what's wrong here?

A

allegansveritatem

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I came across this in the problems in a section labeled: Problems for Discussion in the back of one of the chapters in my precalculus text. I keep getting an impossibly large answer when I work out formula. The parameters I am using are: 90000$ loan at .12 interest over 30 years with payments of 925$ per month. Here is the part of the problem I am asking about:
View attachment 15461

I worked it out again and again and always I got the same results with these parameters: a 90000$ loan, at 12% payable over 30 years. The monthly payments are 925$. 925 times 12 times 30 = 333000$ But when I plug in the numbers in the above formula I find that the poor loanee is going to pay over 3 million dollars at the end of 30 years. I've worked it out again and again and this is what I get:

intsol.PNG

So...what gives? I can only suppose that the formula is incorrect.
 
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pka said:
Why did you double post this question?
Click to expand...
I messed up the posting process I'm afraid. Sorry. I started this last night and came back to it just now without out and things got wild from there. I am still not out of trouble here because I have some bad info in the first paragraph. I will try to fix it now.
 
pka said:
Why did you double post this question?
Click to expand...
Please look at the post again, I think I have repaired it so that the information is correct. I am sorry that the top image is attached rather than embedded...I don't know why. But you can access it easily I guess.
 
I do not see anything wrong with your work. I do see something wrong when the bank charges you so much for a loan!
 
If you think the formula is wrong then derive the correct one.
After the 1st month (after you made your monthly payment) how much did you pay and how much do you owe?
After the 2nd month (after you made your monthly payment) how much did you pay so far and how much do you owe?
After the 2nd month (after you made your monthly payment) how much did you pay so far and how much do you owe?

Do you see a pattern? Remember that after 360 payments you should owe $0. Do you get the same formula?
 
Jomo said:
If you think the formula is wrong then derive the correct one.
After the 1st month (after you made your monthly payment) how much did you pay and how much do you owe?
After the 2nd month (after you made your monthly payment) how much did you pay so far and how much do you owe?
After the 2nd month (after you made your monthly payment) how much did you pay so far and how much do you owe?

Do you see a pattern? Remember that after 360 payments you should owe $0. Do you get the same formula?
Click to expand...
I will try this. Thanks. I hate to give up on this for some reason.
 
Jomo said:
If you think the formula is wrong then derive the correct one.
After the 1st month (after you made your monthly payment) how much did you pay and how much do you owe?
After the 2nd month (after you made your monthly payment) how much did you pay so far and how much do you owe?
After the 2nd month (after you made your monthly payment) how much did you pay so far and how much do you owe?

Do you see a pattern? Remember that after 360 payments you should owe $0. Do you get the same formula?
Click to expand...
I have discovered that the formula given in the body of the problem,i.e.:
interest33.PNG




The above formula is somehow false. If my loan has a 90000$ principle, a .12 rate of interest, with monthly payments for 30 years, then the monthly payment is algebraically determined to be 925$ and the amount to be repaid is:
interrest31.PNG


Now, in order to get 333000$ with the above formula I can't use 30 as a factor in the exponent, instead I have to do this: Here I was going to insert another image but the program will not allow it. So I will just say that I found that instead of 12 times 30 in the exponent I had to have 12 times 12.7806. So in this case the fault in not in myself but in my book that I have been an underling of error for the last 2 days while fooling with this problem.
 

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Your equations are correct. If you borrow $90 K at 12% APR, your monthly payment will be $ 925.75 (for 360 payments).

If you put $90 K in a bank(?) at 12% APR - it will grow to $ 3235467.72 after 30 yrs.
This amount has a present value of $ 90 K.

This is the charm of compounding!!
 
allegansveritatem said:
I have discovered that the formula given in the body of the problem,i.e.:
View attachment 15514




The above formula is somehow false. If my loan has a 90000$ principle, a .12 rate of interest, with monthly payments for 30 years, then the monthly payment is algebraically determined to be 925$ and the amount to be repaid is:
View attachment 15515


Now, in order to get 333000$ with the above formula I can't use 30 as a factor in the exponent, instead I have to do this: Here I was going to insert another image but the program will not allow it. So I will just say that I found that instead of 12 times 30 in the exponent I had to have 12 times 12.7806. So in this case the fault in not in myself but in my book that I have been an underling of error for the last 2 days while fooling with this problem.
Click to expand...
Can you tell us where you got 12.7806 from?
 
Subhotosh Khan said:
Your equations are correct. If you borrow $90 K at 12% APR, your monthly payment will be $ 925.75 (for 360 payments).

If you put $90 K in a bank(?) at 12% APR - it will grow to $ 3235467.72 after 30 yrs.
This amount has a present value of $ 90 K.

This is the charm of compounding!!
Click to expand...
Right, especially if you can get 12% interest---but it is not so good when that 12% is attached to a loan.
 
To understand what is going on, construct a simple spread sheet.

In the first column, show the amount owed at the start of the month. Now calculate interest due for the month in the second column. In the third column, show the monthly payment made at the end of the month as a negative. In the fourth column calculate the sum of the first three columns, which gives the amount owed at the end of the month, which obviously is the amount owed at the start of the next month. Now replicate this 360 times. At the bottom, add up the 360 payments to verify the total paid, and add up the total interest.

In that very first month, you pay 925, but interest is 900. You have reduced the amount owed by a negligible amount. So, for many years, the payment is almost all interest. 12 payments of 925 total 11,100 a year, and over 30 years that clearly adds up to over 3 million. Now compare what you pay in total per month over the life of the loan and per month for terms of 20 and 15 years.
 
JeffM said:
To understand what is going on, construct a simple spread sheet.

In the first column, show the amount owed at the start of the month. Now calculate interest due for the month in the second column. In the third column, show the monthly payment made at the end of the month as a negative. In the fourth column calculate the sum of the first three columns, which gives the amount owed at the end of the month, which obviously is the amount owed at the start of the next month. Now replicate this 360 times. At the bottom, add up the 360 payments to verify the total paid, and add up the total interest.

In that very first month, you pay 925, but interest is 900. You have reduced the amount owed by a negligible amount. So, for many years, the payment is almost all interest. 12 payments of 925 total 11,100 a year, and over 30 years that clearly adds up to over 3 million. Now compare what you pay in total per month over the life of the loan and per month for terms of 20 and 15 years.
Click to expand...
Well, the terms of the loan are these: $90000 principle, monthly payments of 925 for 30 years at an interest rate of 12%. Now 30 times 925 times 12 is: $333000. I mean, this transaction is not described by the formula at the top of this thread. That formula would have the poor sap of a loanee paying 3 million instead of the already egregious sum of $333000.
 
allegansveritatem said:
I input an incomplete equation in the above reply so here is what I actually did to get 12.7806:
View attachment 15523
Click to expand...
Let's assume that you are correct that the x in this equation is 12.7806. It is good that you were able to solve for x but what is much more important is how you can get this value for x without knowing what the whole expression equals (in your case it is 333333). Can you see or figure out how x is 12.7806? If not, then you have really gotten nowhere. What is the meaning of x (or 12x) in the equation is what is important. Do you know what the x means? For the record x is NOT 12.786
 
Name things

[MATH]b = \text {the amount borrowed at time of purchase.}[/MATH]
[MATH]p = \text {periodic payment.}[/MATH]
[MATH]r = \text {the interest rate per compounding period expressed as a decimal.}[/MATH]
[MATH]n = \text {the number of compounding periods before the loan is fully paid off.}[/MATH]
Here is the equation

[MATH]b = \dfrac{p * \{1 - (1 + r)^{-n}\}}{r} \iff p = \dfrac{b * r}{1 - (1 + r)^{-n}}.[/MATH]
This relatively simple formula gets a bit more complex when (as is usually the case in the US with respect to mortgage loans) a loan's term is stated in years and the loan's nominal interest rate is quoted as an annual rate even though the compounding period is a month. So let's adjust.

[MATH]b = \text {the amount borrowed at time of purchase.}[/MATH]
[MATH]p = \text {periodic payment.}[/MATH]
[MATH]r = \text {the nominal interest rate per year expressed as a decimal.}[/MATH]
[MATH]y = \text {the number of years before the loan is fully paid off.}[/MATH]
Notice that we have moved from compounding periods to years. To get back to compounding periods, the equation becomes

[MATH]b = \dfrac{p * \left \{1 - \left (1 + \dfrac{r}{12} \right )^{-12y} \right \}}{\dfrac{r}{12}} \iff p = \dfrac{b * \dfrac{r}{12}}{1 - \left ( 1 + \dfrac{r}{12} \right )^{-12y}}.[/MATH]
Now let's work that mess out for your problem.

[MATH]\dfrac{925 * \left \{1 - \left (1 + \dfrac{0.12}{12} \right )^{-12 * 30} \right \}}{\dfrac{0.12}{12}} = \dfrac{925 * \{1 - (1.01^{-360}\}}{0.01} \approx[/MATH]
[MATH]92500 * (1 - 0.02781668921) \approx 89926.96.[/MATH]
925 is not in fact the correct monthly payment for a 90000 loan at 12% for 30 years, but it is reasonably close. Let's figure out what it actually would be.

[MATH]\dfrac{90000 * \dfrac{0.12}{12}}{1 - \left \{1 + \left (1 + \dfrac{0.12}{12} \right)^{-12 * 30} \right \}} =\dfrac{90000 * 0.01}{1 - 1.01^{-360}} \approx[/MATH]
[MATH]\dfrac{900}{1 - 0.02781668921} \approx 925.75.[/MATH]
Even this is not exact: 4.67 would remain due. I again advise you to work the thing out on a spreadsheet to convince yourself that it works.

Now as for the formula you got from your book. I am not at all sure what it is supposed to be getting at. I have read the text, and it makes no sense. (Understand that I worked in banking for over 42 years so I am used to fairly inarticulate renderings of math.)
 
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In your first post you say "at .12 interest". Do you mean 12% interest or 0.12% interest?
 
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