Sales & Commission

[Explain this!]

A salesman receives a $30 commission for selling $200 worth of items sold. How much would he receive if his sales are $100 worth?

The calculation would be as follows $30/$200 * $100 = $15.

What is the reasoning or logic for determining the answer using the above calculation?

If $30/$200 is reduced to commission per $1 the calculation would be as follows: $0.15/$1.00 * $100.

This is $0.15 for every $1.00 worth of items sold or for every $1.00 in $100.

What does $30/$200 * $100 indicate or represent in regards to the commission the salesman receives?

 

[lev888]

A salesman receives a $30 commission for selling $200 worth of items sold. How much would he receive if his sales are $100 worth?

The calculation would be as follows $30/$200 * $100 = $15.

What is the reasoning or logic for determining the answer using the above calculation?

If $30/$200 is reduced to commission per $1 the calculation would be as follows: $0.15/$1.00 * $100.

This is $0.15 for every $1.00 worth of items sold or for every $1.00 in $100.

What does $30/$200 * $100 indicate or represent in regards to the commission the salesman receives?

There is an unstated assumption that the commission is directly proportional to sales. This means commission = k * sales, where k is a constant. How do we find k if we are given sales and corresponding commission amounts? k = ?
Then you can use k to find the commission for any other sales amount.

 

[JeffM]

You write “if $30/$200 is reduced to commission per $1, the calculation would be as follows $0.15/$1.00 * 100.”

True.

What is also true is that for any three numbers a, b, c such that

[MATH]a = b \text { and } a * c = d \implies b * c = d.[/MATH]
[MATH]\dfrac{30}{200} = 0.15.[/MATH]
[MATH]\therefore \dfrac{30}{200} * 100 = \dfrac{0.15}{1} * 100.[/MATH]
EDIT: I agree with lev about the unstated assumption. My answer is based on the assumption‘s being valid. Of course, if the assumption is not valid, the question cannot be answered.

 

[Explain this!]

There is an unstated assumption that the commission is directly proportional to sales. This means commission = k * sales, where k is a constant. How do we find k if we are given sales and corresponding commission amounts? k = ?
Then you can use k to find the commission for any other sales amount.

Thank you for your reply!

So $30 = k * $200

$30/$200 = k

As you indicated: "Then you can us k to find the commission for any other sales amount."
If sales is then $100, then k * $100 = $30/$200 * $100 where k = $30/$200.

From what you indicated in you reply, and if I understand correctly, this where the calculation, $30/$200 * $100 is derived.

 

[Explain this!]

You write “if $30/$200 is reduced to commission per $1, the calculation would be as follows $0.15/$1.00 * 100.”

True.

What is also true is that for any three numbers a, b, c such that

[MATH]a = b \text { and } a * c = d \implies b * c = d.[/MATH]
[MATH]\dfrac{30}{200} = 0.15.[/MATH]
[MATH]\therefore \dfrac{30}{200} * 100 = \dfrac{0.15}{1} * 100.[/MATH]
EDIT: I agree with lev about the unstated assumption. My answer is based on the assumption‘s being valid. Of course, if the assumption is not valid, the question cannot be answered.

I want tot thank you for your reply.

I see this as follows. $30/$200 * $100 would suggest or represent that $200 divides into $100 1/2 times. This is seen in $30/1 * 1/$200 * $100.
!/200 of $100 is the same as dividing $100 by 200 if I am not mistaken.
In order to keep the $30/$200 proportional with $?/$100, it is necessary to take 1/2 or $30. I'm not sure why it is important to keep $30/$200 proportional by multiplying $30 by 1/2 with $15/$100: $15/$100 = $30/$200.

This explanation is less complicated but perhaps still a correct one.

 

[JeffM]

I want tot thank you for your reply.

I see this as follows. $30/$200 * $100 would suggest or represent that $200 divides into $100 1/2 times. This is seen in $30/1 * 1/$200 * $100.
!/200 of $100 is the same as dividing $100 by 200 if I am not mistaken.
In order to keep the $30/$200 proportional with $?/$100, it is necessary to take 1/2 or $30. I'm not sure why it is important to keep $30/$200 proportional by multiplying $30 by 1/2 with $15/$100: $15/$100 = $30/$200.

This explanation is less complicated but perhaps still a correct one.

The point is that we have three numbers, 30 = a, 1/200 = b, and 100 = c.

It is a fundamental rule of arithmetic

[MATH]abc = acb = bac = bca = cab = cba[/MATH].

How you permute factors makes no difference to the result.

[MATH]30 * \dfrac{1}{200} * 100 = \dfrac{30}{200} * 100 = 0.15 * 100 = 15.[/MATH]
[MATH]30 * \dfrac{1}{200} * 100 = \dfrac{1}{200} * 100 * 30= \dfrac{100}{200} * 30 = 0.5 * 30 = 15.[/MATH]
One way is not right and the other wrong. They are both right because they are equivalent. Whichever way makes the most intuitive sense to you is the one for you to use.

The reason that many people would do this problem starting with 30 /200 is that the problem presents itself that way: 30 dollar commission relative to 200 dollars in sales. So many people would start by writing that ratio down and then applying it to 100. You prefer to think that the significant ratio is 100 relative to 200 and apply that ratio to 30. It makes no mathematical difference in what order you multiply numbers: 3 times 7 gives the exact same result as 7 times 3. You are worried about a psychological difference with no mathematical significance.

 

[Steven G]

I would reduce $30/$200 to just 3/20. Note that the $ signs also canceled out. You can even express 3/20 as a percent (so you know your commission as a percentage). (3/20) = (3/20)*1 = (3/20)*100% = 15%.
So you work on a 15% commission.

 

[JeffM]

I would reduce $30/$200 to just 3/20. Note that the $ signs also canceled out. You can even express 3/20 as a percent (so you know your commission as a percentage). (3/20) = (3/20)*1 = (3/20)*100% = 15%.
So you work on a 15% commission.

Being persnickety, I would not say that dollar signs cancel out. First of all, what is the mathematical proof that units of measurement do cancel. Units of measurement are not the same as numbers. Second, these are not comparable units. We are talking about dollars of commission and dollars of sales.

Treating units of measurements as cancellable, what we have is

[MATH]\dfrac{30 \text { dollars of commission}}{200 \cancel { \text { dollars of sales}}} * 100 \cancel { \text {dollars of sales}} = 15 \text { dollars of commission.} [/MATH]
I suspect that it may have been matching types of dollars that led the OP to 30 * (100/200).

 

[Steven G]

The dollars of sales are the same and they do cancel out.

If they do not cancel out, then what do we call what happens?

What is dimensional analysis?

I am not saying that I am using the right terminology but the units do go away like numbers do.

 

[JeffM]

@Jomo
I am I suppose making several points.

What does it mean to say that “dollars of sales cancel“? I was a banker most of my professional life and never saw any dollars cancelling each other. We did not need to put ten dollar bills and five dollar bills into separate vaults to avoid having the vault substitute one two-dollar bill for each matched pair of ten and five dollar bills. The so-called cancelling of units does not represent any physically observable event whatsoever.

This “cancelling“ hides any number of assumptions that must be true before application of dimensional analysis gives correct results. One of those assumptions is what lev mentioned in post 2, namely that the appropriate mathematics involves proportionality. The value of dimensional analysis is that it prevents gross errors when proportionality is relevant.

Another point I am making is that when dealing with financial matters a great many different concepts are associated with monetary units, and it is very easy for dimensional analysis to lead you astray unless you say dollars of what: dollars of interest do not cancel dollars of principal, and dollars in year ten do not cancel dollars in year 5.

Now I have used dimensional analysis for over 60 years. It is a very useful technique. But it is applicable only when a number of conditions are met; it is not just cancelling words like “dollar.” Using it in financial matters requires great care. I almost never “solve” a problem using dimensional analysis. I use it as a check on solutions.