Expert: Abe Mantell - 1/18/2008

Question
QUESTION: Hello:

If the sales tax on merchandise sold in stores is 6%. The item is multiplied by the sales tax of 6%. For example, the sales tax on pencils worth $1.00 is 6% X $1.00 is $0.06 or 6 cents plus $1.00 for a total of $1.06.

If the tax is 6%/$1.00, why doesn't the dollar sign as a unit cancel after the multiplication?
Isn't it 6%/$1.00 X $1.00 or whatever the dollar amount is?
The dollar signs should cancel, but they do not.

I thank you for your reply.


ANSWER: Hello Kenneth,

It is not 6% per dollar, it is just "6%"...which amounts
to 6 cents per dollar.  Remember, percentages are unitless.
So, the percentage of any quantity retains the units of the
original quantity.  6% of $1 = (6/100) x $1.00 = $0.06

See?

Abe

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for your reply, but I do have two follow-up questions:

1. If percentages are unitless, is there a situation in which they are not?

2. I have the understanding that it is incorrect to have a number without units divided by a number having units as in (6/100)/$1.00, but  a number with units can be divided by a number without units as in 6 gallons/2 equals 3 gallons?

I thank you for your reply.  

Answer
1. Percentages, by definition, are unitless.

2. 6 gallons/2 = 3 gallons...that represents an ordinary
-- division, as in spliting the 6 units into 2 equal parts
-- (giving 3 gallons each).

BTW: I see you've been asking around, eh?  ;-)
http://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/kenneth1.html