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/
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