Monthly Pension: 1.1% * average 5 years highest salary * years of service

[KWF]

A company uses the following calculation to determine the monthly payment for its employees as they retire:


1.1% * average 5 years highest salary * years of service


If an employee worked 33 years and his average 5 year salary is $30,705.17, what is his monthly payment?


I used the following to determine his monthly payment, but I am not sure how the units cancel especially the years.


1.1% * $30,705.17 * 33 years divided by 12 months/year


I determined his payment as $928.83 per month, but the year units do not cancel unless the 1.1% is per year.


Is the correct calculation 1.1%/year * $30,705.17 * 33 years divided by 12 months/year so that the units cancel properly?

The $30,705.17 is an average for 5 years of working. It is not $30,705.17 per year.

 

[KWF]

.011 * 30,705 * 33 = ~11,146
According to opening statement, that would be "the monthly payment"!

So, that being ridiculously high, formula is probably this:
1.1% * average 5 years highest salary * years of service / 12.
11,146 / 12 = ~929 is the result (as you got).

However, 30,705.17 / 12 = ~2,558:
soooo quite a drop in income!

In other words, your problem is probably faulty...as usual!!

1. What to you mean "In other words, your problem is probably faulty...as usual!!" ?

2. Would the amount, $11,145.97... be divided by 12 months not 12 months/year?

3. Why not 12 months/year?

 

[JeffM]

A company uses the following calculation to determine the monthly payment for its employees as they retire:


1.1% * average 5 years highest salary * years of service


If an employee worked 33 years and his average 5 year salary is $30,705.17, what is his monthly payment?

If we assume that you have correctly given the formula for the monthly payment, then the monthly pension given the facts cited is 0.011 * 30705.17 * 33 = 11,145.98 per month, or 133,751.76 per year. It seems very unlikely that someone is paid over 4 times more than their annual salary as a pension. (I want in that pension program.) What seems likely is that you have given us the formula for the annual pension. In other words, you did not formulate the problem correctly, something that happens a lot. We have to adjust to that.

I used the following to determine his monthly payment, but I am not sure how the units cancel especially the years.


1.1% * $30,705.17 * 33 years divided by 12 months/year

Good: you thought about using dimensional analysis.

The dimensions of what we are trying to calculate are \(\displaystyle m\ \dfrac{\text {pension dollars}}{\text {month}}.\)

The dimensions of what results from the formula are \(\displaystyle y\ \dfrac{\text {pension dollars}}{\text {year}}.\)

To go from one to the other

\(\displaystyle m\ \dfrac{\text{pension dollars}}{\text {month}} = \left (y\ \dfrac{\text {pension dollars}}{\text {year}} \right ) * \left ( \dfrac{1 \text { year}}{12 \text { months}} \right ) = \dfrac{y}{12} \ \dfrac{\text{pension dollars}}{\text {month}}.\)

Does this answer your question, or are you worried about the dimensional analysis of how to calculate the annual pension payment?

 

[KWF]

If we assume that you have correctly given the formula for the monthly payment, then the monthly pension given the facts cited is 0.011 * 30705.17 * 33 = 11,145.98 per month, or 133,751.76 per year. It seems very unlikely that someone is paid over 4 times more than their annual salary as a pension. (I want in that pension program.) What seems likely is that you have given us the formula for the annual pension. In other words, you did not formulate the problem correctly, something that happens a lot. We have to adjust to that.


Good: you thought about using dimensional analysis.

The dimensions of what we are trying to calculate are \(\displaystyle m\ \dfrac{\text {pension dollars}}{\text {month}}.\)

The dimensions of what results from the formula are \(\displaystyle y\ \dfrac{\text {pension dollars}}{\text {year}}.\)

To go from one to the other

\(\displaystyle m\ \dfrac{\text{pension dollars}}{\text {month}} = \left (y\ \dfrac{\text {pension dollars}}{\text {year}} \right ) * \left ( \dfrac{1 \text { year}}{12 \text { months}} \right ) = \dfrac{y}{12} \ \dfrac{\text{pension dollars}}{\text {month}}.\)


Does this answer your question, or are you worried about the dimensional analysis of how to calculate the annual pension payment?

Thanks for the reply!

I am mostly interested in the dimensional analysis of the problem. I can determine the monthly amount, but I am not sure how the 'year' units cancel. Can it be determined by the original calculation that is in the word problem?

1.1% * average 5 years highest salary ($30,705.17 * (33 yrs.) years of service divided by 12 months equals monthly payment

I do not understand how the 'years' cancel and $928.83/month becomes the correct answer. The units are dollars, years, and month. I divided 1.1% * ($30,705.17 * 33 yrs. by 12 months/year, but the 'year' units do not cancel.

Are two separate calculations involved?

1. 1.1% * $30,705.17 * 33 yrs. = $11,145.9767... This seems to represent 11,145.976 dollar-years unless 1.1% is per year(?)

2. $11,145.9767 dollar-years divided by 12 months/year, but the units do not cancel properly. (?)

 

[Dr.Peterson]

A company uses the following calculation to determine the monthly [annual] payment for its employees as they retire:

1.1% * average 5 years highest salary * years of service

If an employee worked 33 years and his average 5 year salary is $30,705.17, what is his monthly payment?

I am mostly interested in the dimensional analysis of the problem. I can determine the monthly amount, but I am not sure how the 'year' units cancel. Can it be determined by the original calculation that is in the word problem?

Often there are unstated units, so you have to clarify, perhaps by assigning units to a conversion factor just to make things work out. Sometimes, units are not really involved at all.

Here is how I see it. Each year, they will be giving retirees 1.1% of a year's salary for each year of service. That's it. It's just a single calculation, with no individual units indicated. Working backward, we can add units something like this:

Annual pension = 0.011(S dollars)(Y years) in dollars /year

so (if you really want to) you can think of the 1.1% as a conversion factor measured in years^-2:

Pension = 0.011 years^-2 * $30,705.17 * 33 years = $11,145.98 /year

The units attached to the conversion factor don't mean anything in particular by themselves. This is quite common!

Finding the monthly amount then is easy:

Monthly pension = annual pension / 12 = $11,145.98 /year * 1 year/12 months = $928.83/month

Or, if you prefer,

Pension for 1 month = $11,145.98 /year * 1/12 year = $928.83

 

[JeffM]

Thanks for the reply!

I am mostly interested in the dimensional analysis of the problem. I can determine the monthly amount, but I am not sure how the 'year' units cancel. Can it be determined by the original calculation that is in the word problem?

1.1% * average 5 years highest salary ($30,705.17 * (33 yrs.) years of service divided by 12 months equals monthly payment

I do not understand how the 'years' cancel and $928.83/month becomes the correct answer. The units are dollars, years, and month. I divided 1.1% * ($30,705.17 * 33 yrs. by 12 months/year, but the 'year' units do not cancel.

Are two separate calculations involved?

1. 1.1% * $30,705.17 * 33 yrs. = $11,145.9767... This seems to represent 11,145.976 dollar-years unless 1.1% is per year(?)

2. $11,145.9767 dollar-years divided by 12 months/year, but the units do not cancel properly. (?)

To calculate the pension dollars per year, let's define a as the relevant average salary dollars per year and s as the number of years of service. We previously defined y as pension dollars per year.

Numerically we simply multiply 0.011as. If we want to do dimensional analysis, however, we have to ask what dimensions apply to the constant 0.011. So let's see

\(\displaystyle 0.011 * \left ( a \ \dfrac{\text {salary dollars}}{\text {year}} \right ) * (s\ \text {years}) = 0.011as\ \text {salary dollars}.\)

That is clearly wrong if we want to end up with pension dollars per year. In dimensional analysis, everything must have units so we must figure out what units to apply to the constant to get rid of unwanted units and insert needed units.

\(\displaystyle \left ( 0.011 * \dfrac{\dfrac{\text {pension dollars}}{\text {year}}}{\text {salary dollars}} \right ) \left (a\ \dfrac{\text {salary dollars}}{\text {year}} \right ) * (s\ \text {years}) = 0.011as\ \dfrac{\text {pension dollars}}{\text {year}}.\)

In dimensional analysis, we know the units we start with and the units we want to end up. These determine the units to be assigned to any constant.

 

[KWF]

Often there are unstated units, so you have to clarify, perhaps by assigning units to a conversion factor just to make things work out. Sometimes, units are not really involved at all.

Here is how I see it. Each year, they will be giving retirees 1.1% of a year's salary for each year of service. That's it. It's just a single calculation, with no individual units indicated. Working backward, we can add units something like this:

Annual pension = 0.011(S dollars)(Y years) in dollars /year

so (if you really want to) you can think of the 1.1% as a conversion factor measured in years^-2:

Pension = 0.011 years^-2 * $30,705.17 * 33 years = $11,145.98 /year

The units attached to the conversion factor don't mean anything in particular by themselves. This is quite common!

Finding the monthly amount then is easy:

Monthly pension = annual pension / 12 = $11,145.98 /year * 1 year/12 months = $928.83/month

Or, if you prefer,

Pension for 1 month = $11,145.98 /year * 1/12 year = $928.83

Thank you for your explanation:

I now want to understand the calculation as follows:

1.1%/year * $30,705.17 * 33 years

or

1.1% *$30,705.17/year * 33 years

I think that the year belongs with the salary average amount since it represents an annual average salary.

The dollar units remain after the years cancel. Now divide by 12 months to get the monthly amount, $928.83/month.

Clearly, 1.1% * $30,705.17 = $337.756... The dollar units have not canceled.

$337.756... * 33 years = $11,145.976... The product now has dollar-years. The years need to cancel.

 

[Dr.Peterson]

Thank you for your explanation:

I now want to understand the calculation as follows:

1.1%/year * $30,705.17 * 33 years

The dollar units remain after the years cancel. Now divide by 12 months to get the monthly amount, $928.83/month.

Let's modify that in line with JeffM's version, which is better than mine (but ignoring the distinction of kinds of dollars for simplicity):

Pension = 1.1%/year * $30,705.17/year * 33 years = $11145.98/year

Monthly pension = $11145.98/year * 1 year/12 months = $928.83/month

But you could also focus on dollar amounts rather than rates, and say:

Pension for 1 year = 1.1% * $30,705.17/year * 33 years = $11145.98

Pension for 1 month = $11145.98 /12 months = $928.83