Unit Cancelation: Exec gets 1/4 of $100,000.00 as severance pay annually

KWF

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An executive receives a severance pay of one fourth of his annual salary of $100,000.00. for 10 years. How much would he receive annually?

0.25 * $100,000/year * 10 years = $250,000.00

Would it be incorrect to use the phrase "per year" since the years have cancelled from the above calculation? Would it be just $250,000 or $250,000 per year?
 
An executive receives a severance pay of one fourth of his annual salary of $100,000.00. for 10 years. How much would he receive annually?

0.25 * $100,000/year * 10 years = $250,000.00

Would it be incorrect to use the phrase "per year" since the years have cancelled from the above calculation? Would it be just $250,000 or $250,000 per year?

The question is how much annually, not total.
 
An executive receives a severance pay of one fourth of his annual salary of $100,000.00. for 10 years. How much would he receive annually?

0.25 * $100,000/year * 10 years = $250,000.00

Would it be incorrect to use the phrase "per year" since the years have cancelled from the above calculation? Would it be just $250,000 or $250,000 per year?
You recently asked a question that I answered by explaining that if you are going to use dimensional analysis, every number must have dimensions.

I further explained that you have to figure out what dimensions to use to get an answer with the correct dimensions. In other words, what dimensions must be placed on 0.25 to make things come out right. Dimensional analysis is not a substitute for thinking.

What is easiest here is to think annually from the start.

\(\displaystyle \left ( 0.25 \ \dfrac{\dfrac{\text {severance dollars}}{\text {year}}}{\dfrac{\text {salary dollars}}{\text {year}}} \right ) * \left ( 100000 \ \dfrac{\text {salary dollars}}{\text{year}} \right ) =\)

\(\displaystyle 25000 \ \dfrac{\dfrac{\text {severance dollars}}{\text {year}} * \dfrac{\text {salary dollars}}{\text{year}}}{\dfrac{\text {salary dollars}}{\text {year}}} = 25000 \ \dfrac{\text {severance dollars}}{\text {year}}.\)

You can do it by calculating the total severance and then dividing by 10 years, but that makes the dimensional analysis and the arithmetic even more complicated.
 
I hope you see, from what everyone has been saying, that "for 10 years" is irrelevant to the question, which is only about how much he gets each of those ten years.

Dimensional analysis doesn't help particularly! It only gives you more rope to get tangled in, by hiding the actual words used.
 
An executive receives a severance pay of one fourth of his annual salary of $100,000.00. for 10 years. How much would he receive annually?
I am surprised that no one pointed out that this problem has no solution.
It says that An executive receives a severance pay of one fourth of his annual salary of $100,000.00. My question is does the executive receive one fourth of his annual salary of $100,000.00 every hour, every month, every year, every century...? Fair enough, it probably means every year but it does NOT say this. As others have pointed out it does not matter (except to the executive) for how long he/she gets the money for as it does not affect the yearly amount.
 
I am surprised that no one pointed out that this problem has no solution.
It says that An executive receives a severance pay of one fourth of his annual salary of $100,000.00. My question is does the executive receive one fourth of his annual salary of $100,000.00 every hour, every month, every year, every century...? Fair enough, it probably means every year but it does NOT say this. As others have pointed out it does not matter (except to the executive) for how long he/she gets the money for as it does not affect the yearly amount.

I want to thank you for for indicating this error to me! I have the more accurate version below:

An executive receives a severance pay of one fourth of his annual salary of $100,000.00 every year for 10 years. How much would he receive annually?

I can see that 0.25 * $100,000/year for 10 years = $250,000

Is it correct to indicate that he gets $250,000.00 or $250,000/year?
 
I want to thank you for for indicating this error to me! I have the more accurate version below:

An executive receives a severance pay of one fourth of his annual salary of $100,000.00 every year for 10 years. How much would he receive annually?

I can see that 0.25 * $100,000/year for 10 years = $250,000

Is it correct to indicate that he gets $250,000.00 or $250,000/year?

No.

It's $25,000.00 per year; $250,000 is the total over 10 years. The former is the answer to the question; the latter is not.

Is what you're quoting now the actual wording of the problem as given to you?
 
10 years is ten (10) times "annually".

"How much annually" is just one (1).

I am still confused. It seems like a simple solution to be determined, but the answers seem to be totally confusing.

"10 years is ten (10) times "annually".

"How much annually" is just one (1)
 
No.

It's $25,000.00 per year; $250,000 is the total over 10 years. The former is the answer to the question; the latter is not.

Is what you're quoting now the actual wording of the problem as given to you?

I made up the question so that someone could explain how the year units cancel. I cannot make sense of any of the replies. Can you try again and explain what the correct reply/answer is and why?

I see 1/4 * $100,000/year for 10 years as having the years cancel and the solution is $250,000, but the correct reply is $250,00/year

Short replies make no sense to me, and why should they?
 
I made up the question so that someone could explain how the year units cancel. I cannot make sense of any of the replies. Can you try again and explain what the correct reply/answer is and why?

I see 1/4 * $100,000/year for 10 years as having the years cancel and the solution is $250,000, but the correct reply is $250,000/year

Short replies make no sense to me, and why should they?

No, the correct answer to the correct question (where you added in "every year" for clarity) is not either of those you state, but $25,000/year.

You wrote a confusing question, by mentioning the 10 years but asking about the annual amount. If the question were, "What is the total amount he will get?", then your answer of $250,000 would be correct. But you asked "How much would he receive annually?", so the answer has to be in $/year, and does not involve the 10 years. It is just 1/4 * $100,000/year = $25,000/year. The units cancel just fine either way.

Just to keep this from being a short answer, let me state the whole thing again:

The data given are:
An executive receives a severance pay of one fourth of his annual salary of $100,000.00 every year for 10 years.

The natural question that could be asked is:
How much would he receive (total) over the 10 years?

The answer to that is:
1/4 * $100,000/year * 10 years = $250,000

The question you posed is different:
How much would he receive annually?

For this, we don't care how long he is being given the money; "annually" means each year, not total. So the answer is:
1/4 * $100,000/year = $25,000/year
 
That's the problem statement. As written, it means:
1/4 of 100,000 = 25,000 received each year

That's it. Nothing else required.
Why are you complicating the issue?
AND where did you get that silly problem: from your math teacher?

Why would you ignore the units in this so called "silly" question?
 
No, the correct answer to the correct question (where you added in "every year" for clarity) is not either of those you state, but $25,000/year.

You wrote a confusing question, by mentioning the 10 years but asking about the annual amount. If the question were, "What is the total amount he will get?", then your answer of $250,000 would be correct. But you asked "How much would he receive annually?", so the answer has to be in $/year, and does not involve the 10 years. It is just 1/4 * $100,000/year = $25,000/year. The units cancel just fine either way.

Just to keep this from being a short answer, let me state the whole thing again:

The data given are:
An executive receives a severance pay of one fourth of his annual salary of $100,000.00 every year for 10 years.

The natural question that could be asked is:
How much would he receive (total) over the 10 years?

The answer to that is:
1/4 * $100,000/year * 10 years = $250,000

The question you posed is different:
How much would he receive annually?

For this, we don't care how long he is being given the money; "annually" means each year, not total. So the answer is:
1/4 * $100,000/year = $25,000/year

I think that I am starting to understand now. I am interrupting the question incorrectly. I am wanting the solution for the $25,000/year ( 1/4 * $100,000/year) for 10 years and that would be $250,000 not $250,000/year

Would this make sense? 1/4 * $100,000/year = $25,000/year. He gets $25,000/year for 10 years. This would total $250,000 over the ten year period.

Would it be incorrect to indicate this as $250,000/10 years?
 
I think that I am starting to understand now. I am interrupting the question incorrectly. I am wanting the solution for the $25,000/year ( 1/4 * $100,000/year) for 10 years and that would be $250,000 not $250,000/year

Would this make sense? 1/4 * $100,000/year = $25,000/year. He gets $25,000/year for 10 years. This would total $250,000 over the ten year period.
Yes, doing it in these two steps is good.

Would it be incorrect to indicate this as $250,000/10 years?

That would be a valid thing to say. In effect, you are multiplying $25,000/year by 10 years/decade to get $250,000/decade.
 
I made up the question so that someone could explain how the year units cancel. I cannot make sense of any of the replies. Can you try again and explain what the correct reply/answer is and why?

I see 1/4 * $100,000/year for 10 years as having the years cancel and the solution is $250,000, but the correct reply is $250,00/year

Short replies make no sense to me, and why should they?
Short replies assume that you are intelligent and willing to work. The people who respond are unpaid volunteers and have other demands on their name. However, right now I am laid up with a broken bone in my foot so I have extra time.

I have now told you twice that, if you want to use dimensional analysis, all the numbers must have units attached and that the attached units must make sense in the context of the computation being made. Dimensional analysis prevents mistakes when the units associated with the various numbers are obvious, but takes careful thought when it is not obvious what units should be attached to what numbers. That is why Dr. Peterson said dimensional analysis gets in the way with respect to this problem.

Let's start with why dimensional analysis works. It always starts from an equation involving units. For example,

\(\displaystyle 12x \text { inches } = x \text { feet } \implies \dfrac{12x \text { inches}}{x \text { feet}} = \dfrac{x \text { feet}}{x \text { feet}} \implies 12 \ \dfrac{\text {inches}}{\text {foot}} = 1.\)

\(\displaystyle \text {BUT } 12x \text { inches } = x \text { feet } \implies \dfrac{12x \text { inches}}{12x \text { inches}} = \dfrac{x \text { feet}}{12x \text { inches}} \implies 1 = \dfrac{1}{12} \ \dfrac{\text{foot}}{\text {inch}}.\)

\(\displaystyle \therefore 12 \ \dfrac{\text {inches}}{\text {foot}} = 1 = \dfrac{1}{12} \ \dfrac{\text{foot}}{\text {inch}} \implies 12 \ \dfrac{\text {inches}}{\text {foot}} = \dfrac{1}{12} \ \dfrac{\text{foot}}{\text {inch}}.\)

You can multiply or divide by 1 without changing anything of substance, which is why dimensional analysis is a useful technique to keep you from dividing when you should multiply or from multiplying when you should divide. But I hope it is clear now why you must have units attached to any number other than 1, or else you end up with nonsense such as

\(\displaystyle 12 \ \dfrac{\text {inches}}{\text {foot}} = \dfrac{1}{12} \ \dfrac{\text{foot}}{\text {inch}} \implies 12 = \dfrac{1}{12}.\)

You have numbers such as 0.25 with no units attached so you are simply doing dimensional analysis wrong.

What units should be attached to 0.25? To figure that out, you have to set up one or more equations. What equations? Well that depends on the exact terms that the lawyers put into the severance contract. It might be that severance shall be a lump sum equal to 0.25 * 10 * annual salary. (Stupid way to write a contract: easier to write 2.5.) Then the equation would be

\(\displaystyle \text {lump sum severance in dollars} = 0.25 * 10 * \text { annual salary in dollars } \implies\)

\(\displaystyle 1 = 2.5\ \dfrac{\text {lump sum severance in dollars}}{\text {annual salary in dollars.}}\)

If you multiply that by 100,000 annual salary in dollars you get a lump sum severance in dollars of 250,000, and the salary dollars cancel but the severance dollars remain.

Of course, the legal text might read that total severance shall be paid over ten years in annual installments, each equal to 0.25 * annual salary.

In that case, we could ask two different questions. One is the the total amount of the severance. The other is what is the annual installment. What would the relevant equation be in each case? Therefore what would the units be attached to each number? How would the computation work out if you attached the proper units to each relevant number?
 
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