# Cancellation of Variables

#### Explain this!

##### Junior Member
Is the following solution correct?

31,500/y ÷ 12m = 31,500 y * 1/12m = 2,625/my

1.1% * 2,625/my = 28.875/my

36.5y * 28.875/my = 1,053.9375/m

I am mostly concerned that the variable" y" has cancelled correctly.

#### Otis

##### Senior Member
Is the following solution correct ...
I can't say it's a correct solution (because you didn't post a problem), but there's nothing wrong with the work you've shown. All of your simplifications are correct.

By the way, it's standard notation when typing ratios to put grouping symbols around denominators that contain more than one factor or term. For example, some people (and most software) may interpret 1/12m to mean 1/12th of m. Grouping symbols eliminate that ambiguity.

1/(12m)

28.875/(m*y)

We put grouping symbols around numerators, too, when they contain more than one term.

(x + 4)/y $$\;$$ not $$\;$$ x + 4/y

#### Explain this!

##### Junior Member
I can't say it's a correct solution (because you didn't post a problem), but there's nothing wrong with the work you've shown. All of your simplifications are correct.

By the way, it's standard notation when typing ratios to put grouping symbols around denominators that contain more than one factor or term. For example, some people (and most software) may interpret 1/12m to mean 1/12th of m. Grouping symbols eliminate that ambiguity.

1/(12m)

28.875/(m*y)

We put grouping symbols around numerators, too, when they contain more than one term.

(x + 4)/y $$\;$$ not $$\;$$ x + 4/y

If the "y" and "m" are replaced by year and month, would the solution be 1,053.9375/month?

I think that the calculation would be as follows:

31,500/year ÷ 12 months = 2,625/month-year

1.1% * 2,625/month-year = 28.875/month-year

36.5 years * 28.875/month-year = 1,053.9375/month

#### Otis

##### Senior Member
If the "y" and "m" are replaced by year and month, would the solution be 1,053.9375/month?
I cannot comment on solutions because you have not posted an exercise. I don't know what values your words 'year' and 'month' might represent.

Please post the entire exercise, so we can see what you're working on.

#### Explain this!

##### Junior Member
I cannot comment on solutions because you have not posted an exercise. I don't know what values your words 'year' and 'month' might represent.

Please post the entire exercise, so we can see what you're working on.

I do not have an exercise to post. The value of the units, year and month, are seen in the calculation!

1,500/year ÷ 12 months = 2,625/month-year

1.1% * 2,625/month-year = 28.875/month-year

36.5 years * 28.875/month-year = 1,053.9375/month

Just work through the calculation one line at a time to verify that the solution is 1,053.9375/month.

#### Dr.Peterson

##### Elite Member
I do not have an exercise to post. The value of the units, year and month, are seen in the calculation!

31,500/year ÷ 12 months = 2,625/month-year

1.1% * 2,625/month-year = 28.875/month-year

36.5 years * 28.875/month-year = 1,053.9375/month

Just work through the calculation one line at a time to verify that the solution is 1,053.9375/month.
Clearly you are trying to do something; those numbers (1500, 1.1%, ...) come from somewhere! So you have not stated the problem you are "solving". We would expect to see something like, "If I earn a total of $31,500 over y years, and ... , how much ...?" As has been said, the work is nominally correct (as long as you are taking 2,625/my as 2,625/(my), which is not universally accepted); so if the numbers you use are correct for whatever you are trying to do, your answers are correct. Unfortunately, I can't think of a problem for which your work would be appropriate. In particular, if as you just said, you are earning$31,500 per year, then your work is entirely wrong. What you originally wrote takes the letters as variables, but now you are using them as units. There is a big difference between units and variables! (See here.) I don't even know what a "month-year" would mean! Instead, you would divide by "12 months per year" to get the amount you earn per month; then maybe multiply by 1.1% to find the amount per month at that tax rate or something; and so on.

Do you see yet why we need to see what you are trying to accomplish, whether it is a textbook exercise or a personal calculation?

#### Explain this!

##### Junior Member
Clearly you are trying to do something; those numbers (1500, 1.1%, ...) come from somewhere! So you have not stated the problem you are "solving". We would expect to see something like, "If I earn a total of $31,500 over y years, and ... , how much ...?" As has been said, the work is nominally correct (as long as you are taking 2,625/my as 2,625/(my), which is not universally accepted); so if the numbers you use are correct for whatever you are trying to do, your answers are correct. Unfortunately, I can't think of a problem for which your work would be appropriate. In particular, if as you just said, you are earning$31,500 per year, then your work is entirely wrong. What you originally wrote takes the letters as variables, but now you are using them as units. There is a big difference between units and variables! (See here.) I don't even know what a "month-year" would mean! Instead, you would divide by "12 months per year" to get the amount you earn per month; then maybe multiply by 1.1% to find the amount per month at that tax rate or something; and so on.

Do you see yet why we need to see what you are trying to accomplish, whether it is a textbook exercise or a personal calculation?
This is the best that I can do to explain the calculation:

An individual is going to retire after 35.6 years of service, and his annual pension is determined by the following calculation.
Determine his monthly amount that he will receive:

1.1% * $31,500 * 36.5 years The$31,500 represents his last 5 year average salary. The 36.5 years is his number of years he as worked. I do not know what the 1.1% represents other that no person retiring will receive 100% of the above calculation as a pension.

Instead of multiplying straight across from left to right, I wanted to divide the $31,500 by 12 months and then multiply by 1.1% and then by 36.5 years. I just wanted to verify that I would get the same result. If$31,500 represents $31,500/year and this is divided by 12 months the result is$2,625/month-year. Multiply by 1.1% and then by 35.6 years. The "year" units cancel.
I think the correct amount is indeed $1,053.9375/month. This is the best that I can do to explain this calculation. #### Dr.Peterson ##### Elite Member This is the best that I can do to explain the calculation: An individual is going to retire after 35.6 years of service, and his annual pension is determined by the following calculation. Determine his monthly amount that he will receive: 1.1% *$31,500 * 36.5 years

The $31,500 represents his last 5 year average salary. The 36.5 years is his number of years he as worked. I do not know what the 1.1% represents other that no person retiring will receive 100% of the above calculation as a pension. Instead of multiplying straight across from left to right, I wanted to divide the$31,500 by 12 months and then multiply by 1.1% and then by 36.5 years. I just wanted to verify that I would get the same result. If $31,500 represents$31,500/year and this is divided by 12 months the result is $2,625/month-year. Multiply by 1.1% and then by 35.6 years. The "year" units cancel. I think the correct amount is indeed$1,053.9375/month.

This is the best that I can do to explain this calculation.
Thanks. Presumably you meant 36.5 years, not 35.6. (Typos in math are dangerous!)

So what you are saying is that the monthly amount is defined as 1.1% of the average annual salary over the last 5 years, times the total number of years worked (that is, 1.1% of the total amount he would have earned if he had always made as much per year as in the last 5 years). The meaning of the 1.1% is that each year he will get 12*1.1% = 13.2%, or about 1/8 of the effective amount he has been paid over his lifetime. That's substantial, for anyone who has worked over 8 years.

This amount, for this example, is 0.011 * 31,500 * 36.5 = $12,647.25 (per month), which comes to$151,767 per year.

Note that the units as they stand don't cancel as we like them to, because there is an implicit rate (the 1.1%) that needs its own units. With units, it turns out to be 0.011/month * 31,500 dollars/year * 36.5 years= $12,647.25/month. Now, you for some reason want to do the same calculation (?) in a different way, and you wonder if your rearrangement is valid. You can tell that it is not, of course, just by doing both calculations and seeing that they give different results, since equivalent calculations must give the same result. You appear to want to replace the annual salary with the monthly salary. Your result is, of course, 1/12 of the actual monthly pension -- assuming that what you said about the calculation is correct. Now, if you are motivated by thinking (as I do) that the pension should not be almost 5 times the pre-retirement pay, then the place to look is at the actual wording of whatever information you were given about how the pension is calculated. (That, by the way, is the "assignment" we have been asking for!) If it says, as you imply, that the calculation you show is said to be both the annual pension and the monthly pension, then you have to ask someone what it actually means. But just reworking the equation without justification is not a valid response. #### Otis ##### Senior Member ... The value of the units, year and month, are seen in the calculation! ... Sorry -- you had not defined your symbols, your intent was not clear and I don't always like guessing. #### Explain this! ##### Junior Member Thanks. Presumably you meant 36.5 years, not 35.6. (Typos in math are dangerous!) So what you are saying is that the monthly amount is defined as 1.1% of the average annual salary over the last 5 years, times the total number of years worked (that is, 1.1% of the total amount he would have earned if he had always made as much per year as in the last 5 years). The meaning of the 1.1% is that each year he will get 12*1.1% = 13.2%, or about 1/8 of the effective amount he has been paid over his lifetime. That's substantial, for anyone who has worked over 8 years. This amount, for this example, is 0.011 * 31,500 * 36.5 =$12,647.25 (per month), which comes to $151,767 per year. Note that the units as they stand don't cancel as we like them to, because there is an implicit rate (the 1.1%) that needs its own units. With units, it turns out to be 0.011/month * 31,500 dollars/year * 36.5 years=$12,647.25/month.

Now, you for some reason want to do the same calculation (?) in a different way, and you wonder if your rearrangement is valid. You can tell that it is not, of course, just by doing both calculations and seeing that they give different results, since equivalent calculations must give the same result.

You appear to want to replace the annual salary with the monthly salary. Your result is, of course, 1/12 of the actual monthly pension -- assuming that what you said about the calculation is correct.

Now, if you are motivated by thinking (as I do) that the pension should not be almost 5 times the pre-retirement pay, then the place to look is at the actual wording of whatever information you were given about how the pension is calculated. (That, by the way, is the "assignment" we have been asking for!) If it says, as you imply, that the calculation you show is said to be both the annual pension and the monthly pension, then you have to ask someone what it actually means. But just reworking the equation without justification is not a valid response.
Thanks for the reply, but I am really confused! Your mathematical level/reasoning is much higher than I can comprehend.

Multiplying straight across from left to right gives the same answer ($1,053.9375/month) after dividing the result by 12 months as I have determined by first dividing$31.500/year by 12 months and then multiplying the remaining numbers. I'm not sure about the 1.1% having units. I know that the 36.5 years and the \$31,500/year are most likely correct for having units.

I did make an error with the 35.6 years from my last posting. Yes, typographical errors are dangers in mathematics.

#### Dr.Peterson

##### Elite Member
My math is not really the important point. It's that you have to find out the official statement about how the calculation is done, and base your reasoning on that. Please look that up and quote it for us. We can't talk further without that.

As I suggested, I think it is quite likely that your revised calculation (which is not equivalent to what you started with) is what is intended, and someone somewhere has misinterpreted it.