Expert: Josh - 1/16/2008

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.

1. 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.

2. If percentages are unitless in this situation, is there a reason?

I thank you for your reply and answer.  

Answer
Dear Kenneth,

1. The multiplier in front of the amount should be a scalar quantity. I know that tax rates are often quoted in X cents per dollar, it is best to write 6% as 0.06. There is no need for any unit.

2. We don't bring a story into the calculation. Mathematical operations are true regardless of the context. It is up to the person who carries out the calculation to know what they mean by one unit and what is being modeled.

In applied mathematics, see physics and chemistry for instance, units may be included to impart meaning on different quantities. This is sometimes useful in analysis.

On its own, a percentage is relative to 1, i.e., one unit of something without the need of specifying what it is.