.4 mm equal to 185 m. so that 600 stadia equal one degree and are combined at 8 to a mia chilioi or thousand which measures the boustredon or path of yoked oxen as a distance of a thousand orguia, taken as one orguia wide which defines an aroura or thousand of land and at 10 agros or chains equal to one nautical mile of 1850 m.

Several centuries later, Marinus and Ptolemy used 500 stadia to a degree, but their stadia were composed of 600 Remen of 370 mm and measured 222 m, so the measuRement of the degree was the same.

The same is also true for Eratosthenes, who used 700 stadia of 157.5 m or 300 Egyptian royal cubits to a degree, and for Aristotle, Posidonius, and Archimedes, whose stadia likewise measured the same degree.

The 1771 Encyclopædia Britannica mentions a measure named acæna which was a rod ten (Greek) feet long used in measuring land. — Preceding unsigned comment added by 142.0.102.117 (talk) 20:48, 18 May 2014 (UTC)

Dilution ratio and dilution factor

Hi there,

I am confused the following line in the article "There is often confusion between dilution ratio (1:n meaning 1 part solute to n parts solvent) and dilution factor (1:n+1) where the second number (n+1) represents the total volume of solute + solvent. "

I thought n was already solute + solvent as the example (1mL+4mL water = 5 has shown). That line just adds more confusion. About the difference between dilution ratio and dilution factor; here is what I know. When in biology, we are ask to prepare a 1:100 dilution - I assume that this is the Dilution Ratio The Dilution Factor - for me, it is the 100 above. So if I write 1:a dilution ratio, then a is the dilution factor. For example, a 7.56 M of acid following a 1:100 dilution ratio, the answer would be .0756M which is easily obtained by having 7.56/100 where the denominator is the dilution factor. Of course one may argue that the same result can be obtain by having the stock concentration * Dilution ratio. Essentially, we are adding 1mL of acid + 99 mL of water to dilute the acid.

Would like an academic to review my workings. Flowright138 (talk) (contributions) 10:28, 14 March 2012 (UTC)

This sort of a problem comes up when studying climate change and rising sea levels. Ice loads the poles as a solid, melts into the sea as fresh water diluting salt water. The coefficients of expansion of the fresh and salt water are influenced by temperature and pressure. Warm salt water sinks to the bottom where there is more pressure but cold fresh water stays at the top for a while, then becoming warmer and more saline begins to sink. As the ice on the land melts and flows into the sea it relieves the load on the land while adding to the load on the thinner ocean plates so that there is isostatic rebound reflecting the changed loading.

Now the acidity of the oceans may be affected by the levels of atmospheric carbon in the greenhouse gas emissions so that's another factor to take into account, also winds tend to pile the seas up in the direction of the prevailing wind and relieve them in the opposite direction but weather events such as el Nino can reverse the prevailing winds affecting sea levels. Also surface water near land is attracted more strongly gravitationally than surface water over great depths. The dilution ration thus has to take into account state, relative frozen or liquid height when floating, temperature, pressure,gravitational attraction, acidity, lateral loading from wind and all of these factors become variables in your dilution ratio equation. Given a body of water the size of the earths oceans its probably also necessary to consider the relative humidity of that portion of the water which is in suspension in the atmosphere, and perhaps also tidal forces.Your formula will eventually have as many or more variables than the computation of seismic forces caused by the changed loading on the mantle.Metaphysical Engineering (talk) 19:17, 26 May 2014 (UTC)