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### Rate vs. Ratio

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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?
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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.

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/
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Associated Topics:
Middle School Ratio and Proportion

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