Expert: Abe Mantell - 1/3/2010

Question
QUESTION: Hello:

If a 20 ounce fountain drink costs $0.89, and the sales tax is 7%, the conventional method for determining the sales tax would be to multiply $0.89 by the sales tax of 7% to equal $0.0623.

However, if I wanted to convert the sales tax to represent the cost per dollar, would the following calculation be correct?

(7% X $1.00)/$1.00) X $0.89/20 ounces = $0.07/$1.00 X $0.89/20 ounces = $0.0623/20 ounces

Total cost would be $0.9523 or $0.95/20 ounces.

I thank you for your reply.

ANSWER: Hello Kenneth,

While you get the correct number, technically it is not correct since the units
do not match.  $0.07/$1.00 is unit-less, so the right hand-side should be as well.
Here is another way, where units are consistent:
Drop ounces from the equation: $0.07/$1.00 = x/$0.89, where x=the sales tax in $'s,
then x=$0.0623

OK?

Abe


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

QUESTION: Hello Abe:

I want to thank you for your reply and answer.

I am somewhat confused by " it is not correct since the units do not match.  $0.07/$1.00 is unit-less, so the right hand-side should be as well." Why is this requirement necessary?

There is no right hand side of the equation yet if you are referring to the right hand side of the equal sign. It is true that the units cancel from $0.07/$1.00. As a result, the equation becomes 0.07/1 X $0.89/20 ounces. And this now equals $0.0623/20 ounces. The 20 ounces are not needed, but they are understood since a drink must contain some unit of measure.  If the equation is stripped down, it would be 0.07 X $0.89 = $0.0623, the sales tax.

I thank you for your follow-up reply.

ANSWER: Yes, the right-hand side (RHS) of an equation refers to the RHS of the equals sign, similarly
the left-hand side (LHS) refers to the LHS of the equals signs.

An equation "equates" two quantities or expressions, the LHS and the RHS.  Thus, they MUST
be equal...if dealing with quantities that include units, the units must also match.
For example, it would be incorrect to say that 5 apples equals 5 oranges, even though 5=5.
We would not say that 10 dollars equals 10 cents, would we?  Similarly, $0.07/$1.00 is
equal to just 0.07 (no units, since the dollars "cancel")...so the RHS must also be unitless.

For 0.07 X $0.89 = $0.0623, that is fine, since the 0.07 is unitless (it is really a percent),
so a unitless quantity times $'s equals $'s.

I hope this helps.

Abe


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

QUESTION: If you look at your proportion example you have $0.07/$1.00 = x/$0.89.
In order to solve for "x", a simple method is to cross multiply as follows:

$0.07 X $0.89 = $1.00 X ?

This becomes my original calculation which you indicated is "technically incorrect." The calculation is from my first message to you minus the units for the ounces.

$0.07/$1.00 X $0.89 = X

It is true that the dollar sign units cancel from $0.07/$1.00 but not from the $0.89. The final calculation becomes 0.07 X $0.89 = $0.0623  

Answer
You also had "/20 ounces"...so if you remove that from your equation, I am OK with it.
If you keep it in, then *technically* the units are consistent...but the numbers are OK.
(Yes, the $ signs "cancel" in $0.07/$1.00, leaving the $ sign from the $0.89).

Abe