Rate vs. Ratio

Date: 10/23/2001 at 10:50:02
From: Stacy Courtright
Subject: Rate vs. Ratio

Dr. Math -

In a class with other math teachers, we were discussing the relation 
of rate versus ratio. Are all rates ratios and are all ratios rates?

What is your take on this and how would you explain this?

Date: 10/23/2001 at 12:51:59
From: Doctor Peterson
Subject: Re: Rate vs. Ratio

Hi, Stacy.

I've seen different ways to distinguish these concepts, so I looked 
around to find some samples.

This page

   Math League, Ratio and Proportion
   http://www.mathleague.com/help/ratio/ratio.htm   

says

   A ratio is a comparison of two numbers. 

   A rate is a ratio that expresses how long it takes to do something,
   such as traveling a certain distance. To walk 3 kilometers in one
   hour is to walk at the rate of 3 km/h. The fraction expressing a
   rate has units of distance in the numerator and units of time in
   the denominator.

This approach is more or less what I use in real life; ratio is the 
general concept, and rate refers _usually_ to time (though not always, 
as I'll mention below). So all rates are ratios.

But it's common to make a different distinction:

   Jim Reed, Rate and Ratio
   http://dev1.epsb.edmonton.ab.ca/math14_Jim/math7/strand1/1208.htm   

(which, ironically, has a link to the page above) says

   Rate: A comparison of 2 measurements with different units. 

   Ratio: A comparison of numbers with the same units so units are
   not required. 

This makes ratio and rate both specific terms, so that neither is a 
special case of the other. That seems very orderly, but I don't like 
it, because (a) generic terms are very useful, in order to be able to 
talk about how we work with ANY entity of this sort, and (b) this 
doesn't fit common usage, as in "tax rate" or "interest rate," where 
both numbers have the same unit. Of course, math doesn't generally 
worry too much about common usage, but since I see no real 
mathematical advantage in this distinction, we might as well be 
realistic.

The next,

   MathSteps, Rates
   http://www.eduplace.com/math/mathsteps/6/e/   

says

   A ratio is a comparison of two numbers or measurements. The
   numbers or measurements being compared are called the terms of
   the ratio. A rate is a special ratio in which the two terms are
   in different units. For example, if a 12-ounce can of corn
   costs 69 cents, the rate is 69 cents for 12 ounces. The first term 
   of the ratio is measured in cents; the second term in ounces.

This makes ratio generic again, satisfying one of my objections, but 
requires rates to have different units. We could allow that to be the 
mathematical definition, and not worry about common usage; but it 
doesn't quite stand up to the sorts of things a mathematician thinks 
about. If you have a "rate" of one ounce per pound, does that become a 
"ratio" of 1:16 when you change units to ounces per ounce? In my mind, 
a change of units should not change what the ratio is called; it's a 
ratio between two quantities, not between the numbers currently being 
used to represent them. If we use one ounce of pigment per pound of 
paint, is that a rate or a ratio? I might call it a rate when I'm 
mixing it, because I think of it as "per" or "for each," but a ratio 
when I'm using it, because it just describes the mixture!

Finally, let's check a dictionary. According to Merriam-Webster,

   http://m-w.com/   

we have

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

   Rate 3 a : a fixed ratio between two things b : a charge, payment,
   or price fixed according to a ratio, scale, or standard: as 
   (1) : a charge per unit of a public-service commodity (2) : a
   charge per unit of freight or passenger service (3) : a unit
   charge or ratio used by a government for assessing property taxes
   (4) British : a local tax
   4 a : a quantity, amount, or degree of something measured per unit
   of something else b : an amount of payment or charge based on
   another amount; specifically : the amount of premium per unit of
   insurance

Though this isn't necessarily a mathematician's or teacher's 
definition, it supports my sense that "ratio" is generic, while "rate" 
applies to several slightly different kinds of ratios. A rate 
generally involves a "something else," either two different kinds of 
units (such as distance per time), or just two distinct things 
measured with the same unit (such as interest money per loaned money).

I don't know whether that helps, but at least it may let you move on 
to more significant issues.

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