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

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