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 \( \$ \square \) more each month. (Round to the nearest cent as needed.)

To determine the new monthly payment over one year using the payment formula \( PMT = \frac{P\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-n t}} \), we first need to identify our variables. Here, \( P = 5900 \), \( r = 0.13 \), \( n = 12 \) (monthly payments), and \( t = 1 \). Using these values, we calculate: 1. **Monthly Interest Rate:** \( \frac{0.13}{12} = 0.0108333 \) 2. **Total Number of Payments:** \( 12 \cdot 1 = 12 \) Now substitute these values into the PMT formula: \[ PMT = \frac{5900 \times 0.0108333}{1 - (1 + 0.0108333)^{-12}} \] Calculating the denominator: \[ (1 + 0.0108333)^{-12} \approx 0.887431 \] So, \[ 1 - 0.887431 \approx 0.112569 \] Now substituting back: \[ PMT \approx \frac{5900 \times 0.0108333}{0.112569} \approx \frac{63.9163}{0.112569} \approx 567.1 \] Thus, your new monthly payment will be approximately \( \$ 567.10 \). Next, we find the difference in monthly payment compared to paying off over two years: \[ 567.10 - 280.57 = 286.53 \] So, you will pay **\$286.53 more each month**. Now, let’s determine the total interest paid over one year: \[ \text{Total payment over one year} = 567.10 \times 12 = 6816.20 \] \[ \text{Total interest payment} = 6816.20 - 5900 = 916.20 \] Now, comparing to the original total interest over two years of \( 833.68 \): \[ \text{Amount less paid in total interest} = 833.68 - 916.20 = -82.52 \] So, you'll actually pay **\$82.52 more in total interest** when paying it off over one year instead of two. Finally, to wrap it up: You will pay **\$286.53 more each month** and actually end up paying **\$82.52 more in total interest**.