# [SOLVED] how long will it take to pay off this debt?

• Mar 7th 2009, 11:02 AM
whitelight
[SOLVED] how long will it take to pay off this debt?
PMT = P x (APR)
n
_________
(-ny)]
[1-(1+APR)
n

I need to solve for the y, where y is part of a negative exponent of the bottom half of that equation. I've tried to extract the y, and isolate it on one side of this equation, but I'm totally stymied.

The problem is: I'm borrowing $300,000 to buy a townhouse, and can afford to make mortage payments of$2,000 per month and can obtain a loan from the bank w/ and APR of 7.2%.

How many years will it take to pay of the debt?
• Mar 7th 2009, 04:26 PM
Soroban
Hello, whitelight!

Your formula is incorrect . . .

Quote:

I'm borrowing $300,000 to buy a townhouse, and can afford to make mortage payments of$2,000 per month
and can obtain a loan from the bank w/ and APR of 7.2%.

How many years will it take to pay off the debt?

Formula: . $\displaystyle A \;=\;P\,\frac{i(1+i)^n}{(1+i)^n - 1}$

. . where: .$\displaystyle \begin{Bmatrix}A &=& \text{periodic payment} \\ P &=& \text{Principal} \\ i &=&\text{periodic interest rate} \\ n &=& \text{number of periods} \end{Bmatrix}$

We have: .$\displaystyle A\:=\: 2,\!000,\quad P \:=\: 300,\!000, \quad i \:=\frac{7.2\%}{12} \:=\:0.006$

. . . .and: . $\displaystyle n \:=\: 12y$, where $\displaystyle y$ = number of years.

The formula becomes: .$\displaystyle 2,\!000 \;=\;300,\!000\,\frac{0.006(1.006)^{12y}}{(1.006) ^{12y} - 1}$

Divide by 2000: .$\displaystyle 1 \;=\;\frac{0.9(1.006)^{12y}}{(1.006)^{12y}-1} \quad\Rightarrow\quad (1.006)^{12y} - 1 \;=\;0.9(1.006)^{12y}$

. . $\displaystyle (1.006)^{12y} - 0.9(1.006)^{12y} \;=\;1 \quad\Rightarrow\quad (1 - 0.9)(1.006)^{12y} \;=\;1$

. . $\displaystyle 0.1(1.006)^{12y} \;=\;1 \quad\Rightarrow\quad (1.006)^{12y} \:=\:\frac{1}{0.1} \:=\:10$

Take logs: .$\displaystyle \ln(1.006)^{12y} \;=\;\ln(10) \quad\Rightarrow\quad 12y\cdot\ln(1.006) \:=\:\ln(10)$

Therefore: .$\displaystyle y \;=\;\frac{\ln(10)}{12\ln(1.006)} \;=\;32.07619391$

. . It will take about 32 years, 1 month.

• Mar 8th 2009, 01:38 PM
whitelight
calculator keystrokes, Texas Instruments 30X
thanks so much for this - very helpful.

Strange - I'm getting a syntax error when I try to calculate the logs.

What are the keystrokes on a typical calculator?