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The following question is what I was working on.

A bank gives a mortgage of $\$450,000$ dollars for a $30$ year loan with $6$% annual interest which requires the person to pay monthly. They require the person to pay $\$2700$ dollars each month. However, the borrower decides to pay $\$3500$ per month. How long will it take for this person to pay off the mortgage?

I was thinking that this is a problem with annuity and stuff, but I was not able to really understand the problem.

So, it would be helpful to have the following.

  1. Using the present value formula for annuity, I got a much smaller number than $\$2700$ so that the money builds up to $\$450,000$. So, I am not sure where the $\$2700$ came from.

  2. Using the final value formula for annuity, paying $\$2700$ dollars each month for $30$ years does not match anything close to $\$450,000$. I am not quite understanding how banks are making money off this loan!

  3. I did try the following.

The cost that the bank charges equals

$$(1.06)\big(450K+(450K-42K)+(450K-2*42K)+\cdots+(450K-(n-1)42K)\big)$$

which simplifies to

$$(1.06)\big(450K(n)-42K(n)(n-1)/2\big)$$

which has to equal to

$$3500n$$

Where $n$ is the number of years that it takes to pay off the cost.

I got about $22$ years and numerically it sounds plausible, but I am not certain. Can someone clarify this for me?

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    $\begingroup$ Hint: $450000 \dfrac{0.005\cdot 1.005^{360}}{1.005^{360}-1} = 2697.98... \approx 2700$ $\endgroup$
    – Macavity
    Commented Oct 24, 2013 at 4:39
  • $\begingroup$ This is an annuity problem, but it looks like you are using simple interest. There are very few problems where we use annuities and simple interest so I urge you to rethink the problem in terms of compound interest. If you are still confused about annuity concepts I'd encourage you to ask another question about that specifically. $\endgroup$
    – Tyler
    Commented Oct 26, 2013 at 1:14
  • $\begingroup$ @Macavity: Thank you so much. I was actually able to solve it. >>Tyler: It turns out that I was not thinking about paying the borrowed money. I was only paying for the charged money. The number being plausible seems like it was a coincidence. $\endgroup$
    – hyg17
    Commented Oct 26, 2013 at 2:27

1 Answer 1

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In present value terms, you're trying to find where

$450000 =3500a_{\overline{n}|i=.005}$

Solve for $n$.

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  • $\begingroup$ Thank you. I got exactly what I needed. $\endgroup$
    – hyg17
    Commented Oct 26, 2013 at 2:28

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