I have the following problem, taken out of Giordano, Fox, and Horton's A First Course in Mathematical Modeling:
Your current credit card balance is $\$12,000$ with a current rate of $19.9\%$ per year. Interest is charged monthly. Determine what monthly payment p will pay off the card in two years, and in four years.
Now, assume that each month you charge $\$105$. Determine what monthly payment $p$ will pay off the card in two years, and in four years.
This problem wasn't too tricky, but I got stuck toward the end, when assuming that at time $t=0$, payment $p=0$. Allow me to show my work:
Let $a_t$ denote the amount of the debt at time $t$. We are given that $a_0 = \$12,000.$ Since interest is charged monthly, we know that the interest rate is $i = \frac{0.199}{12} = 0.0165833333…$. Since we are charging an additional $\$105$ per month, and intend to make payment $p$ to pay everything off, our balance can be modeled accordingly:
$$a_{t+1} = a_t(1+i)+105-p$$
$$a_{t+1} = 1.01658333…\cdot a_t+105-p$$
The solution to this equation will take on the form:
$$a_t = 1.01658333…^t \cdot c – \frac{105-p}{1-1.01658333…}$$
$$a_t = 1.01658333…^t \cdot c + \frac{105-p}{0.01658333…}$$
for some scalar $c$. To find the value of $c$, we will suppose that at time $t=0$, no payments would be made:
$$a_0 = 1.01658333…^0 \cdot c + \frac{105-0}{0.01658333…}$$
$$12000= c + \frac{105}{0.01658333…}$$
$$\implies c \approx 5668.3417085.$$
My question is this: was I right to assume that at $t=0, \space p=0$? Otherwise, I'm not sure how else to find $c$.
Any help or insight would be greatly appreciated!
Best Answer
No, $p$ is a constant by assumption. Two simple thoughts: the additional $105$ charged each month just has to be added to the payment to pay off the loan if you don't charge any more, so you can solve the problem without the $105$ and add it in at the end. Don't be so quick to put in the value of $i$. You will save yourself a lot of typing and at the end have a formula that you can use for other values of $i$. You might see problems like this again.
Your first line under the solution to this equation is wrong. Increasing $p$ will increase $a_t$, which cannot be right. What you should do is write the equation you did for $a_0$ but without setting $p=0$. Then write the equation $a_{24}=0$ for the two year term. That gives you two equations in two unknowns $c,p$ that you can solve. For the four year term, change it to $a_{48}=0$