To determine how much more you would need to pay each month and how much less total interest you would pay when opting to pay off your credit card balance over one year instead of two, we can follow these steps:
### Step 1: Identify Given Information
- Principal balance, [tex]\( P = \$3800 \)[/tex]
- Annual interest rate, [tex]\( r = 12\% = 0.12 \)[/tex]
- Payment period for two years: 24 months
- Monthly payment for two years: [tex]\( \$178.88 \)[/tex]
- Total interest paid over two years: [tex]\( \$493.12 \)[/tex]
- Payment period for one year: 12 months
### Step 2: Calculate Monthly Interest Rate
To calculate the monthly interest rate, divide the annual rate by 12:
[tex]\[
\text{Monthly interest rate} = \frac{0.12}{12} = 0.01
\][/tex]
### Step 3: Calculate the Monthly Payment for One Year
Using the PMT formula for monthly payments:
[tex]\[
\text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}}
\][/tex]
Where:
- [tex]\( P = 3800 \)[/tex]
- [tex]\( r = 0.12 \)[/tex]
- [tex]\( n = 12 \)[/tex] (months in a year)
- [tex]\( t = 1 \)[/tex] year
Calculate the monthly payment:
[tex]\[
\text{Monthly payment for one year} = \frac{3800 \times 0.01}{1 - (1 + 0.01)^{-12}}
\][/tex]
### Step 4: Calculate the Total Paid and Total Interest for One Year
Once we have the monthly payment, we calculate the total amount paid over one year:
[tex]\[
\text{Total paid for one year} = \text{Monthly payment for one year} \times 12
\][/tex]
Then subtract the principal to find the total interest paid:
[tex]\[
\text{Total interest paid for one year} = \text{Total paid for one year} - 3800
\][/tex]
### Step 5: Determine Additional Monthly Payment and Difference in Total Interest
Find the additional amount you would need to pay per month by comparing the one-year and two-year monthly payments:
[tex]\[
\text{Additional monthly payment} = \text{Monthly payment for one year} - 178.88
\][/tex]
Calculate the reduction in total interest by comparing the total interest paid over one year and two years:
[tex]\[
\text{Less total interest} = 493.12 - \text{Total interest paid for one year}
\][/tex]
### Final Answer
By working through these calculations, we get:
[tex]\[
\text{Additional monthly payment} = \$158.75
\][/tex]
[tex]\[
\text{Less total interest} = \$241.62
\][/tex]
Thus,
- You will pay [tex]\(\$ 158.75\)[/tex] more each month.
- You will pay [tex]\(\$ 241.62\)[/tex] less in total interest.
