Ratios as Fractions

I’m having trouble understanding how fractions relate to ratios. A ratio like 3:5 isn’t directly related to the fraction 3/5, is it? I see how that ratio could be expressed in terms of the two fractions 3/8 and 5/8, but 3/5 doesn’t seem to relate (or be useful) when considering a ratio of 3:5.

Many textbooks I’ve seen, when introducing the topic of ratios, say something along the lines of “3:5 can be expressed in many ways, it can be expressed directly in words as ‘3 parts to 5 parts’, or it can be expressed as a fraction 3/5, or it can be…” and so on. Some textbooks will clarify that 3/5, when used this way, isn’t “really” a fraction, its just representing a ratio. This makes absolutely no sense to me. Why express 3:5 as 3/5 at all?

• Fractions are ratios. Aug 8 '14 at 11:44
• From wikipedia: "In layman's terms a ratio represents, for every amount of one thing, how much there is of another thing. For example, supposing one has 8 oranges and 6 lemons in a bowl of fruit, the ratio of oranges to lemons would be 4:3 (which is equivalent to 8:6) while the ratio of lemons to oranges would be 3:4. Additionally, the ratio of oranges to the total amount of fruit is 4:7 (equivalent to 8:14). The 4:7 ratio can be further converted to a fraction of 4/7 to represent how much of the fruit is oranges". Aug 8 '14 at 11:47
• @Gitgud If there are 4 oranges and 7 lemons, then 4/11 of the fruit is oranges. I think this is the OP's source of confusion. Aug 8 '14 at 11:52
• Fractions represent rational numbers They are called rational numbers because they are ratios.
– MJD
Aug 8 '14 at 11:52
• Why is it that when basic questions about the nature of mathematics or physics appear, more often than not lengthy discussions about apples and oranges are involved? Aug 8 '14 at 16:21

The notation $a:b$ emphasizes a relative relationship between $a$ and $b$, and the notation $\frac{a}{b}$ emphasizes an operation on two elements $a$ and $b$.

But ultimately the two symbols represent the same thing (at least, when using integers): a relative size between $a$ and $b$. If you think carefully about what it means for two ratios to be equivalent, you'll find that the definition of equality of $\frac{a}{b}$ and $\frac{c}{d}$ is just that the ratios $a:b$ and $c:d$ are equal.

Actually, I do believe that I've seen posters from certain countries actually use "$:$" for division, to the confusion of the rest of us.

One difference in these two notations is that you can link a lot of ratios together at once like this: $1:2:4:7$. This expresses a bunch of ratios at once: $1:2$, $2:4$, $4:7$, $1:4$, $2:7$ etc. If these were ratios of ingredients in some mixture recipe, then you could rather handily increase and decrease the size of your recipe as you desired using this notation.

But this does not translate over to the slash notation, which becomes problematic if you're thinking of the slash as an operation.

This is a bit of a reach, but one way to think of it is that $a:b$ is kind of like "a division operation you are postponing." This is why you can stack them together because no operation is intended. (If you used slashes, the urge would be to carry out the operations until you have a single fraction, but this would require parentheses to make the expression unambiguous.)

In general I would not compare ratios with fractions, because they are different things. As rschwieb mentioned in the comments: when you have apples, lemmons and oranges, you can say that the ratio is $3:4:5$. The only "link" you can make to fractions is that you can now say that this the same as saying the ratio is $\frac35:\frac45:1$. From which you can very easily see that there are $\frac35$ as much apples as there are oranges. You see, the fraction is used to express a relation between two "elements" of your whole whatever.

The key thing is this: when you have a bowl with $3$ oranges and $5$ apples, the ratio $3:5$ gives you the relation between the amount of oranges and the amount of apples. And so does the number $\frac35$ within this context. Confusion arrises from the fact that we are trained to immedately conclude that a fraction, alway gives you a relation between the whole of someting and a part of the whole. This is not always the case though as you can see from the ratio story.

N.B. I would like to add that I personally feel that someone or some method that is teaching mathematics should avoid the comparrison of ratios with fractions, because it generally leads to confusion as it did for the OP. Someone in the process of learning ratios is very likely not ready to be confronted with such a subtle, yet major difference in the application of fractions.

• I don't know about the N.B. This seems like a case where you should bite the bullet, not dodge the bullet. Maybe not on the week you introduce ratios and fractions, but this would certainly worthwhile to draw comparisons after the students are familiar with both. Avoiding something like this is sheltering them from thinking, which isn't something we want to do :) But yeah, that all comes with the caveat that the comparison isn't presented prematurely to students who are still getting the hang of fractions and ratios. Aug 8 '14 at 12:19
• @rschwieb Yes it does. Thanks for the heads up. And yes you are right that the keyword in my N.B. part is prematurely :) Aug 8 '14 at 12:33

"Why express 3:5 as 3/5 at all?"

There is a way 3/5 can be a useful representation of the ratio 3:5, as described in this excerpt from wikipedia (I added the boldface):

"If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of the pieces of fruit are oranges. If orange juice concentrate is to be diluted with water in the ratio 1:4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1/4 the amount of water, while the amount of orange juice concentrate is 1/5 of the total liquid. In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason."

I think I've fallen into that beginner mistake category. As long as it is clear what is being compared to what, representing a ratio of 3:5 as a fraction (3/5), can make sense and be useful. I was jumping to the conclusion that a fraction is always comparing "number of parts" to "total number of parts that make a whole". You can also have a fraction which directly compares "number of parts of type A" to "number of parts of type B".

So for example, say you are told there are 12 apples in a bin (this bin contains both apples and oranges). You are also told that the ratio of oranges to apples is 1:6. You are asked, "how many oranges must be in this bin?" You can find the answer by turning the ratio into a fraction, 1/6. You can understand this fraction as, "there are 1/6 as many oranges in this bin as there are apples." So doing the math, 12 * 1/6 = 2. There must be 2 oranges in the bin.