# Quotient vs Ratio vs Fraction

I am mathematician for whom English is the second language. In general I feel like I do not have major problems keeping up with math vocabulary, whether I am reading an article or giving a lecture. That being said, there is one question which have been bothering me for quite a long time. I am not exactly clear on the nuances of the meaning the three words from the title. To be more specific, the essences of my concerns can be articulated as

Can I use words "quotient", "ratio", and "fraction" interchangeably?

I am mostly concerned about using them within math academia environment, although more general rules would also be interesting to see. In particular, which of the words from the list would be appropriate to use when talking about dividing one function by another one?

Ultimately, even laconic yes/no answer will be greatly appreciated. However, I am also somewhat curious to see detailed explanation of how each of these words is different from the others.

PS As mentioned in the comment section by @BrianDonovan, my list of three words can probably be extended with the word "proportion".

EDIT: Following advices of @Rathony, I include the outline of Wikipedia definitions of each of these words:

• In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second
• A fraction (⋯) represents a part of a whole or, more generally, any number of equal parts.
(⋯)
The word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers (⋯)
• (⋯) a quotient (⋯) is the result of division.

Let me point out once again that I am particularly interested in the differences in use of these words in math academic environment, e.g. about the case when both enumerator and denominator of a fraction are general functions or algebraic expressions of some sort.

• Might proportion belong in the list too? Dec 21 '15 at 15:38
• @BrianDonovan It surely does, thank you! I did not even think of this one =\
Dec 21 '15 at 15:40
• Related question in Mathematics Educators beta, How to explain the difference between the fraction a / b and the ratio a : b?. I think this question is on-topic there.
– user140086
Dec 21 '15 at 15:40
• @FumbleFingers I don't see any difference between the OP's question and this one.
– user140086
Dec 21 '15 at 15:58
• I would advise you to post your question to the site I linked. They are experts in mathematics and you said you are a mathematician. I hope you didn't take it personally. I just wanted to guide you to write a better/more on-topic question. You can always ask on-topic questions related with English Language and its Usage here. Hope to see you again here. Merry Christmas!!!
– user140086
Dec 21 '15 at 17:13

Common usage observes distinctions that mathematics does not. The numeral 2 is a valid quotient, but not a fraction or ratio, although 2/1 and 2:1 are implied mathematically. Quotients may be in decimal form, which is not a proper fraction or ratio. Quotient is almost exclusively used in explicitly mathematical contexts.

In normal usage, fractions compare parts to wholes. "A small fraction of Green Party candidates support declaring Jupiter as an enemy planet" compares the in group (Jupiter is an enemy) from the whole (all Green Party candidates).

A ratio is normally used to distinguish between two groups, without requiring that the whole is addressed. "Ironically, Republicans favor mandatory tea drinking in a 2-to-1 ratio over tea party members" is not addressing the whole of all political party members, but specifically comparing the two groups.

• Thank you for the answer, it makes the distinctions between words much clearer for me, particularly within the context of your examples. Does the same logic applies in a classroom when we are talking about dividing functions?
Dec 21 '15 at 17:24
• Yes. You would talk about the quotient as the result of the division, and the fraction as the pre-computed resulted. In h() = f()/g(), h() is the quotient, and f()/g() is the fraction. Ratio is unusual to talk about in that context, unless you're describing change. For example, "The ratio of f^2() to g() approaches infinity faster than f() to g() for f() > 1", means you don't have to redefine a separate h() and h2() (for when f() is squared). Dec 21 '15 at 17:54

A quotient is the result derived from dividing two numbers. For example, dividing 6 by 3 gives a quotient of 2.

A ratio is the quantity of one thing relative to the quantity of another. For example, Georgia Tech's guy-to-girl ratio is 2:1. This means that there are twice as many guys as there are girls at Georgia Tech. Another way to look at this is saying that for every girl at Georgia Tech, there are two guys.

A fraction is very similar to ratio but not quite. A fraction is a way to describe a quantity in parts of a whole, a whole, or more than a whole. But it is similar to a ratio since it describes quantities in parts, or relative to another quantity (what constitutes a "whole"). For example, after the pizza is cut up into eight pieces, eating one will remove 1/8 of the pizza. One piece is only one of eight pieces that make up the "whole" pizza.

So I guess the short answer is no, that you cannot use these words interchangeably. They each cater to different scenarios. Perhaps, the closest to being used interchangeably are ratio and fraction.

• Thank you for the prompt answer! I am also curious whether these words are still not entirely interchangeable when used to describe fraction composed of functions, not just integer numbers?
Dec 21 '15 at 16:10
• Are you talking about 1/f(x)? If not, please provide an example. Dec 21 '15 at 16:23
• I had in mind a bit more general case, something like f(x)/g(x) for arbitrary functions f and g
Dec 21 '15 at 16:26

Ratio is a relationship between numbers or magnitudes with respect to their relative size. Specifically one number can be multiple of the other, part of it or parts of it (two times, one third part, three fourth parts etc). Quotient is a number that names the ratio between two numbers (quotient equal to 1/3 indicates the ratio of one third part). Quotient (number) has the same ratio to 1 as dividend has to divisor. So proportionaly dividend:divisor=quotient:1. Proper fraction is a number less than 1. The numerator has the same ratio to the denominator as fraction has to 1. For example 2 is two thirds of 3. 2/3 is two thirds of 1.

• Welcome to ELU.SE.This site strives to provide objective answers. Take the site tour or have a look at the help center to find out more about good answers. It would be great if you could find references to bolster your answer. Sep 29 '16 at 12:35

Firstly, let us take a look at the definition of these 3 terms.

Quotient

noun

a result obtained by dividing one quantity by another

Ratio

noun

the quantitative relation between two amounts showing the number of times one value contains or is contained within the other

e.g. the ratio of women's job to men's is 8:1

Fraction

noun

a numerical quantity that is not a whole number

e.g. 1/2, 0.5

Although they might seem similar mathematically but their definition and expression differ from one another. I believe they cannot be used loosely in place of one another, the context or the meaning of the sentence might differ. For example:

Can I have the quotient of the number of pies sold to cupcakes?

Can I have the ratio of the number of pies sold to cupcakes?

Can I have the fraction of the number of pies sold to cupcakes?

In this case, I'm sure you can now tell the difference of the three words, and whether they can be used interchangeably with one another.

• Your last examples could be improved by providing answers to each question. What I mean is, if 10 pies and 5 cupcakes were sold, the quotient of pies and cupcakes is 2, the ratio is 2:1, and the fraction is kind of nonsensical in this context, but if you really wanted to force it, you could say 1/2, i.e. half as many cupcakes as pies. The point is, the questions have very different answers, because these three words are not interchangeable. Dec 21 '15 at 15:58
• Thank you for the answer. To be completely honest I do not think I get the nuances of the differences among words' meanings. I feel like each of three example sentences have slightly different tone to it, but I fail to formally articulate unique characteristics of each of the words in question.