Your credit card has a balance of $$ \$ 5900 $$ and an annual interest rate of $$ 13 \% $$. You decide to pay off the balance over two years. If there are no further purchases charged to the card, you must pay $$ \$ 280.57 $$ each month, and you will pay a total interest of $$ \$ 833.68 $$. Assume you decide to pay off the balance over one year rather than two. How much more must you pay each month and how much less will you pay in total interest? Use PMT $$ =\frac{P\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-n t}} $$ to determine the regular payment amount. You will pay $$ \$ 246.48 $$ more each month. (Round to the nearest cent as needed.) You will pay $$ \$ \square $$ less in total interest. (Round to the nearest cent as needed.)

Question

UpStudy AI · Accepted Answer

We start with a balance of  
$$
P=\$5900,
$$  
an annual interest rate of  
$$
r=13\%,
$$  
and a number of compounding periods per year  
$$
n=12.
$$

For a payment plan over $$ t $$ years the regular payment is given by  
$$
\text{PMT}=\frac{P\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-n t}}.
$$

For the two‐year plan the numbers are:

- $$ t=2 $$ years so the total number of months is $$ 24 $$.
- It is given that the monthly payment is  
$$
\text{PMT}_{2year}=\$280.57.
$$
- Thus the total amount paid is  
$$
280.57\times24=\$6733.68.
$$
- The total interest paid is  
$$
6733.68-5900 = \$833.68.
$$

Now we consider paying off the balance over one year ($$ t=1 $$).

The monthly payment for a one‐year plan is given by  
$$
\text{PMT}_{1year}=\frac{5900\left(\frac{0.13}{12}\right)}{1-\left(1+\frac{0.13}{12}\right)^{-12}}.
$$
Although one could compute the value from the formula, the problem tells us that switching from two years to one year increases the monthly payment by  
$$
\Delta \text{PMT}=\$246.48.
$$
That is,
$$
\text{PMT}_{1year} = 280.57 + 246.48 = \$527.05.
$$

Next, we find the total payments and interest for the one‐year plan:

- Total amount paid in one year is  
$$
527.05\times12 = \$6324.60.
$$
- The total interest for one year is  
$$
6324.60 - 5900 = \$424.60.
$$

Then the difference in total interest paid between the two plans is:
$$
833.68 - 424.60 = \$409.08.
$$

Thus, by choosing the one‐year plan:
- You must pay an extra $$\$246.48$$ each month.
- You will pay $$\$409.08$$ less in total interest.
You must pay an extra \$246.48 each month and will pay \$409.08 less in total interest.

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