Dividing Two Numbers with and without Units
Date: 09/15/2004 at 14:12:07 From: Kenneth Subject: Division With and Without Units It makes sense to divide a number or an amount, for example $500.00, by another number having no units (dollars in this example), such as 10, as in $500.00/10 = $50.00, and it also makes sense to divide $500.00 by $10.00, $500.00/$10.00 = 50, both amounts having units. However, what does it indicate when the divisor has the units (dollars) but not the dividend, as in 500 is divided by $10.00? Does 500/$10.00 make sense? 500/$10 as a ratio makes sense. This ratio could represent the number of 500 items for the cost of $10.00. 500 divided by $10.00 equals what? 50 or $50.00 will not provide a correct quotient. 500/$10.00 cannot equal either $50.00 or 50. I believe that $500.00/$10.00 represent measurement division and that $500.00/10 represent partition division. 500/$10.00 does not match either one of these definitions for division.
Date: 09/15/2004 at 15:36:25 From: Doctor Rick Subject: Re: Division With and Without Units Hi, Kenneth. Your observations are correct. It does sometimes make sense to divide a dimensionless quantity by a quantity that has units. What are the units of the quotient in this case? We see it sometimes in stores (especially dollar stores): "4 for $1". We can also say "4 per dollar". In science, such units are often written as "dollar^-1" (that's dollar with an exponent of -1) because 1/x = x^(-1). I can't say I've seen dollar^(-1) in use anywhere, but I'd understand it. What we have here is essentially a rate--which can perhaps be called a third kind of division according to your thinking. For instance, if a car goes 100 miles in 2 hours, its average speed is 50 miles per hour, or miles/hour. The result of dividing miles by hours is the new unit miles/hour. Similarly, if a factory produces 4,000 widgets in an 8-hour shift, its average rate of production is 500 widgets/hour: when we divide a number of widgets by a number of hours, we get widgets/hour. We don't usually think of "widgets" as a unit, so we can say instead that the production rate of widgets is 500 units/hour. Here, "units" stands for a dimensionless quantity--a number of widgets, or of anything else. One example of a real-life unit of the form we're discussing is the Hertz. This unit of frequency represents the rate at which something happens, such as one period of an electromagnetic wave; thus the old name was "cycles per second", but you would often see the units as "sec^(-1)" (1/second, or "units per second") because cycles are not really a unit. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 09/15/2004 at 21:01:26 From: Kenneth Subject: Thank you (Division With and Without Units) Hello Doctor Rick: I want to thank you for the reply and information. The Math Forum provides a great service for those seeking help and assistance!
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