Math CentralQuandaries & Queries


Question from Kenneth:


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, is there a reason?

I thank you for your reply.

Hi Kenneth,

A percent is just another way of writing a fraction, it is a fraction with a denominator of 100. Percent means per one hundred or over one hundred so 6% is 6/100 = 0.06. In a specific situation where there are units involved. as in your case with dollars, the tax is $6 per $100, that is you pay $6 tax for every $100 you spend. As a fraction this is $6/$100 = 0.06 or 6%.

I hope this helps,

Kenneth wrote back


I have a follow-up question for Harley.

I want to thank you for the reply and answer. I suppose that it is incorrect to say or write that the rate is 6% per $1 but instead $0.06 per $1.00 or $0.06/$1.00 because the terminology 6% per $1.00 would not be correct. It is 6 % OF $100 and this is $0.06/$1.00.

Do I have the correct idea?

I thank you for your follow-up reply and answer.


The preposition per in this situation means for each and designates a rate as in 60 miles per hour where the units are miles divided by hours or $125 per pound where the units are dollars divided by pounds. The prefix per in percent has the same meaning. Cent comes from the Latin word centum which means hundred so 6 percent or 6% means 6 for every 100 or 6/100 = 0.06. Since 6 and 100 are pure numbers without units so is 6%. Thus 6% per $100 would have units a pure number divided by dollars which is not what you want.

The preposition of in mathematics designated multiplication as in one-third of six is two, in other words 1/3 × 6 = 2. Thus 6% of $100 is 6/100 × $100 = $6 and the units are dollars as you want. Hence 6% of $100 is exactly what you want.


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