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Have you ever thought about what these two words really mean? They are often used together as a phrase, "ratio and proportion", but are they in fact different terms for the same mathematical concept? If asked the difference by a pupil, how might you respond?

I have to admit that I did not consciously reflect on the exact meaning of ratio or proportion until I began my teacher training. I suspect that this won't come as a particular surprise, but should I be worried about the definitions of these terms? One of the difficulties we are often faced with in the classroom is familiarising children with the mathematical definition of a word which is also
used in everyday language. Perhaps this applies to some extent to ratio and proportion? Somewhere in the depths of my mind, I seem to remember being told that ratio compares part to part whereas proportion compares part to whole. But what does that really mean? Is this helpful? And is it the whole story?

Let's look at ratio first. In my mind, ratio is the comparison between two or more quantities. According to the Oxford English Dictionary online, ratio is 'the relation between two similar magnitudes in respect of quantity, determined by the number of times one contains the other (integrally or fractionally)'. For example, on a bottle of orange squash it
might say "dilute one part concentrate to four parts water". The amount of water required is given in terms of the amount of concentrate. The National Numeracy Framework suggests that, when first introduced to children, this idea might be better expressed as "for every 1 part concentrate, we need 4 parts water". This ratio can be illustrated very clearly
using simple pictures:

In the classroom, "for every " can be modelled by actually drawing 4 "water rectangles" next to every "concentrate rectangle" so that pupils will be able to decide how many parts water are needed for a certain number of parts concentrate. At a higher level, they will then be in a position to confirm whether or not a given pictorial representation describes the same ratio. Of course, having orange and white counters or cubes would be another way of depicting the concentrate and water. It is not too much of a leap then to introduce slightly different vocabulary for the same thing ? "4 for every 1" can also be expressed as "4 to every 1".

In discussion with colleagues, we also realised that when we talk about ratios, it is perfectly appropriate to ignore units. We might say that the ratio of apples to pears is 3 to 1 and this goes against the grain in terms of consistency of units. Surely this can only make the understanding of ratio more problematic?

Mathematical dictionaries often include the word "fraction" in their definition of ratio. For example, the Dictionary of Mathematics published by McGraw-Hill (2003) defines the ratio of two quantities, A and B, as 'their quotient or fraction A/B'. So how does proportion fit in with this? The Numeracy Framework indicates that by the end of Year 6, children should be able to 'relate fractions to simple proportions'. So, it seems that fractions are related to proportion too.

Looking again at the Oxford English Dictionary, we find proportion defined as 'a portion or part in its relation to the whole; a comparative part, a share; sometimes simply, a portion, division, part'. At first sight, this would seem to agree with my original hunch. If we look back at the image above, we can describe the same situation in terms of proportion: there is 1 part concentrate in every 5 parts. Put slightly differently, we might say 1 in every 5 parts is concentrate. This time we are relating the amount of concentrate (1 part) to the whole (5 parts).

However, if we turn to a mathematical dictionary again, we are told that 'the proportion of two quantities is their ratio' (McGraw-Hill, 2003). The Collins Dictionary of Mathematics (2002) expands on this a little outlining proportion as the 'relationship between four numbers or quantities in which the ratio of the first pair equals the ratio of the second pair'. I think this latter mathematical definition might encompass my everyday use of the word 'proportion' and this is not comparing part to whole at all.

Where does this leave us? I struggle to make any conclusions from the above -the boundaries between the two seem very blurred to me. At best, I feel happy with my understanding of ratio but it appears that the word proportion is used in two different ways. I would be very keen to hear your own thoughts on this matter which I would add to this article. Perhaps you have clear definitions in your own mind that might help?

If you can shed light on "ratio and proportion", let us know by emailing primary.nrich@maths.org

Borowski, E. J. & Borwein, J. M. eds (2002) Dictionary of Mathematics . Glasgow: Harper Collins Publishers.

DfEE (1999) National Numeracy Strategy Framework for teaching mathematics from Reception to Year 6 . Sudbury: DfEE.

Geller, E. ed (2003) Dictionary of Mathematics. New York: McGraw-Hill Education.

Feedback

Thank you to Phil West, a teacher at MEF High School in Istanbul who sent us his comments on ratio and proportion. Click here to read what Phil had to say.

Veronica Bates from Colchester also sent us her thoughts which you can read here. Many thanks.

Thank you also to Bruce Moody, from New Zealand, who has contributed these ideas.

Debby Sanjaya, from Indonesia, emailed to say:

I just want to share my understanding. I think proportion is a ratio of the quantity of a part in its relation to the whole." She suggests that a proportion is different to ratio in that "... a proportion can not be less than 0:1 and can not be more than 1:1.

Someone who remained anonymous wrote:

I would argue, ratio is part to part and proportion is part to whole. They mean different things.

I would argue, ratio is part to part and proportion is part to whole. They mean different things.

What is 25% as a ratio? I would say it is 1:3 Others claim 1:4. I disagree.

I claim the ratio is 1 to 3 or 1: 3 and the proportion is 1 in 4. ":" mean to and not in."

Mark also sent us an email. You can read what he had to say here.

Someone else found a useful bit of information on http://www.math10.com/en/algebra/ratio-proportion/proportion-2.html

John Block wrote:

The Greeks thought all quantities were rational (note the root word "ratio") and could be written in the form A/B, where A and B are relatively prime integers. This understanding seems to point to the Dictionary of Mathematics definition of a ratio being A/B.

A proportion or proportional situation occurs when two things are related in such a way that the ratios of corresponding parts are equal. It would then seem that the Collins Dictionary of Mathematics has the better definition, "... the ratio of the first pair equals the ratio of the second pair". This also seems to stress the importance of similarity between the objects or situations.

Still, there is some ambiguity in the use of the word "proportion". For instance, two situations can be inversely proportional, then A/B does not = k, but AB = k. Perhaps this is merely an incorrect use of the word "proportional". Regardless, "proportional" in a description is always assumed to be "directly" proportional unless another word, "inversely" is present. I think proportionality
and the ability to express proportionality mathematically is a major idea in a student's maths education.

Another closely related topic is rate or unit rate. Is a rate a ratio? Not in terms of consistency of units, but in terms of proportional situations it is. Is not 80 km/ 1 hr the same as 160 km/ 2 hr ? Or in a problem 80 km/ 1 hr = x km / 2 hrs.? Thinking of rates as ratios, which will fit into the proportion model, helps students solve problems involving rate.

Helen Lord wrote:

Hi I read your article with interest - the way I have come to terms with these two in my mind is that the ratio focuses on the relationship of the split of the whole into parts whilst the proportion shows the 'gap'- or 'space between' which needs to be maintained - almost like it provides the track or parallel lines the figures must travel along to maintain their relationship - am I way off the
mark?

Ruth Campbell, a mathematics PGCE student at Leicester University emailed to say:

I would say that ratio is the relationship between two quantities. Proportion is the a subset of this, in that the two quantities are a part and the whole of the same thing. The concept of proportional extends this idea to any two quantities that have a constant multiplicative relationship. This is obviously closer to the usage of ratio than proportion, but the word “rational” had already been
allocated elsewhere. So ratio is now just one way of representing a proportional relationship, with other ways being fraction, equation, graph, bar diagram etc.