# Percentages & Rates

#### Explain this!

##### Junior Member
At 5% per unit (5%/unit), how much does a salesman make if he sells 30 units at $10.00 per unit? I tried this, but I do not think that it is not correct: 5%/unit X 30 units X$10.00/units = 5%/unit X $300.00 =$15/unit. I do not think this is correct. "Units" cancel except in the percentage.

#### lev888

##### Senior Member
A percentage of X dollars should equal to dollars. Not sure what "/units" is about.

#### Explain this!

##### Junior Member
A percentage of X dollars should equal to dollars. Not sure what "/units" is about.
I saw the following: : "Let’s say that you pay someone on the sales team 5% for every unit up to $100,000." I based my question on this information. I could have misinterpreted it. I assumed 5% for every unit is the same as 5%/unit. I read it at the following: https://smallbiztrends.com/ I'm curious as to why a rate of commission is called a rate. 5%/unit appears to be a rate, but I cannot get it to function as one. #### lev888 ##### Senior Member I saw the following: : "Let’s say that you pay someone on the sales team 5% for every unit up to$100,000."

I based my question on this information. I could have misinterpreted it. I assumed 5% for every unit is the same as 5%/unit.

I read it at the following: https://smallbiztrends.com/

I'm curious as to why a rate of commission is called a rate. 5%/unit appears to be a rate, but I cannot get it to function as one.
Rate is a ratio of 2 quantities, which can have the same or different units.
E.g. different units: velocity, measured in miles/hour.
Same units: commission $/$.
When units are the same they cancel and we end up with dimensionless ratio (we discussed this before). A dimensionless ratio can be converted to a fraction with denominator 100. The numerator of such fraction is called .... percentage. E.g. your commission is 1 dollar per $20 of sales:$1/$20 = 1/20 = 5/100 = 5%. #### Subhotosh Khan ##### Super Moderator Staff member I saw the following: : "Let’s say that you pay someone on the sales team 5% for every unit up to$100,000."

I based my question on this information. I could have misinterpreted it. I assumed 5% for every unit is the same as 5%/unit.

I read it at the following: https://smallbiztrends.com/

I'm curious as to why a rate of commission is called a rate. 5%/unit appears to be a rate, but I cannot get it to function as one.
I went to the website but I could not find the "problem statement".

You say " 5% for every unit " - but that does not make sense. It could have said, 5% of the selling price for every unit, or some such statement associated with money.

#### Explain this!

##### Junior Member
Rate is a ratio of 2 quantities, which can have the same or different units.
E.g. different units: velocity, measured in miles/hour.
Same units: commission $/$.
When units are the same they cancel and we end up with dimensionless ratio (we discussed this before). A dimensionless ratio can be converted to a fraction with denominator 100. The numerator of such fraction is called .... percentage. E.g. your commission is 1 dollar per $20 of sales:$1/$20 = 1/20 = 5/100 = 5%. I want to thank you for your reply.! I understand all that you have indicated in your message. I have the firm understanding that a rate is a certain kind of ratio. When I think of a rate, I think of a fraction with numerator and denominator having different units. If I am told that the boy to girl ratio is 1 to 3, I would not think of this ratio as a rate of boys to girls being a rate of boys to girls of 1 to 3. In fact, most likely no one would call this a rate. This is a mystery to me. because I am thinking that a rate of commission has different units, but I cannot determine what they are. I understand the information in your reply, but I am still puzzled by the rate-ratio definition. If this were called a ratio of commission. I would not be asking these questions. My best guess at this point is that the rate of commission my be based on a percentage annually--5% percent per year, but this may be incorrect as well. Usually rates (such as interest rates) involve some element of time--per day, per year, etc. Thanks again for your efforts in explaining this to me!! #### Subhotosh Khan ##### Super Moderator Staff member At 5% per unit (5%/unit),....... One critical point missing.. At 5% (of "what") per unit I hope that - what - is not weight of the TV, not height of the TV........ 5% of what property of the TV? I am assuming it (property) is the price of the TV (it could be profit generated from sale) - it needs to be stated explicitly. #### Explain this! ##### Junior Member One critical point missing.. At 5% (of "what") per unit I hope that - what - is not weight of the TV, not height of the TV........ 5% of what property of the TV? I am assuming it (property) is the price of the TV (it could be profit generated from sale) - it needs to be stated explicitly. I want to thank you for the information to my posting. I understand your comments. I saw this at the Internet site listed in my first inquiry. I suppose that it would need to me 5% of some dollar amount, as in 5% per$100 dollars per unit or item sold. If the sales person sold 10 units at $20.00 per unit, that would be 5% X$200.00.

Do you think or believe that percentages are unit less? They do not have a denominator that is a unit of measure, as in 5%/year.

#### Explain this!

##### Junior Member
Rate is a ratio of 2 quantities, which can have the same or different units.
E.g. different units: velocity, measured in miles/hour.
Same units: commission $/$.
When units are the same they cancel and we end up with dimensionless ratio (we discussed this before). A dimensionless ratio can be converted to a fraction with denominator 100. The numerator of such fraction is called .... percentage. E.g. your commission is 1 dollar per $20 of sales:$1/$20 = 1/20 = 5/100 = 5%. Are you also indicating to me that a percentage is always unit-less? It's not 5%/$100.00, for example, but (5/100)/$100, which equals$5.00? I hope my math is correct.

#### lev888

##### Senior Member
Are you also indicating to me that a percentage is always unit-less? It's not 5%/$100.00, for example, but (5/100)/$100, which equals $5.00? I hope my math is correct. 5% of blah is 5/100 of blah. 5/100 is unit-less. 5%/$100 is not $5. If you want to calculate 5% of$100 you multiply 5% by $100: 5%*$100 = (5/100)*$100 =$5.
I don't know what 5%/$100.00 is. #### Explain this! ##### Junior Member So, no one would ever use 5%/$100.00 in mathematics even though it indicates 5% or 0.05 divided by $100.00? #### Subhotosh Khan ##### Super Moderator Staff member So, no one would ever use 5%/$100.00 in mathematics even though it indicates 5% or 0.05 divided by $100.00? "... no one would should ever use 5%/$100.00 in mathematics ...." ....................CORRECT

The correct use would be:

5% OF $100.00 =$5

"OF" indicates multiplication, and

"/ " indicates division

#### lev888

##### Senior Member
So, no one would ever use 5%/$100.00 in mathematics even though it indicates 5% or 0.05 divided by$100.00?
You wrote earlier: "I suppose that it would need to me 5% of some dollar amount, as in 5% per $100 dollars per unit or item sold." Can you explain why you first used "5% of" and then a few words later switched to "5% per"? The correct usage is "5% of". If instead of being given a percentage you are told "Your commission is$5 per $100 of sales", then you calculate the total like this: ($5/$100)*$300 = $15. Last edited: #### JeffM ##### Elite Member This is not to detract from the explanations already given to you, but a lot of your confusion comes from not labelling your units carefully. That is absolutely critical when using dimensional analysis. $$\displaystyle 5 \text { percent commission} \equiv \dfrac{5 \text { commission dollars}}{100 \text { sales dollars}}.$$ So $$\displaystyle \dfrac{30 \text { units sold}}{1} * \dfrac{10 \text { sales dollars}}{\text {unit sold}} * \dfrac{5 \text { commission dollars}}{100 \text { sales dollars}} =$$ $$\displaystyle \dfrac{30 \cancel {\text { units sold}}}{1} * \dfrac{10 \cancel {\text { sales dollars}}}{\cancel {\text {unit sold}}} * \dfrac{5 \text { commission dollars}}{100 \cancel{\text { sales dollars}}} =$$ $$\displaystyle \dfrac{30 * 10 * 5}{100} \text { commission dollars} = 15 \text { commission dollars.}$$ #### Explain this! ##### Junior Member Thank you so much! This is excellent! This calculation explains what I could not think of. Thank you again!! #### Explain this! ##### Junior Member "... no one would should ever use 5%/$100.00 in mathematics ...." ....................CORRECT

The correct use would be:

5% OF $100.00 =$5

"OF" indicates multiplication, and

"/ " indicates division

If 5%/$100.00 should not be used, what about 5%/year as seen in banking? Is 5%/year misleading? Should this be 5% of the dollar amount in account per year? I am beginning to think that the % sign should not be regarded as a unit of measure. #### Subhotosh Khan ##### Super Moderator Staff member Thank you for the reply! If 5%/$100.00 should not be used, what about 5%/year as seen in banking? Is 5%/year misleading? Should this be 5% of the dollar amount in account per year?
I am beginning to think that the % sign should not be regarded as a unit of measure.
% sign is NOT a unit of measure - it is UNITLESS.

#### lev888

If 5%/$100.00 should not be used, what about 5%/year as seen in banking? Is 5%/year misleading? Should this be 5% of the dollar amount in account per year? I am beginning to think that the % sign should not be regarded as a unit of measure. Create a problem that involves 5% per year rate and we'll take a look. #### Explain this! ##### Junior Member The only example that I can think of would be as follows. A person can borrow$100.00 at 5% per year. After one year, what is the interest?
5% X $100.00 =$5.00
This example is common. I suppose that from what has been explained to me it would be (5% of $100.00)/year. This would equal$5.00/year. The year unit would not cancel. I am getting the understanding that the % is not a unit of measure in most cases. (?)