Definition of Ratio

Date: 08/21/2003 at 13:59:03
From: Suzanne
Subject: The correct definition of "ratio"

I am a copy editor for a public-relations firm that works with clients 
in high technology. Recently, I lost a battle with other editors on 
the staff who insisted that a construction such as "eight out of ten" 
is a ratio. I said I didn't think it met the mathematical criteria for 
ratio, but since I don't know precisely how "ratio" is defined by 
mathematicians, I was unable to argue persuasively - and don't know 
for a fact that my co-editors aren't right, that "eight out of ten" 
(or 8 out of 10) IS a ratio. Can you help? 

Thank you.

Date: 08/21/2003 at 17:07:19
From: Doctor Peterson
Subject: Re: The correct definition of "ratio"

Hi, Suzanne.

This is as much an English language question as a math question, and 
that makes it very confusing. Words like this are not used as 
consistently as you might expect, even among math teachers or 
mathematicians. (That's partly because mathematicians today don't 
tend to pay much attention to ratios, and therefore don't have to 
define them carefully.)

My first impression is that we TEND to think of ratios as comparisons 
of, say, the number of boys to the number of girls, rather than of a 
part to a whole, but that the term "ratio" does not necessarily 
exclude the latter. So it might not be technically wrong to use it 
that way, but depending on the context there might be clearer ways to 
phrase what you want to say, so as to avoid suggesting that the 
comparison is part to part.

Then I did a little searching.

For word questions, I like to see what a dictionary says, since 
lexicographers know how to make distinctions between words (though 
they often don't understand the mathematical distinctions). Merriam-
Webster (m-w.com) says

  ratio 1 a : the indicated quotient of two mathematical expressions
          b : the relationship in quantity, amount, or size between
              two or more things : PROPORTION

This agrees with my general sense of the word: any quotient can be 
called a ratio, but in particular it tends to compare two distinct 
things. But, of course, we make a distinction between ratio and 
proportion, and they call them synonymous. How do they define the 
latter?

  proportion 3 : the relation of one part to another or to the whole
                 with respect to magnitude, quantity, or degree :
                 RATIO
             4 : SIZE, DIMENSION
             5 : a statement of equality between two ratios in which
                 the first of the four terms divided by the second
                 equals the third divided by the fourth (as in
                 4/2=10/5)

Their definition 5 is the one we use technically, in distinction to 
ratio. But look at their definition 3: when they consider this 
synonymous with ratio, they specifically include relations of part to 
whole as well as part to part. That would say that your example IS a 
ratio.

How about math sites? Mathematicians, as I said, don't deal with this 
much, but math teachers do. Here is one reference I found:

    Ratio. Fraction. What's the Difference?
    http://www.sci.tamucc.edu/txcetp/cr/math/rf/RatioFraction.pdf 

    Due to common notation, students often improperly interchange
    the ideas of ratio and fraction. Through this lesson students
    will learn why the two are different ideas and when they
    actually can overlap.

    Discuss the fact that fractions always illustrate a "part to
    whole" relationship while ratios can be used to illustrate a
    much larger set of relationships; such as part to part and whole
    to part.

This seems to say that a fraction always refers to part of a whole, 
but a ratio can indicate a variety of relationships. There are some 
examples in a worksheet, but no answers. The next page I found 
happened to be the answer sheet, and illustrates what they mean:

    http://www.tamu-commerce.edu/coas/math/FACULTY/WEBSTER/Math351/
HEIDISTU/RatioFra.htm

    2. Two out of every five students in this class plan to be
       middle school teachers. 

               Circle One:     Ratio        Fraction        [Both] 
    Reasoning: 

    This is definitely a ratio of future middle school teachers to
    total students in the class of 2:5.  In addition, since the
    ratio is part to whole, it can also be thought of as a
    fractional relationship. 2/5 of the students in the class plan
    to be middle school teachers.

So "two of every five" is considered a ratio.

How about Math Doctors? Here is what one says, which uses ratio just 
as broadly:

  Ratios as Fractions
  http://mathforum.org/library/drmath/view/58014.html 

  At age 12 you have probably spent 8 years in grades K through 7.
  There is a ratio there, namely, the ratio of "number of years in
  school" to "number of years in your whole life".  For you this
  ratio is 8:12 or 8-to-12. You could simplify that to 4:6 and
  further to 2:3.  The same thing can be done with the fraction 8/12
  which is the fraction of your life you have spent in school 
  (including summers). You reduce that fraction to 4/6 or 2/3.
  Either way, as the ratio 2:3 or the fraction 2/3, it says you
  have spent two thirds of your life as a person of school age.

But this one seems to have a strong preference for letting "ratio" 
only compare two parts of one whole:

  Ratios: Second Number
  http://mathforum.org/library/drmath/view/58027.html 

But I found lots of pages that talk about ratio of part to whole, as 
well as part to part.

So it looks like you've lost - but I'd still like to see the context 
in which you don't think the word "ratio" fits, because even if 
something CAN be called a ratio, there may be reasons not to use the 
word "ratio" in a particular sentence.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/21/2003 at 17:26:19
From: Suzanne
Subject: Thank you (The correct definition of "ratio")

Wow, thank you for the incredibly thorough explanation you so quickly 
provided. I admit to be being math-phobic, but I found your examples 
illuminating. Thanks again, and I'll quit arguing with my co-workers. 
Or find something else to argue about, more likely.