Assume you have a $10,791 credit card bill that you wish to pay off. If the minimum montly payment is $207, and your annual interest rate on your card is 12.9%, how long will it take you to pay off the credit card if you make the minimum payment every month and add no new purchases to the credit card?
+118723 Thanks Alan,
This is a present value of an ordinary annuity problem.
This is another way that you can do it.
A=$10791
R=207
i=0.129/12=0.01075
find n (number of months)
$$\begin{array}{rll}
10791&=&R\times \left[\frac{1-(1+i)^{-n}}{i}\right]\\\\
10791&=&207\times \left[\frac{1-(1+0.01075)^{-n}}{0.01075}\right]\\\\
10791&=&207\times \left[\frac{1-(1.01075)^{-n}}{0.01075}\right]\\\\
(10791*0.01075/207)-1&=&-(1.01075)^{-n}\right]\\\\
1-(10791*0.01075/207)&=&(1.01075)^{-n}\right]\\\\
2.274806518&=&1.01076^n\\\\
ln2.274806518&=&ln1.01076^n\\\\
ln2.274806518&=&n\times ln1.01076\\\\
\frac{ln2.274806518}{ln1.01076}&=&n\\\\
n&=&\frac{ln2.274806518}{ln1.01076}\\\\
n&=&76.79\\\\
&=&\mbox{77 months}\\\\
&=&\mbox{6 years and 5 months}\\\\
\end{array}$$
+33661 Perhaps this will help:
So, rounding up to the nearest month, n = 77 months.
+118723 Thanks Alan,
This is a present value of an ordinary annuity problem.
This is another way that you can do it.
A=$10791
R=207
i=0.129/12=0.01075
find n (number of months)
$$\begin{array}{rll}
10791&=&R\times \left[\frac{1-(1+i)^{-n}}{i}\right]\\\\
10791&=&207\times \left[\frac{1-(1+0.01075)^{-n}}{0.01075}\right]\\\\
10791&=&207\times \left[\frac{1-(1.01075)^{-n}}{0.01075}\right]\\\\
(10791*0.01075/207)-1&=&-(1.01075)^{-n}\right]\\\\
1-(10791*0.01075/207)&=&(1.01075)^{-n}\right]\\\\
2.274806518&=&1.01076^n\\\\
ln2.274806518&=&ln1.01076^n\\\\
ln2.274806518&=&n\times ln1.01076\\\\
\frac{ln2.274806518}{ln1.01076}&=&n\\\\
n&=&\frac{ln2.274806518}{ln1.01076}\\\\
n&=&76.79\\\\
&=&\mbox{77 months}\\\\
&=&\mbox{6 years and 5 months}\\\\
\end{array}$$
+33661 Thanks Melody, your way is much more concise. However, I deliberately wanted to construct the result step by step, in such a way that the questioner might spot the pattern, and understand where the 77 months comes from. I guess that, since a question of this type was asked, the basic formula is already available to the questioner (though that doesn't mean they know how to identify the right one if they've been given several, I suppose!).
Looking again at what I did, I notice I skipped a few steps in summarising the formula for the amount owed at the end of each month; so perhaps my way won't be so useful in practice!
+118723 Hi Alan,
I think it is good to to do questions in more than one way.
I have taught questions like this both ways. At school with high classes they would do it from scratch but I have also taught Commercial Mathematics at a Uni College and they were given pages of formulas that they were also allowed to use in exams. The difficult bit was learning which was the appropriate formula. 
