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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

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!
Associated Topics:
Middle School Division
Middle School Fractions
Middle School Terms/Units of Measurement

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